Abstract
We investigate the Fröhlich polaron model on a three-dimensional torus, and give a proof of the second-order quantum corrections to its ground-state energy in the strong-coupling limit. Compared to previous work in the confined case, the translational symmetry (and its breaking in the Pekar approximation) makes the analysis substantially more challenging.
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1 Introduction
The underlying physical system we are interested in studying is that of a charged particle (e.g., an electron) interacting with the quantized optical modes of a polar crystal (called phonons). In this situation, the electron excites the phonons by inducing a polarization field, which, in turn, interacts with the electron. In the case of a ‘large polaron’ (i.e., when the De Broglie wave-length of the electron is much larger than the lattice spacing in the medium), this system is described by the Fröhlich Hamiltonian [10], which represents a simple and well-studied model of non-relativistic quantum field theory (see [1, 8, 11, 20, 27, 28] for properties, results and further references).
A key parameter that appears in the problem is the coupling constant, usually denoted by \(\alpha \). We study the strong coupling regime of the model, i.e., its asymptotic behavior as \(\alpha \rightarrow \infty \). In this limit, the ground state energy of the Fröhlich Hamiltonian agrees to leading order with the prediction of the Pekar approximation [24], which assumes a classical behavior for the phonon field. This was first proved in [4], using a path integral approach (see also [21, 22], for recent work on the Pekar process [28]). Later, the result was improved in [18], by providing explicit bounds on the leading order correction term.
The object of our study is, precisely, the main correction to the classical (Pekar) approximation of the polaron model, i.e., the leading error term in the aforementioned asymptotics for the ground state energy. Such correction is expected to be of order \(O(\alpha ^{-2})\) smaller than the leading term, and arises from the quantum fluctuations about the classical limit [2]. This claim was first verified rigorously in [9], where both the electron and the phonon field are confined to a bounded domain (of linear size adjusted to the natural length scale set by the Pekar ansatz) with Dirichlet boundary conditions. Such restriction breaks translation invariance and simplifies the structure of the Pekar problem in comparison with the unconfined case, guaranteeing, at least in the case of the domain being a ball [6], uniqueness up to phase of the Pekar minimizers and non-degeneracy of the Hessian of the Pekar functional. We build upon the strategy developed in [9] to treat the ultraviolet singularity of the model, which in turn relies on multiple application of the Lieb–Yamazaki commutator method [19] and a subsequent use of Nelson’s Gross transformation [13, 23].
The key novelty of the present work is to deal with a translation invariant setting. We investigate the quantum correction to the Pekar approximation of the polaron model on a torus, and prove the validity of the predictions in [2] also in this setting. As a first step, we analyze the structure of the set of minimizers of the corresponding Pekar functional, proving uniqueness of minimizers up to symmetries, which was so far known to hold only in the unconfined case [15, 16] and on balls with Dirichlet boundary conditions [6]. The translation invariance leads to a degeneracy of the Hessian of the Pekar functional and corresponding zero modes, substantially complicating the analysis of the quantum fluctuations. In order to ‘flatten’ the surface of minimizers, we introduce a convenient diffeomorphism inspired by formal computations in [14], which effectively allows us to decouple the zero modes.
2 Setting and Main Results
2.1 The Model
We consider a 3-dimensional flat torus of side length \(L>0\). We denote by \(\Delta _L\) the Laplacian on \({\mathbb {T}}^3_L\) and by \((-\Delta _L)^{-1}(x,y)\) the integral kernel of its ‘inverse’, which we define by
An explicit formula for \((-\Delta _L)^{-1}(x,y)\) is given by
which, for any \(x\in {\mathbb {T}}^3_L\), yields an \(L^2\) function of y, its Fourier coefficients being in \(\ell ^2\). Analogously we define \((-\Delta _L)^{-s}\) for any \(s>0\). In the following, we identify \({\mathbb {T}}^3_L\) with the box \([-L/2,L/2]^3\subset {\mathbb {R}}^{3}\), and the Laplacian with the corresponding one on \([-L/2,L/2]^3\) with periodic boundary conditions.
Let
and \(v^x_{L}(y):=v_L(y-x)\). The Fröhlich Hamiltonian [10] for the polaron is given by
acting on \(L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}(L^2({\mathbb {T}}^3_L))\), where \({\mathcal {F}}(L^2({\mathbb {T}}^3_L))\) denotes the bosonic Fock space over \(L^2({\mathbb {T}}^3_L)\). The number operator, denoted by \({\mathbb {N}}\), accounts for the field energy, whereas \(-\Delta _L\) accounts for the electron kinetic energy. The creation and annihilation operators for a plane wave of momentum k are denoted by \(a_k^\dagger \) and \(a_k\), respectively, and they are assumed to satisfy the rescaled canonical commutation relations
In light of (2.1.5), \({\mathbb {N}}\) has spectrum \(\sigma ({\mathbb {N}})=\alpha ^{-2} \{0,1,2,\dots \}\). We note that the definition (2.1.4) is somewhat formal, since \(v_L\not \in L^2({\mathbb {T}}^3_L)\). It is nevertheless possible to define \({\mathbb {H}}_L\) via the associated quadratic form, and to find a suitable domain on which it is self-adjoint and bounded from below (see [12], or Remark 4.4 in Section 4 below).
We shall investigate the ground state energy of \({\mathbb {H}}_L\), for fixed L and \(\alpha \rightarrow \infty \).
Remark 2.1
By rescaling all lengths by \(\alpha \), \({\mathbb {H}}_L\) is unitarily equivalent to the operator \(\alpha ^{-2} \widetilde{{\mathbb {H}}}_L\), where \(\widetilde{{\mathbb {H}}}_L\) can be written compactly as
with the creation and annihilation operators \({\tilde{a}}^{\dagger }\) and \({\tilde{a}}\) now satisfying the (un-scaled) canonical commutation relations 
, and \(\tilde{{\mathbb {N}}}\) the corresponding number operator. Large \(\alpha \) hence corresponds to the strong-coupling limit of a polaron confined to a torus of side length \(L\alpha ^{-1}\). We find it more convenient to work in the variables defined in (2.1.4), however.
Remark 2.2
The Fröhlich polaron model is typically considered without confinement, i.e., as a model on \(L^2({\mathbb {R}}^{3})\otimes {\mathcal {F}}(L^2({\mathbb {R}}^{3}))\) with electron–phonon coupling function given by \((-\Delta _{{\mathbb {R}}^{3}})^{-1/2}(x,y)= (2 \pi ^2)^{-1} |x-y|^{-2}\). In the confined case studied in [9], \({\mathbb {R}}^3\) was replaced by a bounded domain \(\Omega \), and thus the electron–phonon coupling function was given by \((-\Delta _{\Omega })^{-1/2}(x,y)\), where \(\Delta _{\Omega }\) denotes the Dirichlet Laplacian on \(\Omega \). The latter setting, similarly to ours, has the advantage of guaranteeing compactness for the corresponding inverse Laplacian, which is a key technical ingredient both for [9] and our main results. In addition, for generic domains \(\Omega \) the Pekar functional has a unique minimizer up to phase (which is proved in [6] for \(\Omega \) a ball, and enters the analysis in [9] for general \(\Omega \) as an assumption). Compared with [9], setting the problem on the torus (or on \({\mathbb {R}}^3\)) introduces the extra difficulty of having to deal with translation invariance and a whole continuum of Pekar minimizers. Hence the present work can be seen as a first step in the direction of generalizing the results of [9] to the case of \({\mathbb {R}}^3\).
2.2 Pekar Functional(s)
For \(\psi \in H^1({\mathbb {T}}^3_L)\), \(\Vert \psi \Vert _2=1\), and \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\), we introduce the classical energy functional corresponding to (2.1.4) as
where \(h_\varphi \) is the Schrödinger operator
We define the Pekar energy as
In the case of \({\mathbb {R}}^{3}\), it was shown in [4, 18] that the infimum of the spectrum of the Fröhlich Hamiltonian converges to the minimum of the corresponding classical energy functional as \(\alpha \rightarrow \infty \). In [9], it was shown that the same holds for the model confined to a bounded domain with Dirichlet boundary conditions and the subleading correction in this asymptotics was computed. Our goal is to extend the results of [9] to the case of \({\mathbb {T}}^3_L\).
We define the two functionals
and their respective sets of minimizers
Clearly, \({\mathcal {E}}_L\) is invariant under translations and changes of phase and \({\mathcal {F}}_L\) is invariant under translations. It is thus useful to introduce the notation
for \(\psi \in H^1({\mathbb {T}}^3_L)\) and \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\), respectively.
Our first result, Theorem 2.3 (or, more precisely, Corollary 2.4) is a fundamental ingredient to prove our main result, Theorem 2.5. It concerns the uniqueness of minimizers of \({\mathcal {E}}_L\) up to symmetries and shows the validity of a quadratic lower bound for \({\mathcal {E}}_L\) in terms of the \(H^1\)-distance from the surface of minimizers. We shall prove these properties for sufficiently large L.
Theorem 2.3
(Uniqueness of Minimizers and Coercivity for \({\mathcal {E}}_L)\) There exist \(L_1>0\) and a positive constant \(\kappa _1\) independent of L, such that for \(L>L_1\) there exists \(0<\psi _L\in C^{\infty }({\mathbb {T}}^3_L)\) such that
Moreover \(\psi _L^y\ne \psi _L\) for any \(0\ne y\in {\mathbb {T}}^3_L\) and, for any \(L^2\)-normalized \(f\in H^1({\mathbb {T}}^3_L)\),
These properties of \({\mathcal {E}}_L\) translate easily to analogous properties for the functional \({\mathcal {F}}_L\), as stated in the following corollary:
Corollary 2.4
(Uniqueness of Minimizers and Coercivity for \({\mathcal {F}}_L)\) For \(L>L_1\) (where \(L_1\) is the same as in Theorem 2.3) there exists \(\varphi _L\in C^{\infty }({\mathbb {T}}^3_L)\) such that
Moreover, with \(\psi _L\) as in Theorem 2.3, we have
Finally, there exists \(\kappa '>0\) independent of L such that, for all \(\varphi \in L^2({\mathbb {T}}^3_L)\),
and this implies
with \(\tau _L:=\frac{\kappa ' (2\pi /L)^2}{1+\kappa ' (2\pi /L)^2}\).
In the case of \({\mathbb {R}}^{3}\), similar results are known to hold. In particular, the analogue of (2.2.9) was shown in [16] and the analogue of (2.2.10) follows from the results in [15]. In the case of a bounded domain with Dirichlet boundary conditions, an equivalent formulation of Theorem 2.3 was taken as working assumption in [9]. In the case of a ball in \({\mathbb {R}}^{3}\) with Dirichlet boundary conditions, the analogue of Theorem 2.3 was proved in [6]. In both the case of \({\mathbb {R}}^{3}\) and of balls, rotational symmetry plays a key role in the proof of these results. Rotational symmetry is not present in our setting, hence a different approach is required. Our method of proof of Theorem 2.3 relies on a comparison of the models on \({\mathbb {T}}^3_L\) and \({\mathbb {R}}^{3}\), for large L. As a consequence, our analysis does not easily yield quantitative estimates on \(L_1\).
To state our main result, which also holds in the case \(L>L_1\), we need to introduce the Hessian of the functional \({\mathcal {F}}_L\) at its unique (up to translations) minimizer \(\varphi _L\),
An explicit computation gives (see Proposition 3.15)
where \(h_{\varphi _L}\) is defined in (2.2.2), \(\psi _L\) is interpreted as a multiplication operator and 
. Clearly, by minimality of \(\varphi _L\), \(H^{{\mathcal {F}}_L}_{\varphi _L}\geqq 0\), and it is also easy to see that \(H^{{\mathcal {F}}_L}_{\varphi _L}\leqq 1\). We shall show that \(H^{{\mathcal {F}}_L}_{\varphi _L}\) has a three-dimensional kernel, given by \({{\,\mathrm{span}\,}}\{\partial _j \varphi _L\}_{j=1}^3\), corresponding to the invariance under translations of the functional. Note that we could define the Hessian of \({\mathcal {F}}_L\) at any other minimizer \(\varphi _L^y\), obtaining a unitarily equivalent operator \(H^{{\mathcal {F}}_L}_{\varphi _L^y}\).
2.3 Main Result
Recall the definition (2.2.3) for the Pekar energy \(e_L\) as well as (2.2.16) for the Hessian of \({\mathcal {F}}_L\) at its minimizers, for \(L>L_1\). Our main result is as follows:
Theorem 2.5
For any \(L>L_1\), as \(\alpha \rightarrow \infty \)
More precisely, the bounds
hold for some \(C_L>0\) and \(\alpha \) sufficiently large.
The trace appearing in (2.3.1) and (2.3.2) is over \(L^2({\mathbb {T}}^3_L)\). Note that, since \(H^{{\mathcal {F}}_L}_{\varphi _L}\leqq 1\), the coefficient of \(\alpha ^{-2}\) in (2.3.1) is negative.
In the case of bounded domains with Dirichlet boundary conditions, an analogue of Theorem 2.5 was proven in [9] (where logarithmic corrections appear in the bounds that correspond to (2.3.2) as a consequence of technical complications due to the boundary). Showing the validity of an analogous result on \({\mathbb {R}}^{3}\) still remains an open problem, however. Indeed, the constant \(C_L\) appearing in the lower bound in (2.3.2) diverges as \(L\rightarrow \infty \). This is mainly due to the lack of compactness of the resolvent of the full-space Laplacian (which leads, for instance, to a zero lower bound in (2.2.14) and, in particular, a divergence of the effective number of modes in (4.3.2)). On the other hand, our method of proof used in Section 4.1 to show the upper bound in (2.3.2) does apply, with little modifications, to the full space case. In any case, both the upper and lower bound are expected to hold in the case of \({\mathbb {R}}^{3}\) as well [2, 9, 14, 27].
Compared to the results obtained in [9], Theorem 2.5 deals with the additional complication of the invariance under translations of the problem, which implies that the set of minimizers of \({\mathcal {F}}_L\) is a three-dimensional manifold. This substantially complicates the proof of the lower bound in (2.3.2), as we shall see in Section 4.3. In particular, we need to perform a precise local study around the manifold of minimizers \(\Omega _L(\varphi _L)\), which we carry out by introducing a suitable diffeomorphism (inspired by Gross [14]).
Remark 2.6
(Small L Regime) As we show in Lemma 3.2, there exists \(L_0>0\) such that the analogue of Theorem 2.3 for \(L<L_0\) can be proven with a few-line-argument. In this case, \({\mathcal {E}}_L\) is simply non-negative and is therefore minimized by the constant function. In particular, \(e_L = 0\) and \(\varphi _L=0\).
Also an analogue of Theorem 2.5 can be proven in the regime \(L< L_0\), i.e., it is possible to show that for \(L<L_0\) there exists \(C_L>0\) such that
for large \(\alpha \). In this case (unlike the regime \(L>L_1\) where the set of minimizers \({\mathcal {M}}^{{\mathcal {F}}}_L\) is a three-dimensional manifold) \({\mathcal {M}}^{{\mathcal {F}}}_L\) only consists of the 0 function, and this allows to follow essentially the same arguments of [9] (with only small modifications, which are also needed in the regime \(L>L_1\) and hence are discussed in this paper). We shall therefore not carry out the details of this analysis here.
Whether uniqueness of Pekar minimizers up to symmetries holds for all \(L>0\) (i.e., also in the regime \(L_0\leqq L \leqq L_1\)) remains an open problem.
Throughout the paper, we use the word universal to describe any constant (which is generally denoted by C) or property that is independent of all the parameters involved and in particular independent of L, for \(L\geqq L_0\) (for some fixed \(L_0>0\)). Also, we write \(a\lesssim b\) whenever \(a\leqq Cb\) for some universal and positive C. We write \(C_L\) whenever a constant depends on L but is otherwise universal with respect to all other parameters. Finally, we write \(a\lesssim _L b\) whenever \(a\leqq C_L b\) for some positive \(C_L\).
2.4 Proof Strategy and Structure of the Paper
In Section 3 we study the properties of the Pekar functionals \({\mathcal {E}}_L\) and \({\mathcal {F}}_L\) defined in (2.2.4). We start by recalling the relevant properties of the Pekar functionals on \({\mathbb {R}}^3\) in Section 3.1. In the long Section 3.2 we give the proof of Theorem 2.3. Our method of proof relies on showing the convergence, as \(L\rightarrow \infty \), of \({\mathcal {E}}_L\) to its full-space counterpart \({\mathcal {E}}\). Proposition 3.5 in Section 3.2.1 formalizes the precise meaning of this convergence. Then, in Section 3.2.2, we prove a stronger notion of convergence, namely that the Hessian of \({\mathcal {E}}_L\) at any minimizer converges to the Hessian of \({\mathcal {E}}\) at a corresponding minimizer (in the sense of Proposition 3.7); in particular, it is strictly positive above its trivial zero modes for large L. By combining the results obtained in Sections 3.2.1 and 3.2.2, we conclude the proof of Theorem 2.3 in Section 3.2.3. Section 3.3 is dedicated to the investigation of the properties of \({\mathcal {F}}_L\). First, in Section 3.3.1, we show the validity of Corollary 2.4. Subsequently we compute the Hessian of \({\mathcal {F}}_L\) (in Proposition 3.15 in Section 3.3.2) and characterize its kernel (in Proposition 3.18 in Section 3.3.3). Finally, in Section 3.3.3 we introduce a family of weighted norms (see (3.3.46)) which is of key importance in Section 4 and we show, in Lemma 3.17, that the surface of minimizers of \({\mathcal {F}}_L\) locally admits a unique projection w.r.t. any of these norms.
In Section 4 we prove Theorem 2.5. First of all, in Section 4.1 we construct a trial state and use it to obtain an upper bound to the ground state energy of \({\mathbb {H}}_L\). This is carried out using the Q-space representation of the bosonic Fock space \({\mathcal {F}}(L^2({\mathbb {T}}^3_L))\) (see [25]) and follows ideas contained in [9], with only small modifications. The remaining sections are devoted to the lower bound. In Section 4.2, we show that it is possible to apply an ultraviolet cutoff on momenta of size larger than some \(\Lambda \) to \({\mathbb {H}}_L\) at an expense of order \(\Lambda ^{-5/2}\) (see Proposition 4.5). This is proven following closely the approach used in [9]: as a first step we apply a triple Lieb–Yamazaki bound [19] (in Section 4.2.1) and then make use of a Gross transformation [13, 23] (in Section 4.2.2). In Section 4.3 we show the validity of the lower bound in (2.3.2), thus completing the proof of Theorem 2.5. With Proposition 4.5 at hand, we have good estimates on the cost of applying an ultraviolet cutoff to \({\mathbb {H}}_L\) and this allows to reduce the problem to a finite dimensional one (with dimension N diverging as \(\alpha \rightarrow \infty \)). We adopt a similar strategy to [9], using IMS localization to split the space into an inner region close to the surface of minimizers of \({\mathcal {F}}_L\) and an outer region far away from it. The goal is to extract the relevant quantum correction to the ground state energy from the inner region and to show, using the bound (2.2.14), that the outer region contributes only as an error term. Compared to [9], the translation invariance substantially complicates the analysis. In contrast to the case considered in [9], the set of minimizers of \({\mathcal {F}}_L\) is a three-dimensional manifold and does not only consist of a single function. Hence, in order to treat the inner region and decouple the zero-modes of the Hessian of \({\mathcal {F}}_L\), we have to introduce a suitable diffeomorphism (see Definition 4.7 in Section 4.3.1) that ‘flattens’ the manifold of minimizers and the region close to it. It is here where we make use Lemma 3.17, which allows us to understand the local structure of the tubular neighborhood of the surface of minimizers of \({\mathcal {F}}_L\). Another technical complication relates to the metric used to distinguish between the inner and outer region, as simply considering the \(L^2\)-norm is not sufficient for our purposes, and we need the weighted norms defined in (3.3.46) (in particular we apply the IMS localization with respect to a metric which depends with \(\alpha \)).
3 Properties of the Pekar Functionals
In this section we derive important properties of the functionals \({\mathcal {E}}_L\) and \({\mathcal {F}}_L\), introduced in Section 2.2 and defined in (2.2.4). In Section 3.2, we show the validity of Theorem 2.3, relying on the comparison of the models on \({\mathbb {T}}^3_L\) and \({\mathbb {R}}^{3}\) for large L. In Section 3.3, we study the functional \({\mathcal {F}}_L\). In particular, we prove Corollary 2.4 and compute the Hessian of \({\mathcal {F}}_L\) at its minimizers.
Given a function \(f\in L^2({\mathbb {T}}^3_L)\) and \(k\in \frac{2\pi }{L}{\mathbb {Z}}^3\), we denote by \(f_k\) the k-th Fourier coefficient of f. We also denote
We shall use the following definition of fractional Sobolev semi-norms for functions \(f\in L^2({\mathbb {T}}^3_L)\), \(0\ne s\in {\mathbb {R}}\):
Before moving on with the discussion, we recall in the following subsection the definition and relevant properties of the full-space Pekar functional.
3.1 The Full-Space Pekar Functional
Let \(\psi \in H^1({\mathbb {R}}^{3})\) be an \(L^2({\mathbb {R}}^{3})\)-normalized function and \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {R}}^{3})\). Then
where \(h_{\varphi }^{{\mathbb {R}}^{3}}\) is the Schrödinger operator
Comparing with (2.2.1) and (2.2.2), we note the analogy between the definitions and observe that we are slightly abusing notation by denoting both potentials with the same symbol (we do this for simplicity and since ambiguity does not arise). Analogously to (2.2.4), we define
In analogy with (2.2.3), we denote
For our purposes, in the case of \({\mathbb {R}}^{3}\), it is sufficient to focus our discussion on the functional \({\mathcal {E}}\), of which we now recall the main properties. As shown in [16], \({\mathcal {E}}\) admits a unique positive and radially decreasing minimizer \(\varPsi \) which is also smooth, the set of minimizers of \({\mathcal {E}}\) coincides with
and \(\varPsi \) satisfies the Euler–Lagrange equation
with
where T and W defined in (3.2.1) below
Furthermore, as was shown in [15], the Hessian of \({\mathcal {E}}\) at its minimizers is strictly positive above the trivial zero modes resulting from the invariance under translations and changes of phase. This implies the validity of the following Theorem, which is not stated explicitly in [15] but can be obtained by standard arguments (see, e.g., [5, Appendix A] or [7]) as a consequence of the results therein contained.
Theorem A
There exists a constant \(C>0\), such that, for any \(L^2\)-normalized \(f \in H^1({\mathbb {R}}^{3})\)
Our strategy to prove Proposition 3.5 relies on Theorem A and in comparing \({\mathbb {T}}^3_L\) with \({\mathbb {R}}^{3}\) for large L.
3.2 Study of \({\mathcal {E}}_L\)and Proof of Theorem 2.3
To compare \({\mathcal {E}}_L\) and \({\mathcal {E}}\), we prefer to write both of them in the following form, which can be obtained from (2.2.4) and (3.1.3), respectively, by a simple completion of the square,
The next ingredient, needed for the comparison of \({\mathcal {E}}_L\) and \({\mathcal {E}}\), is the following lemma:
Lemma 3.1
There exists a universal constant C such that
Proof
We define \(F_L(x):=-\Delta ^{-1}_L(x,0)\) and \(F(x)=(4\pi )^{-1}|x|^{-1}\) and observe that our statement is equivalent to showing that
By definition, we have \(F_L(x)=\frac{1}{L} F_1(\frac{x}{L})\). Hence, (3.2.4) is equivalent to
Again by definition, \(F_1-F\) is harmonic (distributionally and hence also classically) on \(\left( {\mathbb {R}}^{3}{\setminus } \{{\mathbb {Z}}^3\}\right) \cup \{0\}\) (when \(F_1\), and only \(F_1\), is extended to the whole space by periodicity). Thus we conclude that \(F_1-F\) is in \(C^{\infty }\left( (-1,1)^3\right) \) and, in particular, bounded on \([-1/2,1/2]^3\). \(\square \)
The analogy between (3.2.2) and (3.2.1), combined with Lemma 3.1, clearly suggests that \({\mathcal {E}}_L\) formally converges to \({\mathcal {E}}\) as \(L\rightarrow \infty \). Hence, we set out to show that this convergence can be made rigorous and allows to infer properties of \({\mathcal {E}}_L\) by comparing it to \({\mathcal {E}}\), in the large L regime.
In Section 3.2.1 we derive an important preliminary result, namely Proposition 3.5. It formalizes in a mathematical useful way the concept of \({\mathcal {E}}_L\) converging to \({\mathcal {E}}\). In Section 3.2.2, we study the Hessian of \({\mathcal {E}}_L\), showing that it converges (in the sense of Proposition 3.7) to the Hessian of \({\mathcal {E}}\) and therefore is strictly positive above its trivial zero modes for large L. Finally, in Section 3.2.3 we use the results obtained in Sections 3.2.1 and 3.2.2 to show the validity of Theorem 2.3.
We remark that our approach differs from the one used on \({\mathbb {R}}^{3}\) and on balls to show, for the related \({\mathcal {E}}\)-functional, uniqueness of minimizers and strict positivity of the Hessian (see [15, 16] for the case of \({\mathbb {R}}^{3}\) and [6] for the case of balls). In those cases, rotational symmetry allows to first show uniqueness of minimizers and then helps to derive the positivity of the Hessian at the minimizers. We take somewhat the opposite road: comparing \({\mathcal {E}}_L\) to \({\mathcal {E}}\), we first show that minimizers (even if not unique) all localize around the full-space minimizers (see Proposition 3.5) and that the Hessian at each minimizer is universally strictly positive (see Proposition 3.7) for large L. We then use these two properties to derive, as a final step, uniqueness of minimizers.
3.2.1 Preliminary Results
The next Lemma proves the existence of minimizers for any \(L>0\). Moreover, it shows that there exists \(L_0>0\) such that, for \(L<L_0\), \({\mathcal {E}}_L\) is strictly positive on any non-constant \(L^2\)-normalized function, as already mentioned in Remark 2.6.
Lemma 3.2
For any \(L>0\), \(e_L\) in (2.2.3) is attained, and there exists a universal constant \(C>0\) such that \(e_L>-C\). Moreover, there exists \(L_0>0\) such that, for \(L<L_0\), \({\mathcal {E}}_L(\psi )>0\) for any non-constant \(L^2\)-normalized \(\psi \).
Proof
We consider any \(L^2\)-normalized \(\psi \in H^1({\mathbb {T}}^3_L)\) and begin by observing that in terms of the Fourier coefficients we have
By Parseval’s identity \(|\psi _0|\leqq 1\) and thus, using the Cauchy–Schwarz inequality, we can deduce that
Therefore
We can bound both terms on the r.h.s. in two different ways, one which is good for small L and one which is good for all the other L. Indeed, by applying estimate (3.2.8) and using the Poincaré–Sobolev inequality (see [17], chapter 8) on the zero-mean function \({\hat{\psi }}\), we get
Moreover,
Therefore, we can conclude that
Thus, for \(L< L_0:=C^{-1}\), either \({\psi \equiv {{\,\mathrm{const.}\,}}}\) and \({\mathcal {E}}_L(\psi )=0\) or \({\mathcal {E}}_L(\psi )\gtrsim T_L(\psi )>0\). Moreover, this also implies
which is the analogue of (2.2.10) from Theorem 2.3 in the case \(L< L_0\).
We now proceed to study the more interesting regime \(L\geqq L_0\). By Lemma 3.1, splitting \({{\,\mathrm{dist}\,}}^{-1}_{{\mathbb {T}}^3_L}(x,\cdot )\) into an \(L^{3/2}\) part and the remaining \(L^{\infty }\) part (whose norms can be chosen to be proportional to \(\varepsilon \) and \(\varepsilon ^{-1}\), respectively, for any \(\varepsilon >0\)), and by applying again the Poincaré-Sobolev inequality, we obtain
Moreover, since \(L\geqq L_0\), trivially \( L^{-1} \Vert {\hat{\psi }}\Vert _{L^2({\mathbb {T}}^3_L)}^2\lesssim 1\) and we can conclude that for any \(L^2\)-normalized \(\psi \in H^1({\mathbb {T}}^3_L)\)
From this we can infer that \(e_L\geqq -\,C\) for any L. To show existence of minimizers, we observe that by (3.2.15) any minimizing sequence \(\psi _n\) on \({\mathbb {T}}^3_L\) must be bounded in \(H^1({\mathbb {T}}^3_L)\). Therefore, there exists a subsequence (which we still denote by \(\psi _n\) for simplicity) that converges weakly in \(H^1({\mathbb {T}}^3_L)\) and strongly in \(L^p({\mathbb {T}}^3_L)\), for any \(1\leqq p<6\) to some \(\psi \) (by the Banach–Alaoglu Theorem and the Rellich–Kondrachov embedding Theorem). The limit function \(\psi \) is \(L^2\)-normalized and
by weak lower semicontinuity of the norm. Using the \(L^4\)-convergence of \(\psi _n\) to \(\psi \) and the fact that \(\Vert \cdot \Vert _{\mathring{H}^{-1}({\mathbb {T}}^3_L)}\lesssim L\Vert \cdot \Vert _{L^2({\mathbb {T}}^3_L)}\), we finally obtain
This implies that
and thus that \(\psi \) is a minimizer. Note that, since \({\mathcal {E}}_L(\psi _n)\rightarrow e_L={\mathcal {E}}_L(\psi )\) by definition of \(\psi _n\) and, as shown, \(W_L(\psi _n)\rightarrow W_L(\psi )\), it also holds
which implies that \(\psi _n\) actually converges to \(\psi \) strongly in \(H^1({\mathbb {T}}^3_L)\). \(\square \)
Once we have shown existence of minimizers, we need to investigate more carefully their properties. Some of them are derived in the following Lemma. Recall that
and that, as stated above, we call any property universal which does not depend on \(L\geqq L_0\).
Lemma 3.3
Let \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) (as defined in (2.2.5)). Then \(\psi \) satisfies the following Euler–Lagrange equation
Moreover, \(\psi \in C^{\infty }({\mathbb {T}}^3_L)\), is universally bounded in \(H^2({\mathbb {T}}^3_L)\) (and therefore in \(L^{\infty }({\mathbb {T}}^3_L)\)), has constant phase and never vanishes. Finally, any \(L^2\)-normalized sequence \(f_n\in H^1({\mathbb {T}}^3_{L_n})\) such that \({\mathcal {E}}_{L_n}(f_n)\) is universally bounded, is universally bounded in \(H^1({\mathbb {T}}^3_{L_n})\).
Proof
The fact that sequences \(f_n\in H^1({\mathbb {T}}^3_{L_n})\) of \(L^2\)-normalized functions for which \({\mathcal {E}}_{L_n}\) is universally bounded are universally bounded in \(H^1({\mathbb {T}}^3_{L_n})\) follows trivially from estimate (3.2.15). This immediately yields a universal bound on the \(H^1\)-norm of minimizers.
The Euler–Lagrange equation (3.2.21) for the problem is derived by standard computations omitted here. By Lemma 3.1 and by splitting \(({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_L}(0,\,\cdot \,))^{-1}\) in its \(L^{3/2}\) and \(L^{\infty }\) parts, we have
Therefore, by the universal \(H^1\)-boundedness of minimizers, \(V_{\sigma _{\psi }}\) is universally bounded in \(L^{\infty }({\mathbb {T}}^3_L)\), for any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\). This immediately allows to conclude universal \(\mathring{H}^2\) (and hence \(H^2\)) bounds for functions in \({\mathcal {M}}^{{\mathcal {E}}}_{L}\), using the Euler–Lagrange equation (3.2.21), Lemma 3.2 and the universal \(H^1\)-boundedness of minimizers, which guarantee that
Since \(L\geqq L_0\), universal \(H^2\)-boundedness also implies universal \(L^{\infty }\)-boundedness of minimizers by the Sobolev inequality.
For any \(L>0\), any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) satisfies (3.2.21), is in \(H^1({\mathbb {T}}^3_L)\) and is such that \(V_{\sigma _{\psi }}\in L^{\infty }({\mathbb {T}}^3_L)\). Therefore \(\psi \) also satisfies, for any \(\lambda >0\)
In particular, by a bootstrap argument we can conclude that \(\psi \in C^{\infty }({\mathbb {T}}^3_L)\). Moreover, picking \(\lambda >-\mu ^L_{\psi }+\Vert V_{\sigma _{\psi }}\Vert _{L^{\infty }({\mathbb {T}}^3_L)}\) and using that \((-\Delta _L+\lambda )^{-1}\) is positivity improving, we can also conclude that if \(\psi \geqq 0\) then \(\psi >0\). By the convexity properties of the kinetic energy (see [17], Theorem 7.8), we have that \(T_L(|\psi |)\leqq T_L(\psi )\) which implies that if \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) then \(T_L(\psi )=T_L(|\psi |)\) and also \(|\psi |\in {\mathcal {M}}^{{\mathcal {E}}}_{L}\). Hence both \(\psi \) and \(|\psi |\) are eigenfunctions of the least and simple (by positivity of one of the eigenfunctions) eigenvalue \(\mu ^L_{\psi }=\mu ^L_{|\psi |}\) of the Schrödinger operator \(-\Delta _L+V_{\sigma _{\psi }}\), which allows us to infer that \(\psi \) has constant phase and never vanishes. \(\square \)
We now proceed to develop the tools that will allow to show the validity of Theorem 2.3. We begin with a simple Lemma.
Lemma 3.4
For \(\psi \in H^1({\mathbb {T}}^3_L)\),
Proof
We have
We define
and observe that \((g_i)_0=0\) and \(|(g_i)_k|=\frac{|k_i(\rho _{\psi })_{k}|}{|k|^{7/4}}\leqq \frac{|(\rho _{\psi })_k|}{|k|^{3/4}}\) for \(k\ne 0\). These estimates on the Fourier coefficients of \(g_i\) imply that, for \(i=1,2,3\),
Moreover, using the fractional Sobolev embeddings (see, for example, [3]) and that \(g_i\) has zero mean, we have
Applying these results to (3.2.25) and using Hölder’s inequality two times, the Poincaré–Sobolev inequality and the convexity properties of the kinetic energy (see [17], Theorem 7.8), we conclude that
\(\square \)
Our next goal is to show that \(e_L\rightarrow e_{\infty }\) as \(L\rightarrow \infty \), and that in the large L regime the states that are relevant for the minimization of \({\mathcal {E}}_L\) are necessarily close to the full space minimizer (or any of its translates). This is a key ingredient for the discussion carried out in the following sections, and is stated in a precise way in the next proposition. The coercivity results obtained in [15] are of fundamental importance here as they guarantee that, at least for the full space model, low energy states are close to minimizers.
We recall that the full-space Pekar functional, defined in (3.2.1), admits a unique positive and radial minimizer \(\varPsi \) which is also smooth (see (3.1.5)), and we introduce the notation
Note that \(\varPsi _L\in H^1({\mathbb {T}}^3_L)\), by radiality and regularity of \(\varPsi \).
Proposition 3.5
We have
Moreover, for any \(\varepsilon >0\) there exist \(L_{\varepsilon }\) and \(\delta _{\varepsilon }\) such that for any \(L>L_{\varepsilon }\) and any \(L^2\)-normalized \(\psi \in H^1({\mathbb {T}}^3_L)\) with \({\mathcal {E}}_L(\psi )-e_L<\delta _{\varepsilon }\),
where \(\Theta _L(\psi )\), \(\varPsi _L\), \(\mu ^L_{\psi }\) and \(\mu _{\varPsi }\) are defined in (2.2.7), (3.2.30), (3.2.21) and (3.1.7), respectively.
Proof
We first show that \(\limsup _{L\rightarrow \infty } e_L\leqq e_{\infty }\) by using \(\varPsi _L\) as a trial state for \({\mathcal {E}}_L\). Observe that \({\Vert \varPsi _L\Vert _{L^2({\mathbb {T}}^3_L)}\rightarrow 1}\) and \(T_L(\varPsi _L)\rightarrow T(\varPsi )\) as \(L\rightarrow \infty \). To estimate the difference of the interaction terms we note that \(\varPsi _L(\varPsi -\varPsi _L)=0\) and therefore
By dominated convergence, the last two terms converge to zero as \(L\rightarrow \infty \). On the other hand, by Lemma 3.1 and since \(\varPsi \) is normalized
Moreover, since \({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_L}(x,y)=|x-y|\) for \(x,y\in [-L/4,L/4]^3\) and using the symmetry and the positivity of the integral kernel and the fact that \({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_L}(x,y)\leqq |x-y|\), we get
Finally, by splitting \({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_L}^{-1}(x,\cdot )\) in its \(L^{\infty }\) and \(L^1\) parts and using that \(\varPsi \) is normalized, we can bound the r.h.s. of (3.2.35) by \(\left( C_1\Vert \varPsi _L-\varPsi _{L/2}\Vert _2^2+C_2\Vert \varPsi _L-\varPsi _{L/2}\Vert _{\infty }^2\right) \), which vanishes as \(L\rightarrow \infty \), since \(\varPsi (x)\xrightarrow {|x|\rightarrow \infty }0\). Putting the pieces together, we conclude that
This shows our first claim, since
We now proceed to show that
and the validity of (3.2.32) using IMS localization. We shall show that for any \(L^2\)-normalized sequence \(\psi _n \in H^1({\mathbb {T}}^3_{L_n})\) with \(L_n\rightarrow \infty \) such that
we have
which implies the claim of the proposition.
Pick \(\eta \in C^{\infty }({\mathbb {R}}^{3})\) with \(\text {supp}(\eta )\subset B_1\) and \(\Vert \eta \Vert _2=1\). We denote by \(\eta _R\) the rescaled copy of \(\eta \) supported on \(B_R\) with \(L^2\)-norm equal to 1. As long as \(R\leqq L/2\), \(\eta _R \in C^{\infty }({\mathbb {T}}^3_L)\) and we then consider the translates \(\eta _R^y\) for any \(y\in {\mathbb {T}}^3_L\). Given \(\psi \in H^1({\mathbb {T}}^3_L)\), we also define
By standard properties of IMS localization, for any \(R\leqq L/2\), we have
Moreover, by using that \(|\psi |^2=\int _{{\mathbb {T}}^3_L} |\psi \eta _R^y|^2 \,{\mathrm{d}}y=\int _{{\mathbb {T}}^3_L} |\psi _R^y|^2 \Vert \psi \eta _R^y\Vert ^2 \,{\mathrm{d}}y\) and completing the square
Combining (3.2.42) and (3.2.43), we therefore obtain
Since the integrand on the r.h.s. is equal to the l.h.s. on average (indeed \(\Vert \psi \eta _R^y\Vert _2^2dy\) is a probability measure) there exists \({\bar{y}}\in {\mathbb {T}}^3_L\) such that
This fact has several consequences and it is particularly useful if we apply it to our sequence \(\psi _n\) with a radius \(R=R_n\leqq L_n/2\) (we take for simplicity \(R=L_n/4\)). Indeed, by the above discussion and (3.2.39), we obtain that there exists \({\bar{y}}_n\in {\mathbb {T}}^3_{L_n}\) such that the \(L^2\)-normalized functions
are competitors both for the minimization of \({\mathcal {E}}_{L_n}\) and \({\mathcal {E}}\) (indeed, \({{\bar{\psi }}}_n\) can then be thought of as a function in \(C^{\infty }_c({\mathbb {R}}^{3})\), supported on \(B_{L_n/4}\)) and satisfy
In other words, we can localize any element of our sequence \(\psi _n\) to a ball of radius \(R= L_n/4\) with an energy expense of order \(L_n^{-2}\), and the localized function is close (in the sense of the second line of (3.2.47)) to \(\psi _n\) itself, up to an error again of order \(L_n^{-2}\).
Moreover \(T_{L_n}({{\bar{\psi }}}_n)=T({{\bar{\psi }}}_n)\) and, using Lemma 3.1 and the fact that \({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_{L_n}}(x,y)=|x-y|\) for all \(x,y \in B_{L_n/4}\), we have
Therefore, using (3.2.47)
which shows the first claim in (3.2.40). By Theorem A and (3.2.49), it also follows that
Hence, up to an n-dependent translation and change of phase (which we can both assume to be zero without loss of generality by suitably redefining \(\psi _n\)), \({{\bar{\psi }}}_n\xrightarrow {H^1({\mathbb {R}}^{3})} \varPsi \), and the convergence also holds in \(L^p({\mathbb {R}}^{3})\) for any \(2\leqq p \leqq 6\). From this and the second line of (3.2.47), we would like to deduce that also \(\psi _n\) and \(\varPsi _{L_n}\) are close. We first note that, by a simple application of Hölder’s inequality, it follows that for any \(f\in L^2({\mathbb {T}}^3_L)\) with zero mean
We combine this with (3.2.47) and apply it to the zero mean function \((\rho _{\psi _n}-\rho _{{{\bar{\psi }}}_n})\), obtaining
Applying Lemma 3.4 to \(\psi _n\) and \({{\bar{\psi }}}_n\) (which are uniformly bounded in \(H^1\) by Lemma 3.3) we conclude that \((\rho _{\psi _n}-\rho _{{{\bar{\psi }}}_n})\xrightarrow {L^2}0\).
As a consequence, since \(\psi _n\) and \({{\bar{\psi }}}_n\) have the same phase, \(\psi _n\) and \({{\bar{\psi }}}_n\) are arbitrarily close in \(L^4\). Indeed,
By the identification of \({\mathbb {T}}^3_{L_n}\) with \([-L_n/2,L_n/2]^3\), we finally get \(\Vert \psi _n-\varPsi \Vert _{L^4({\mathbb {R}}^{3})}\rightarrow 0\), if \(\psi _n\) is set to be 0 outside \([-L_n/2,L_n/2]^3\). Moreover, \(\psi _n\) converges to \(\varPsi \) in \(L^p({\mathbb {R}}^{3})\) for any \(2\leqq p <6\), since \(\Vert \psi _n\Vert _2=1\), \(\psi _n\xrightarrow {L^4}\varPsi \), \(\Vert \varPsi \Vert _2=1\) and \(\Vert \psi _n\Vert _p\) is uniformly bounded for any \(2\leqq p \leqq 6\).
To show the second claim in (3.2.40), we need to show that the convergence actually holds in \(H^1({\mathbb {T}}^3_{L_n})\), i.e., that \(\Vert \psi _n-\varPsi _{L_n}\Vert _{H^1({\mathbb {T}}^3_{L_n})}\rightarrow 0\). First, we show convergence in \(H^1(B_R)\) for fixed R. Note that
since
and \(T_{L_n}({{\bar{\psi }}}_n)=T({{\bar{\psi }}}_n) \rightarrow T(\varPsi )\) by \(H^1\) convergence. Moreover, given that \(\psi _n\) is uniformly bounded in \(H^1(B_R)\) and \(\psi _n \rightarrow \varPsi \) in \(L^2(B_R)\), we have \(\psi _n\rightharpoonup \varPsi \) in \(H^1(B_R)\) for any R and this, together with (3.2.54) and weak lower semicontinuity of the norms, implies \(\psi _n\rightarrow \Psi \) in \(H^1(B_R)\) for any R.
Finally, for any \(\varepsilon >0\) there exists \(R=R(\varepsilon )\) such that \(\Vert \varPsi \Vert _{H^1(B_R^c)}\leqq \varepsilon \) and, using strong \(H^1\)-convergence on balls and again (3.2.54), we obtain
which concludes the proof of the second claim in (3.2.40).
Finally, we show the third claim in (3.2.40). This simply follows from the previous bounds, which guarantee that \({\mathcal {E}}_{L_n}(\psi _n)\rightarrow e_{\infty }\) and \(T_{L_n}(\psi _n)\rightarrow T(\Psi )\) and hence
\(\square \)
We conclude this section with a simple corollary of Proposition 3.5.
Corollary 3.6
There exists \(L^*\) such that for \(L>L^*\) and any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) we have \(\psi \ne \psi ^y\) for \(0\ne y\in {\mathbb {T}}^3_L\).
Proof
It is clearly sufficient to show the claim for \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) such that
and for \(y \in {\mathbb {T}}^3_L\) such that \(|y|\geqq L/4\) (indeed, if the claim fails for some \(y'\) such that \(|y'|<L/4\) it also fails for some y such that \(|y|\geqq L/4\)). For any such \(\psi \) and y, Proposition 3.5 and the fact that \(\Psi \ne \Psi ^y\) for any \(y\in {\mathbb {R}}^{3}\) guarantee the existence of \(L^*\) such that for any \(L>L^*\) we have
and this completes the proof. \(\square \)
3.2.2 Study of the Hessian of \({\mathcal {E}}_L\)
In this section we study the Hessian of \({\mathcal {E}}_L\) at its minimizers, showing that it is strictly positive, universally, for L big enough. Positivity is of course understood up to the trivial zero modes resulting from the symmetries of the problem (translations and changes of phase). This is obtained by comparing \({\mathcal {E}}_L\) with \({\mathcal {E}}\) and exploiting Theorem A.
For any minimizer \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\), the Hessian of \({\mathcal {E}}_L\) at \(\psi \) is defined by
An explicit computation gives
with \(Q_{\psi }=1\!\!1- |\psi \rangle \langle \psi |\) and
(We use the same notation for the operator \(X^L_\psi \) and its integral kernel for simplicity.) We recall that \(\mu ^L_{\psi }=T_L(\psi )-2W_L(\psi )\) (see (3.2.21)) and that \(V_{\sigma _{\psi }}= -2(-\Delta _L)^{-1} \rho _{\psi }\) (see (3.2.20)) and we note that \(L^L_{\psi } \psi =0\) is exactly the Euler–Lagrange equation derived in Lemma 3.3.
By minimality of \(\psi \), we know that \({{\,\mathrm{inf\,spec}\,}}L = {{\,\mathrm{inf\,spec}\,}}Q (L-4X) Q =0\), since both operators are clearly nonnegative and \(\psi \) is in the kernel of both of them. Moreover, \(\ker L^L_{\psi }= {{\,\mathrm{span}\,}}\{\psi \}\), since it is a Schrödinger operator of least (simple) eigenvalue 0. The situation is more complicated for \(Q_{\psi }(L^L_{\psi }-4X^L_{\psi }) Q_{\psi }\), whose kernel contains at least \(\psi \) and \(\partial _i \psi \) (by the translation invariance of the problem). Since both \(L^L_{\psi }\) and \(Q_{\psi }(L^L_{\psi }-4X^L_{\psi })Q_{\psi }\) have compact resolvents (they are given by bounded perturbations of \(-\Delta _L\)), they both have discrete spectrum. Our aim is two-fold: first we need to show that the kernel of \(Q_{\psi }(L^L_{\psi }-4X^L_{\psi })Q_{\psi }\) is exactly spanned by \(\psi \) and its partial derivatives, secondly we want to show that the spectral gap (above the trivial zero modes) of both operators is bounded by a universal positive constant.
Before stating the main result of this section, we introduce the relevant full-space objects: let again \(\varPsi \) be the unique positive and radial full-space minimizer of the Pekar functional (3.2.1) and, analogously to (3.2.62), define
We introduce
We emphasize that the results contained in [15] imply that \(\min \{h_{\infty }',h_{\infty }''\}>0\). Moreover, it is easy to see, using that \(V_{\sigma _{\varPsi }}(x)\lesssim -|x|^{-1}\) for large x, that \(L_{\varPsi }\) has infinitely many eigenvalues between 0, its least and simple eigenvalue with eigenfunction given by \(\varPsi \), and \(-\mu _{\varPsi }\), the bottom of its continuous spectrum. Since furthermore \(X_{\varPsi }\) is positive, this implies, in particular, that
which we shall use later.
Proposition 3.7
For any \(L>0\), we define
Then
It is not difficult to show that
simply by considering localizations of the full-space optimizers and using Proposition 3.5. Hence there is actually equality in (3.2.68).
To prove Proposition 3.7 we need the next two Lemmas.
Lemma 3.8
For \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\), the operator \(Y^L_{\psi }\) with integral kernel \(Y^L_{\psi }(x,y):= (-\Delta _L)^{-1}(x,y)\psi (y)\) is universally bounded from \(L^2({\mathbb {T}}^3_L)\) to \(L^{\infty }({\mathbb {T}}^3_L)\). This in particular implies that the operators \(X^L_{\psi }\), defined in (3.2.62), are universally bounded from \(L^2({\mathbb {T}}^3_L)\) to \(L^2({\mathbb {T}}^3_L)\).
Proof
Using Lemma 3.1 and the normalization of \(\psi \), we have
To conclude, we also made use of the fact that the minimizers are universally bounded in \(L^{\infty }\) by Lemma 3.3. \(\square \)
Recall the definition of \(\varPsi _L\) in (3.2.30).
Lemma 3.9
For any \(\varepsilon >0\), there exists \(R'_{\varepsilon }\) and \(L'_{\varepsilon }\) (with \(R_{\varepsilon }'\leqq L_{\varepsilon }'/2\)) such that for any \(L>L'_{\varepsilon }\), any normalized f in \(L^2({\mathbb {T}}^3_L)\) supported on \(B_{R'_{\varepsilon }}^c:=[-L/2,L/2]^3{\setminus } B_{R'_{\varepsilon }}\), and any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) such that
we have
Proof
By definition of \(L^L_{\psi }\) and \(X^L_{\psi }\), we have
By Proposition 3.5, taking \(L_{\varepsilon }'\) sufficiently large guarantees that
Thus we only need to show that 
and 
can be made arbitrary small by taking \(L_{\varepsilon }'\) and \(R_{\varepsilon }'\) sufficiently large. Since f is normalized and supported on \(B_{R'_{\varepsilon }}^c\),
Moreover, using Lemma 3.1, splitting the integral over \(B_t(x)\) and \(B_t^c(x)\) (for some \(t>0\)), and assuming \(x\in B_{R'_{\varepsilon }}^c\), we find
On the other hand, by Lemma 3.8,
Therefore, by applying Proposition 3.5, we can conclude that there exists \(L'_{\varepsilon }\) and \(R'_{\varepsilon }\) such that, for any \(L>L'_{\varepsilon }\) and any \(L^2\)-normalized f supported on \(B_{R'_{\varepsilon }}^c\), we have
which concludes our proof. \(\square \)
Proof of Proposition 3.7
We only show the second inequality in (3.2.68), as its proof can easily be modified to also show the first. Moreover, we observe that the second inequality in (3.2.68) is equivalent to the statement that for any sequence \(\psi _n\in {\mathcal {M}}_{L_n}\) with \(L_n\rightarrow \infty \),
which we shall prove in the following.
We consider \(\psi _n\in {\mathcal {M}}_{L_n}\), \(L_n\rightarrow \infty \), and define
By translation invariance of \({\mathcal {E}}_{L_n}\) and by Proposition 3.5, we can also restrict to sequences \(\psi _n\) converging to \(\varPsi \) in \(L^2({\mathbb {R}}^{3})\) and such that \(\Vert \psi _n-\varPsi _{L_n}\Vert _{H^1\left( {\mathbb {T}}^3_{L_n}\right) }\rightarrow 0\), where \(\varPsi _{L_n}\) is defined in (3.2.30).
Let now \(g_n\) be a normalized function in \(L^2({\mathbb {T}}^3_{L_n})\), orthogonal to \(\psi _n\) and its partial derivatives, realizing \(h_n\) (which exists by compactness, and can be taken to be a real-valued function). We define the following partition of unity \(0\leqq \eta ^1_R,\eta ^2_R\leqq 1\), with \(\eta ^i_R\in C^{\infty }({\mathbb {R}}^{3})\), \(\eta ^i_R(x)=\eta _i(x/R)\) and
We define \(\eta ^i_n:=\eta ^i_{L_n/8}\) and
Standard properties of IMS localization imply that
Clearly, the first summand in the second sum is of order \(O(L_n^{-2})\), by the scaling of \(\eta ^i_n\). For the second summand, we observe that
and proceed to bound the Hilbert–Schmidt norm of both operators (\(i=1,2\)), which will then bound the last line of (3.2.83). We make use of Lemma 3.1 to obtain
Therefore, also the second summand in the error terms is order \(L_n^{-2}\), which allows us to conclude that
By Lemma 3.9 applied to \(g_n^2\) (which is supported on \(B^c_{L_n/4}\)) and (3.2.65), we find
Since the l.h.s. of (3.2.86) is a convex combination and \((L^{L_n}_{\psi _n}-4X^{L_n}_{\psi _n})\) is uniformly bounded from below, (3.2.87) allows to restrict to sequences \(\psi _n\) such that
uniformly in n and
since our claim holds on any sequence for which (3.2.88) and (3.2.89) are not simultaneously satisfied. Using (3.2.88) it is easy to see that \(g_n^1\) is almost orthogonal to \(\psi _n\), in the sense that
Here we used the \(L^2\)-convergence of \(\psi _n\) to \(\varPsi \). Clearly, the same computation (together with the \(H^1\)-convergence of \(\psi _n\) to \(\varPsi \)) shows that \(g_n^1\) is also almost orthogonal to the partial derivatives of \(\psi _n\).
To conclude, we wish to modify \(g_n^1\) in order to obtain a function \({\tilde{g}}_n\) which satisfies the constraints (i.e., is a competitor) of the full-space variational problem introduced in (3.2.64). We also wish to have
Indeed, (3.2.91) together with (3.2.89) and the fact that \({\tilde{g}}_n\) is a competitor on \({\mathbb {R}}^{3}\), would imply that
which finally yields the proof of the Proposition also for sequences \(\psi _n\) satisfying (3.2.88) and (3.2.89).
We have a natural candidate for \({\tilde{g}}_n\), which is simply
with 
. Clearly \({\tilde{g}}_n\) is a competitor for the full space minimization and we are only left with the task of proving that \({\tilde{g}}_n\) satisfies (3.2.91).
We observe that, since \(g_n^1\) is almost orthogonal to \(\psi _n\) and its partial derivatives, and using Proposition 3.5,
Therefore
Hence, the normalization factor does not play any role in the proof of (3.2.91). Moreover
and thus we can conclude that also \({\mathcal {P}}(g_n^1)\) does not play any role in the proof of (3.2.91), since \((L_{\varPsi }-4X_{\varPsi }){\mathcal {P}}\) is a bounded operator (\({\mathcal {P}}\) has finite dimensional range contained in the domain of \((L_{\varPsi }-4X_{\varPsi })\)), \({\mathcal {P}}\) is a projection and \(\Vert {\mathcal {P}}(g_n^1)\Vert _2\rightarrow 0\). With this discussion, we reduced our problem to showing that
Clearly the kinetic energy terms coincide for every n and \(\mu ^{L_n}_{\psi _n}\rightarrow \mu \), by Proposition 3.5. Therefore we only need to prove that
For the first term, using that \(g_n^1\) is supported on \(B_{L_n/4}\), we have
If we define \(\varPsi _R:=\chi _{B_R} \varPsi \) and \((\psi _n)_R:=\chi _{B_R} \psi _n\) we have \(V_{\sigma _{\varPsi }}=V_{\sigma _{\varPsi _R}}+V_{\sigma _{[\varPsi -\varPsi _R]}}\) and \(V_{\sigma _{\psi _n}}=V_{\sigma _{(\psi _n)_R}}+V_{\sigma _{[\psi _n-(\psi _n)_R]}}\). We consider \(R=R(n)=L_n/8\) and observe that
Similar computations, together with Lemma 3.1, yield similar estimates for \(|V_{\sigma _{[\psi _n-(\psi _n)_R]}}(x)|\). Moreover, since \({{\,\mathrm{dist}\,}}_{{\mathbb {T}}^3_{L_n}}(x,y)=|x-y|\) for \(x,y\in B_{L_n/8}\), we have, for any \(x\in B_{L_n/8}\)
Here we used again Lemma 3.1, the convergence of \(\psi _n\) to \(\varPsi \) and the universal \(L^{\infty }\)-boundedness of minimizers. Putting the pieces together we obtain
as desired. The study is similar for 
, hence we shall not write it down explicitly.
We conclude that (3.2.97) holds and, by the discussion above, the proof is complete. \(\square \)
3.2.3 Proof of Theorem 2.3
In this section we first prove universal local bounds for \({\mathcal {E}}_L\) around minimizers. These are a direct consequence of the results on the Hessian in the previous subsection, the proof follows along the lines of [7, 9, Appendix A] and [5, Appendix A]. Such universal local bounds yield universal local uniqueness of minimizers, i.e., the statement that minimizers that are not equivalent (i.e., not obtained one from the other by translations and changes of phase) must be universally apart (in \(H^1({\mathbb {T}}^3_L)\)). Together with Proposition 3.5, this clearly implies uniqueness of minimizers for L big enough, which is the first part of Theorem 2.3. A little extra effort will then complete the proof of Theorem 2.3.
In this section, for any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) and any \(f\in L^2({\mathbb {T}}^3_L)\), we write \(e^{i\theta }\psi ^y=P^{L^2}_{\Theta _L(\psi )}(f)\), respectively \(e^{i\theta }\psi ^y=P^{H^1}_{\Theta _L(\psi )}(f)\), to mean that \(e^{i\theta } \psi ^y\) realizes the \(L^2\)-distance, respectively the \(H^1\)-distance, between f and \(\Theta _L(\psi )\). Note that by compactness these always exist, but they might not be unique. The possible lack of uniqueness is not a concern for our analysis, however.
Proposition 3.10
(Universal local bounds) There exist universal constants \(K_1>0\) and \(K_2>0\) and \(L^{**} >0\) such that, for any \(L>L^{**}\), any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) and any \(L^2\)-normalized \(f\in H^1({\mathbb {T}}^3_L)\) with
we have
Proof
We can restrict to positive \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) and normalized f such that
which clearly implies
Under this assumption, we prove that if (3.2.103) holds then
The general result follows immediately by invariance of \({\mathcal {E}}_L\) under translations and changes of phase.
We denote \(\delta :=f-\psi \) and proceed to expand \({\mathcal {E}}_L\) around \(\psi \):
We recall that \(H^{{\mathcal {E}}_L}_{\psi }\) is simply the quadratic form associated to the Hessian of \({\mathcal {E}}_L\) at \(\psi \) and it is defined in (3.2.61). We denote 
. The last term, which we see as an error contribution, is explicitly given by
Our first goal is to estimate \(|{{\,\mathrm{Err}\,}}_{\psi }(\delta )|\). By (3.2.106) and the normalization of both \(\psi \) and f, we find
Therefore, also using the positivity of \(\psi \), we have
We now apply Lemma 3.8 to obtain
Finally, by (3.2.15),
Recalling (3.2.103), we can estimate
and this implies, combined with (3.2.112) and (3.2.113), that
We now want to bound \(H^{{\mathcal {E}}_L}_{\psi }(\delta )\). We fix \(0<\tau <\min \{h_{\infty }',h_{\infty }''\}\), where \(h_{\infty }'\) and \(h_{\infty }''\) are defined in (3.2.64). Proposition 3.7 implies that there exists \(L^{**}\) such that for \(L>L^{**}\) and \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\), we have
where we define \(Q_{\psi }=1\!\!1-P_{\psi }\) and \(Q_{\psi }':=1\!\!1-P_{\psi }-\sum _{i=1,2,3} P_{\partial _i \psi /\Vert \partial _i \psi \Vert _2}\). We note that, by (3.2.106) and since \(\psi \) is orthogonal in \(L^2\) to its partial derivatives, we have
Therefore, recalling the definition of \(H^{{\mathcal {E}}_L}_{\psi }\) given in (3.2.61),
Moreover, applying (3.2.110),
and we can thus conclude that
On the other hand, by the universal boundedness of \(V_{\sigma _{\psi }}\) in \(L^{\infty }({\mathbb {T}}^3_L)\) and the universal boundedness of \(\mu ^L_{\psi }\) (see Proposition 3.5), we have, for some universal \(C_1>0\),
Similarly, also using Lemma 3.8, for some universal \(C_2>0\),
If we then define \(C:=(\max \{C_1,C_2\}+1)\), we can conclude the validity of the universal bound
By interpolating between (3.2.120) and (3.2.123), we obtain
Using (3.2.115) and (3.2.124) in (3.2.108), we can conclude that there exists a universal constant C such that for any \(L>L^{**}\), any \(0<\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) and any normalized f satisfying (3.2.105),
In particular, for \(K_2\) sufficiently small, we can find a universal constant c such that (3.2.107) holds, as long as
To conclude the proof, it only remains to show that there exists a universal \(K_1\) such that (3.2.126) holds as long as (3.2.103) holds. This can be achieved as follows. We have, using that both \(\psi \) and \(P^{H^1}_{\Theta _L(\psi )}(f)\) are in \({\mathcal {M}}^{{\mathcal {E}}}_{L}\) and thus are universally bounded in \(H^2({\mathbb {T}}^3_L)\) (by Lemma 3.3) and recalling (see (3.2.105)) that \(\psi =P^{L^2}_{\Theta _L(\psi )}(f)\),
Therefore, for some universal C
and it suffices to take \(K_1\leqq \left[ (-C+\sqrt{C^2+4c})/2\right] ^2\) to conclude our discussion.
\(\square \)
We are ready to prove Theorem 2.3.
Proof of Theorem 2.3
Fix \(K_1\) as in Proposition 3.10. Using Proposition 3.5, we know that there exists \(L_{K_1/2}\) such that, for any \(L>L_{K_1/2}\) and any \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\), we have
We claim that (2.2.9) holds with \(L_1:=\max \{L_{K_1/2}, L^*, L^{**}\}\), where \(L^*\) is the same as in Corollary 3.6 and \(L^{**}\) is the same as in Proposition 3.10.
Let \(L>L_1\) and \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\). Since \(L>L_1\geqq L^*\), we have \(\psi ^y\ne \psi \) for any \(0\ne y\in {\mathbb {T}}^3_L\). Moreover, since \(L>L_1\geqq L_{K_1/2}\) and using the triangle inequality, for any other \(\psi _1\in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) we have
Since \(L>L_1\geqq L^{**}\), we can apply Proposition 3.10, finding
i.e., \(\psi _1\in \Theta _L(\psi )\), and (2.2.9) holds for \(L>L_1\).
For \(\psi \in {\mathcal {M}}^{{\mathcal {E}}}_{L}=\Theta _L(\psi )\), and \(L>L_1\), we now show the quadratic lower bound (2.2.10), independently of L. Lemma 3.3, which guarantees universal \(H^1\)-boundedness of minimizers, and estimate (3.2.15) ensure, by straightforward computations, that there exists \(0<\kappa ^*<1/2\) such that, if \(f\in L^2({\mathbb {T}}^3_L)\) is normalized and satisfies
then f is universally bounded in \(H^1({\mathbb {T}}^3_L)\) and must satisfy
where \(\delta _{K_1}\) is the \(\delta _{\varepsilon }\) from Proposition 3.5 with \(\varepsilon =K_1\). On the other hand, Proposition 3.5 and Proposition 3.10 combined with the fact that we have taken \(L_1\geqq L_{K_1/2}\) (and that trivially \(L_{K_1/2}\geqq L_{K_1}\)), guarantee that any \(L^2\)-normalized f satisfying (3.2.133) must satisfy
Therefore the bound (2.2.10) from Theorem 2.3 holds with the universal constant \(\kappa _1:=\min \{\kappa ^*,K_2\}\) and our proof is complete. \(\square \)
This concludes our study of \({\mathcal {E}}_L\). We now move on to the study of the functional \({\mathcal {F}}_L\).
3.3 Study of \({\mathcal {F}}_L\)
This section is structured as follows. In Section 3.3.1 we prove Corollary 2.4. In Section 3.3.2, we compute the Hessian of \({\mathcal {F}}_L\) at its minimizers, showing the validity of (2.2.16). This allows to obtain a more precise lower bound for \({\mathcal {F}}_L\) (compared to the bounds (2.2.13) and (2.2.14) from Corollary 2.4), which holds locally around the 3-dimensional surface of minimizers \({\mathcal {M}}^{{\mathcal {F}}}_L=\Omega _L(\varphi _L)\). Finally, in Section 3.3.3, we investigate closer the surface of minimizers \(\Omega _L(\varphi _L)\) and the behavior of the functional \({\mathcal {F}}_L\) close to it. In particular, we show that the Hessian of \({\mathcal {F}}_L\) at its minimizers is strictly positive above its trivial zero modes and derive some key technical tools, which we exploit in Section 4.
3.3.1 Proof of Corollary 2.4
In this section, we show the validity of Corollary 2.4. We need the following Lemma. Recall that in our discussion constants are universal if they are independent of L for \(L\geqq L_0>0\).
Lemma 3.11
For \(\psi ,\phi \in H^1({\mathbb {T}}^3_L)\), \(\Vert \psi \Vert _2=\Vert \phi \Vert _2=1\),
Proof
We define \(f(x):=|\psi (x)|+|\phi (x)|\) and \(g(x):=|\psi (x)|-|\phi (x)|\). By the Hardy-Littlewood-Sobolev and the Sobolev inequality (see for example [3] for a comprehensive overview of such results on the torus), and using the normalization of \(\phi \) and \(\psi \) we have
which proves the Lemma. \(\square \)
Proof of Corollary 2.4
With \(\psi _L\) as in Theorem 2.3, let \(\varphi _L:=\sigma _{\psi _L}\in C^{\infty }({\mathbb {T}}^3_L)\). Observing that
and using Theorem 2.3 we can immediately conclude that in the regime \(L>L_1\)
It is also immediate, recalling the definition of \({\mathcal {G}}_L\) in (2.2.1) and that \(\psi _L>0\) (as proven in Theorem 2.3), to conclude that \(\psi _L\) must be the unique positive ground state of \(h_{\varphi _L}\).
To prove (2.2.13), we first of all observe that if \(\varphi \in L^2({\mathbb {T}}^3_L)\), we have
Therefore, it is sufficient to restrict to \(\varphi \) with zero-average and show that in this case
Using Theorem 2.3, we obtain
for some \(y\in {\mathbb {T}}^3_L\). We now apply Lemma 3.11 and use that \(\varphi _L^y=\sigma _{\psi _L^y}\) (see (3.3.3)), obtaining with a simple completion of the square
where \(F=1\!\!1+\kappa '(-\Delta _L)^{1/2}\). Dropping the first term and minimizing over \(\psi \) yields our claim. Finally, (2.2.14) immediately follows from (2.2.13) and the spectral gap of the Laplacian, using the fact that \(\varphi _L\) and all its translates have zero average since \(\varphi _L=\sigma _{\psi _L}\). \(\square \)
3.3.2 The Hessian of \({\mathcal {F}}_L\)
For any \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\), we introduce the notation
and observe that \({\mathcal {F}}_L\), defined in (2.2.4), can equivalently be written as
We compute the Hessian of \({\mathcal {F}}_L\) at its minimizers using standard arguments in perturbation theory, showing the validity of expression (2.2.16). We need the following two Lemmas.
Lemma 3.12
For \(L\geqq L_0>0\), any \(\varphi \in L^2({\mathbb {T}}^3_L)\) and any \(T>0\)
for some constant \(C_T>0\) with \(\lim _{T\rightarrow \infty } C_T = 0\). Here \(\varphi \) is understood as a multiplication operator, \(\Vert \cdot \Vert \) denotes the operator norm on \(L^2({\mathbb {T}}^3_L)\), and
Note that
which clearly makes the two norms equivalent. Nevertheless, we find it more natural to work with a bound of the form (3.3.11), where \(C_T\) is independent of L.
Lemma 3.12 implies that, for any \(\varphi \in L^2({\mathbb {T}}^3_L)+L^{\infty }({\mathbb {T}}^3_L)\), the multiplication operator associated with \(\varphi \) is infinitesimally relatively bounded with respect to \(-\Delta _L\). More precisely, for any \(\delta >0\), there exists \(C\left( \delta , \Vert \varphi \Vert _{L^2({\mathbb {T}}^3_L)+L^{\infty }({\mathbb {T}}^3_L)}\right) \) depending on \(\varphi \) only through \(\Vert \varphi \Vert _{L^2({\mathbb {T}}^3_L)+L^{\infty }({\mathbb {T}}^3_L)}\), such that for any \(f \in \text {Dom}(-\Delta _L)\)
Whenever infinitesimal relative boundedness holds with a constant \(C(\delta )\) uniform over a class of operators, we will say that the class is uniformly infinitesimally relatively bounded. In this case, Lemma 3.12 ensures that multiplication operators associated to functions in \((L^2+L^{\infty })\)-balls are uniformly infinitesimally relatively bounded with respect to \(-\Delta _L\).
Proof
We first observe that, by self-adjointness of \((-\Delta _L + T)^{-1}\), it is sufficient to show that the claimed bound holds for \(\Vert \varphi (-\Delta _L+T)^{-1}\Vert \). For any \(f, \varphi \in L^2({\mathbb {T}}^3_L)\) and any decomposition of the form \(\varphi =\varphi _1+\varphi _2\) with \(\varphi _1\in L^2({\mathbb {T}}^3_L)\) and \(\varphi _2\in L^{\infty }({\mathbb {T}}^3_L)\) we have
Moreover,
Therefore, picking \(C_T:=\max \left\{ T^{-1}, C T^{-1/2} \right\} \) yields
optimizing over \(\varphi _1\) and \(\varphi _2\) completes the proof. \(\square \)
Lemma 3.13
For \(\varphi \in L^2({\mathbb {T}}^3_L)\)
Proof
We write \(f_1=\chi _{[0,1)}\) and \(f_2=\chi _{[1,+\infty )}\) and
Clearly \((-\Delta _L)^{-1/2} \varphi =\varphi _1+\varphi _2\). Moreover
This concludes the proof. \(\square \)
Lemmas 3.12 and 3.13 together yield the following Corollary, whose proof is omitted as it is now straightforward.
Corollary 3.14
For any \(\varphi \) such that \(\Vert (-\Delta _L+1)^{-1/2}\varphi \Vert _2\) is finite, the multiplication operator \(V_{\varphi }\) (defined in (2.2.2)) is infinitesimally relatively bounded with respect to \((-\Delta _L)\). Moreover, for \(T> 0\) there exists \(C_T\) such that
In particular, Corollary 3.14 implies that the family of multiplication operators associated to \(\{V_{\varphi } \,|\, \Vert (-\Delta _L+1)^{-1/2}\varphi \Vert _2 \leqq M\}\) is uniformly infinitesimally relatively bounded with respect to \(-\Delta _L\) for any M.
With these tools at hand we now investigate \({\mathcal {F}}_L\) close to its minimum and, in particular, compute the Hessian of \({\mathcal {F}}_L\) at its minimizers. We follow very closely the analogous analysis carried out in [9]. By translation invariance of the problem, it is clearly sufficient to perform the computation with respect to \(\varphi _L\), where \(\varphi _L\) is the same as in Corollary 2.4.
Proposition 3.15
For \(L> L_1\) let \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\) be such that
for some \(\varepsilon _L>0\) small enough. Then
where
and \(\psi _L\), which we recall (see (2.2.12)) is the (positive) ground state of \(h_{\varphi _L}\), is understood, in the expressions for \(K_L\) and \(J_L\), as a multiplication operator.
Note that this implies that \(H^{{\mathcal {F}}_L}_{\varphi _L}=1\!\!1-K_L\), as claimed in (2.2.16). In particular, \(K_L\leqq 1\!\!1\) by minimality of \(\varphi _L\). It is also clear, by definition, that \(K_L\geqq 0\). We emphasize that \(J_L\) is trace class, being the square of 2 \((-\Delta _L+1)^{-1/2}\psi _L(-\Delta _L)^{-1/2}\), which is Hilbert-Schmidt since \(\psi _L\) is in \(L^2\), as a function of x, and \(f(k):=(|k|^2+1)^{-1/2}|k|^{-1}\) is in \(L^2\), as a function of k. From the trace class property of \(J_L\), together with the boundedness of \((-\Delta _L+1)^{1/2}\frac{Q_{\psi _L}}{h_{\varphi _L}-e(\varphi _L)} (-\Delta _L+1)^{1/2}\) (which follows from Corollary 3.14), we immediately infer the trace class property of \(K_L\). We even show in Lemma 3.16 that \(J_L,K_L\lesssim _L (-\Delta _L+1)^{-2}\).
We shall in the following denote by \(K_L^y\), respectively \(J_L^y\), the unitary equivalent operators obtained from \(K_L\) and \(J_L\) by a translation by y. Note that \(K_L^y\) and \(J_L^y\) appear if one expands \({\mathcal {F}}_L\) with respect to \(\varphi _L^y\) instead of \(\varphi _L\). Moreover, the invariance under translations of \({\mathcal {F}}_L\) implies that
We show in Section 3.3.3 that these two sets coincide. Finally, even though both \(\varepsilon _L\) and the estimate (3.3.23) in Proposition 3.15 depend on L, with a little extra work one can show that the bound is actually uniform in L (for large L). For simplicity we opt for the current version of Proposition 3.15, as it is sufficient for the purpose of our investigation, which is set on a torus of fixed linear size \(L>L_1\).
Proof
We shall denote \(h_0:=h_{\varphi _L}\). By assumption (3.3.22) and since \(\varphi _L \in L^2({\mathbb {T}}^3_L)\), we can apply Corollary 3.14 to \(\varphi _L\) and to \((\varphi -\varphi _L)\). This way we see that \(V_{\varphi -\varphi _L}\) is uniformly infinitesimally relatively bounded with respect to \(h_{0}\) for any \(\varphi \) satisfying (3.3.22).
It is clear that \(h_{0}\) admits a simple and isolated least eigenvalue \(e(\varphi _L)\). Standard results in perturbation theory then imply that there exist \(\varepsilon _L>0\) and a contour \(\gamma \) around \(e(\varphi _L)\) such that for any \(\varphi \) satisfying (3.3.22) \(e(\varphi )\) is the only eigenvalue of \(h_{\varphi }=h_0+V_{\varphi -\varphi _L}\) inside \(\gamma \). (For fixed \(\varphi \), the statement above is a standard result in perturbation theory, see [26, Theorem XII.8]; moreover it is also possible to get a \(\varphi \)-independent \(\gamma \) encircling \(e(\varphi )\) (see [26, Theorem XII.11]) since \(V_{\varphi -\varphi _L}\) is uniformly infinitesimally relatively bounded with respect to \(h_0\).) We can thus write
Moreover, by the uniform infinitesimal relative boundedness of \(V_{\varphi -\varphi _L}\) with respect to \(h_0\), we have
for \(\varepsilon _L\) sufficiently small. For any \(z\in \gamma \), we can thus use the resolvent identity in the form
The first term is analytic inside the contour \(\gamma \) and hence it gives zero after integration when inserted in (3.3.26). Inserting the second term of (3.3.28), which is rank one, in (3.3.26) and using Fubini’s Theorem to interchange the trace and the integral, we obtain
For simplicity, we introduce the notation
Because of (3.3.27), both A and B are smaller than 1 in norm, uniformly in \(z\in \gamma \). We shall use the identity
We insert the various terms in (3.3.29) and do the contour integration. The term \(1\!\!1\) gives \(e(\varphi _L)\). The term A, recalling (see (2.2.12)) that \((-\Delta _L)^{-1/2} \rho _{\psi _L}=\varphi _L\), yields
A standard calculation shows that the term \(A(A+B)\) gives
Furthermore, since \(Q_{\psi _L}\psi _L=0\), the term \(B(1\!\!1-B)^{-1}\) yields zero. Recalling that \({\mathcal {F}}_L(\varphi )=\Vert \varphi \Vert ^2+e(\varphi )\) we obtain from (3.3.29)
We observe that, since \(\gamma \) is uniformly bounded and uniformly bounded away from \(e(\varphi _L)\), we can get rid of the integration, i.e., it suffices to bound
with the r.h.s. of (3.3.23) to conclude the proof. We note that
and that, by infinitesimal relative boundedness of \(V_{\varphi _L}\) with respect to \((-\Delta _L)\) and since \(\gamma \) is uniformly bounded away from \(e(\varphi _L)\), there exists some constant \(C_L>0\) such that
Therefore,
Since
it follows that
where we used the relative boundedness of \(h_{\varphi }\) w.r.t to \(-\Delta _L\) and Corollary 3.14. This yields the right bound for (I). Similar estimates yield the right bounds for \(\Vert A(1\!\!1-A)^{-1}\Vert \) and \(\Vert (1\!\!1-A)^{-1}B(1\!\!1-B)^{-1}\Vert \lesssim _L\Vert B\Vert \), concluding the proof. \(\square \)
As a final result of this subsection, we prove the following Lemma about the operators \(K_L\) and \(J_L\):
Lemma 3.16
Let \(K_L\) and \(J_L\) be the operators defined in (3.3.24). We have
Proof
We prove the result for \(J_L\). By the relative boundedness of \(h_{\varphi _L}\) with respect to \(-\Delta _L\) the same proof applies to \(K_L\). We shall show that \((-\Delta _L+1)(-\Delta _L)^{-1/2}\psi _L (-\Delta _L+1)^{-1/2}\) is bounded as an operator on \(L^2({\mathbb {T}}^3_L)\). In fact, for \(f\in L^2({\mathbb {T}}^3_L)\),
where we used that \(\psi _L\in C^{\infty }({\mathbb {T}}^3_L)\) and that \(\sum _{0\ne k\in \frac{2\pi }{L} {\mathbb {Z}}^3}\frac{(|k|^2+1)^2}{|k|^2(|k-\xi |^2+1)^{3}}\lesssim |\xi |^2+1\). Therefore
as claimed. \(\square \)
3.3.3 Local Properties of \({\mathcal {M}}^{{\mathcal {F}}}_L\) and \({\mathcal {F}}_L\)
For \(L>L_1\) we introduce the notation
which is going to be used throughout this section and Section 4. According to Theorem 2.3, the condition \(L> L_1\) guarantees that \(\psi _L^y\ne \psi _L\) for any \(\psi _L \in {\mathcal {M}}^{{\mathcal {E}}}_{L}\) and any \(y\ne 0\), which implies that \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\) is three dimensional (i.e that the partial derivatives of \(\varphi _L\) are linearly independent); if not, there would exist \(\nu \in {\mathbb {S}}^2\) such that \(\partial _{\nu } \psi _L=0\) and this would imply \(\psi _L=\psi _L^y\) for any y parallel to \(\nu \).
For technical reasons, we also introduce a family of weighted norms which will be needed in Section 4. For \(T\geqq 0\), we define
where \(W_T\) acts in k-space as multiplication by
Note that 
and \(\Vert \varphi \Vert _{W_{\infty }}=\Vert \varphi \Vert _2\).
For the purpose of this section we could formulate the following Lemma only with respect to \(\Vert \cdot \Vert _2=\Vert \cdot \Vert _{W_{\infty }}\), but we opt for this more general version since we shall need it in Section 4:
Lemma 3.17
For any \(L>L_1\), there exists \(\varepsilon '_L\) (independent of T) such that for any \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\) with \({{\,\mathrm{dist}\,}}_{W_T}(\varphi ,\Omega _L(\varphi _L))\leqq \varepsilon '_L\) there exist a unique couple \((y_{\varphi },v_{\varphi })\), depending on T, with \(y_{\varphi }\in {\mathbb {T}}^3_L\) and \(v_{\varphi }\in ({{\,\mathrm{span}\,}}_{i=1,2,3} \{W_T\partial _i \varphi _L\})^{\perp }\), such that
As Proposition 3.15 above, we opt for an L-dependent version of Lemma 3.17 for simplicity, as it is sufficient for our purposes. We nevertheless believe it is possible to prove a corresponding statement that is uniform in L. Note that Lemma 3.17 is equivalent to the statement that there exists a T-independent \(\varepsilon '_L\) such that the \(W_T\)-projection onto \(\Omega _L(\varphi _L)\) is uniquely defined in an \(\varepsilon _L'\)-neighborhood of \(\Omega _L(\varphi _L)\) with respect to the \(W_T\)-norm, and that, for any \(\varphi \) therein, \(\varphi _L^{y_{\varphi }}\) characterizes the \(W_T\)-projection of \(\varphi \) onto \(\Omega _L(\varphi _L)\), so that
Proof
We begin by observing that the Lemma is equivalent to showing that for any \(\Vert \cdot \Vert _{W_T}\)-normalized \({v \in ({{\,\mathrm{span}\,}}_{i=1,2,3} \{W_T\partial _i \varphi _L\})^{\perp }}\), any \(\varepsilon \leqq \varepsilon _L'\) and any \(0\ne y\in {\mathbb {T}}^3_L\) we have
Indeed, if the Lemma holds then \(\varphi =\varphi _L+\varepsilon v \) does not admit other decompositions of the form (3.3.48), which implies that, for any \(y\ne 0\), (3.3.50) holds (otherwise there would exist \(y\ne 0\) minimizing the \(W_T\)-distance of \(\varphi \) from \(\Omega _L(\varphi _L)\) and such y would necessarily yield a second decomposition of the form (3.3.48)). On the other hand, if the statement (3.3.50) holds and the Lemma does not, then there exists \(\varphi \) such that \({{\,\mathrm{dist}\,}}_{W_T}(\varphi ,\Omega _L(\varphi _L))\leqq \varepsilon '_L\) and also such that \((y_1,v_1)\) and \((y_2,v_2)\) yield two different decompositions of the form (3.3.48) for \(\varphi \) (note that at least one decomposition of the form (3.3.48) always exist, as there exist at least one element of \(\Omega _L(\varphi _L)\) realizing the \(W_T\)-distance of \(\varphi \) from \(\Omega _L(\varphi _L)\)). By considering \(\varphi ^{-y_1}\) (respectively \(\varphi ^{-y_2}\)) we find \(\Vert v_1\Vert _{W_T}>\Vert v_2\Vert _{W_T}\) (respectively \(\Vert v_2\Vert _{W_T}>\Vert v_1\Vert _{W_T}\)), which is clearly a contradiction. We shall hence proceed to prove the statement (3.3.50).
Taylor’s formula and the regularity of \(\varphi _L\) imply the existence of a T-independent constant \(C^1_L\) such that
As remarked after (3.3.45), the kernel of \(\Pi ^L_{\nabla }\) is three-dimensional, hence there exists a constant \(C^2_L\) independent of T such that
Therefore, using that \(v\perp _{W_T} \nabla \varphi _L\) in combination with (3.3.51) and (3.3.52), we find, for
that
i.e., that (3.3.50) holds for y satisfying (3.3.53). Furthermore, we have
and this implies that (3.3.50) holds for any y such that
Using again (3.3.52) and (3.3.51), there exist \(C^3_L,c^1_L,c^4_L>0\) independent of T such that
where the second line simply follows from \(\Vert \cdot \Vert _{W_T}\geqq \Vert \cdot \Vert _{W_0}\), the fact that \(\varphi _L\ne \varphi _L^y\) for any \(0\ne y \in [-L/2,L/2]^3\) and the continuity of \(\varphi _L\). Combining (3.3.56) and (3.3.57), we conclude that (3.3.50) holds if either \(|y|>c^1_L\) or
Picking \(\varepsilon _L'\) sufficiently small, the fact that (3.3.50) holds both under the conditions (3.3.53) and (3.3.58) shows that it holds for any \(y\in {\mathbb {T}}^3_L\), and this completes the proof. \(\square \)
We conclude our study of the Pekar functional \({\mathcal {F}}_L\) by showing that \(\ker (1\!\!1-K_L)={{\,\mathrm{span}\,}}\{\partial _j \varphi _L\}_{j=1}^3={{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\). Since clearly \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\subset \ker (1\!\!1-K_L)\), this is a consequence of the following Proposition:
Proposition 3.18
Recalling the definition of \(\tau _L\) from Corollary 2.4, we have
Proof
We need to show that for all normalized \(v\in {{\,\mathrm{ran}\,}}(1\!\!1-\Pi ^L_{\nabla })\) the bound
holds. Using Lemma 3.17 in the case \(T=\infty \), for any such v and \(\varepsilon \) small enough, denoting \(\varphi =\varphi _L+\varepsilon v\), we obtain
Moreover, since \(\Vert (-\Delta _L+1)^{-1}(\varphi -\varphi _L)\Vert \leqq \varepsilon \Vert v\Vert _2=\varepsilon \), for \(\varepsilon \) small enough we can expand \({\mathcal {F}}_L(\varphi )\) with respect to \(\varphi _L\) using Proposition 3.15. Combining this with (2.2.14), we arrive at
Since \(\varepsilon \) can be taken arbitrarily small, the proof is complete. \(\square \)
4 Proof of Main Results
In this Section we give the proof of Theorem 2.5. In Section 4.1 we prove the upper bound in (2.3.2). In Section 4.2 we estimate the cost of substituting the full Hamiltonian \({\mathbb {H}}_L\) with a cut-off Hamiltonian depending only on finitely many phonon modes, a key step in providing a lower bound for \({{\,\mathrm{inf\,spec}\,}}{\mathbb {H}}_L\). Finally, in Section 4.3, we show the validity of the lower bound in (2.3.2).
The approach used in Sections 4.1 and 4.2 follows closely the one used in [9], even if, in our setting, minor complications arise in the proof of the upper bound due the presence of the zero modes of the Hessian. For the lower bound in Section 4.3, however, a substantial amount of additional work is needed to deal with the translation invariance, which complicates the proof significantly.
4.1 Upper Bound
In this section we construct a trial state, which will be used to obtain an upper bound on the ground state energy of \({\mathbb {H}}_L\) for fixed \(L>L_1\). This is carried out using the Q-space representation of the bosonic Fock space \({\mathcal {F}}(L^2({\mathbb {T}}^3_L))\) (see [25]). Even though the estimates contained in this section are L-dependent, we believe it is possible, with little modifications to the proof, to obtain the same upper bound with the same error estimates uniformly in L.
Our trial state depends non-trivially only on finitely many phonon variables, and we proceed to describe it more in detail. If one picks \(\Pi \) to be a real finite rank projection on \(L^2({\mathbb {T}}^3_L)\), then
The first factor \({\mathcal {F}}(\Pi L^2({\mathbb {T}}^3_L))\) can isomorphically be identified with \(L^2({\mathbb {R}}^N)\), where N is the complex dimension of \({{\,\mathrm{ran}\,}}\Pi \). In particular, there is a one-to-one correspondence between any real \(\varphi \in {{\,\mathrm{ran}\,}}\Pi \) and \(\lambda =(\lambda _1,\dots ,\lambda _N)\in {\mathbb {R}}^N\), explicitly given by
where \(\{\varphi _i\}_{i=1}^N\) denotes an orthonormal basis of \({{\,\mathrm{ran}\,}}\Pi \) consisting of real-valued functions. The trial state we use corresponds to the vacuum in the second factor \({{\mathcal {F}}((1\!\!1-\Pi )L^2({\mathbb {T}}^3_L))}\) and shall hence be written only as a function of x (the electron variable) and \(\lambda \) (the finitely many phonon variables selected by \(\Pi \)). We begin by specifying some properties we wish \(\Pi \) to satisfy. Consider \(\varphi _L\) from Corollary 2.4 and define \(\Pi \) to be a projection of the form \(\Pi =\Pi '+\Pi ^L_{\nabla }\), where \(\Pi ^L_{\nabla }\) is defined in (3.3.45) and \(\Pi '\) is an \((N-3)\)-dimensional projection onto \(({{\,\mathrm{span}\,}}\{\partial _j \varphi _L\}_{j=1}^3)^{\perp }={{\,\mathrm{ran}\,}}(1\!\!1-\Pi ^L_{\nabla })\) that will be further specified later but will always be taken so that \(\varphi _L \in {{\,\mathrm{ran}\,}}\Pi \). Our trial state is of the form
where
-
\(x\in {\mathbb {T}}^3_L\) and \(\varphi \) is a real element of \({{\,\mathrm{ran}\,}}\Pi \) (identified with \(\lambda \in {\mathbb {R}}^N\) as in (4.1.2)),
-
\(G(\varphi )\) is a Gaussian factor explicitly given by
(4.1.4) -
\(\eta \) is a ‘localization factor’ given by
$$\begin{aligned} \eta (\varphi )=\chi \left( \varepsilon ^{-1}\Vert (-\Delta _L+1)^{-1/2}(\varphi -\varphi _L)\Vert _{L^2({\mathbb {T}}^3_L)}\right) , \end{aligned}$$(4.1.5)for some \(0<\varepsilon <\varepsilon _L\) (with \(\varepsilon _L\) as in Proposition 3.15), where \(0\leqq \chi \leqq 1\) is a smooth cut-off function such that \(\chi (t)=1\) for \(t\leqq 1/2\) and \(\chi (t)=0\) for \(t\geqq 1\),
-
\(\psi _{\varphi }\) is the unique positive ground state of \(h_{\varphi }=-\Delta _L+V_{\varphi }\).
We note that our state actually depends on two parameters (N and \(\varepsilon \)) and, of course, on the specific choice of \(\Pi '\). We choose \(\{\varphi _i\}_{i=1,\dots , N}\) to be a real orthonormal basis of eigenfunctions of \([\Pi (1\!\!1-K_L)\Pi ]\) corresponding to eigenvalues \(\mu _i=0\) for \(i=1,2,3\) and \(\mu _i\geqq \tau _L>0\) for \(i=4,\dots ,N\). Recalling Proposition 3.18, this amounts to choosing \(\{\varphi _i\}_{i=1,2,3}\) to be a real orthonormal basis of \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\) and \(\{\varphi _i\}_{i=4,\dots ,N}\) to be a real orthonormal basis of eigenfunctions of \([\Pi '(1\!\!1-K_L)\Pi ']\). With this choice, we have (with a slight abuse of notation)
where \(\varphi _L\cong \lambda _L=(0,0,0,\lambda ^L_4,\dots ,\lambda _N^L)\), since \(\varphi _L \in {{\,\mathrm{ran}\,}}\Pi \) by construction, and the first three coordinates are 0 since \(\varphi _L \in \left( {{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\right) ^{\perp }\).
We first show that even if G does not have finite \(L^2({\mathbb {R}}^N)\)-norm, \(\Psi \) does due to the presence of \(\eta \). We define
and
Then, on \(T_{\varepsilon }\), noting that \(\Pi ^L_{\nabla }\varphi _L =0\), we have
and this implies, using the normalization of \(\psi _{\varphi }\), that
We spend a few words to motivate our choice of \(\Psi \). The absolute value squared of \(\Psi \) has to be interpreted as a probability density over the couples \((\varphi ,x)\), with \(\varphi \) being a classical state for the phonon field and x the position of the electron. In the electron coordinate, our \(\Psi \) corresponds to the ground state of \(h_{\varphi }\) for any value of \(\varphi \). This implies, by straightforward computations, that the expectation value of the Fröhlich Hamiltonian in \(\Psi \) equals the one of \(e(\varphi )+{\mathbb {N}}\), \(e(\varphi )\) being the ground state energy of \(h_{\varphi }\) and \({\mathbb {N}}\) the number operator. Moreover, because of the factor \(\eta \), we are localizing our state to the regime where the Hessian expansion of \(e(\varphi )\) from Proposition 3.15 holds. To leading order, this effectively makes our system formally correspond to a system of infinitely many harmonic oscillators with frequencies given by the eigenvalues of \((1\!\!1-K_L)^{1/2}\), with a Gaussian ground state. To carry out this analysis out rigorously, we need to choose a suitable finite rank projection \(\Pi \), as detailed in the remainder of this section.
We are now ready to delve into the details of the proof. It is easy to see that the interaction term appearing in the Fröhlich Hamiltonian acts in the Q-space representation as the multiplication by \(V_{\varphi }(x)\). Therefore, since \(\Psi \) corresponds to the vacuum on \((1\!\!1-\Pi )L^2({\mathbb {T}}^3_L)\) and only depends on x through the factor \(\psi _{\varphi }(x)\), the g.s. of \(h_{\varphi }\), it follows that
where \(\varphi = \Pi \varphi \cong \lambda \in {\mathbb {R}}^N\) and the inner product on the r.h.s. is naturally interpreted as the one on \(L^2({\mathbb {T}}^3_L)\otimes L^2({\mathbb {R}}^{N})\). In the Q-space representation, the number operator takes the form
Using the fact that \(\eta \) is supported on the set \(T_{\varepsilon }\) defined in (4.1.7), we can use the Hessian expansion from Proposition 3.15 to obtain bounds on \(e(\lambda )\). Consequently, for a suitable positive constant \(C_L\),
with
We shall now proceed to first show that B only contributes as an error term and then to rewrite A as the sum of a leading order energy correction term and an error term. We recall that by Lemma 3.16
Therefore, since \(\eta \) is supported on \(T_{\varepsilon }\), we have
To treat A a bit more work is required. A direct calculation shows that
The previous identity, together with straightforward manipulations involving integration by parts, shows that
where the first term is clearly a leading order energy correction whereas \(A_1\) and \(A_2\) have to be interpreted as error terms, as we now proceed to show. By standard first order perturbation theory (using that the phase of \(\psi _{\varphi }\) is chosen so that it is the unique positive minimizer of \(h_{\varphi }\)) we have
where we recall that 
. This implies that, for fixed \(\varphi \),
where \(\psi _{\varphi }\) is interpreted as a multiplication operator in the last two expressions. Since \((-\Delta _L+1)^{1/2} \left( \frac{Q_{\psi _{\varphi }}}{h_{\varphi }-e(\varphi )}\right) ^2 (-\Delta _L+1)^{1/2}\) is uniformly bounded over the support of \(\eta \) (the potential \(V_{\varphi }\) being uniformly infinitesimally relatively bounded with respect to \(-\Delta _L\) by Corollary 3.14) and recalling that \(\psi _{\varphi }\) is normalized by definition, we get
In summary, we conclude that
Finally, we proceed to bound \(A_2\). Recalling the definition of \(\eta \) and \(T_{\varepsilon }\), we see that
where we used that \(\eta \) is supported on \(T_{\varepsilon }\) and that \(\chi \) is smooth and compactly supported. Therefore, using also the normalization of \(\psi _{\varphi }\), we obtain
We now need to bound \(\Vert 1\!\!1_{T_{\varepsilon }}G\Vert _{L^2({\mathbb {R}}^N)}\) in terms of \(\Vert \Psi \Vert = \Vert \eta G\Vert _{L^2({\mathbb {R}}^N)}\). We define
and observe that on \(S_{\nu }\cap T_{\varepsilon }\) we have, by the triangle inequality,
and that on \(S_{\nu }^c\)
where we used that \(\left[ \Pi (1\!\!1-K_L)\Pi \right] ^{1/2}\geqq \tau _L^{1/2} \Pi '\) (with \(\tau _L\) being the constant appearing in Proposition 3.18). We then have, using (4.1.27), that
We now perform the change of variables \((\lambda _1,\lambda _2,\lambda _3)=3(\lambda '_1,\lambda '_2,\lambda '_3)\) in the first integral and the change of variables \(\lambda -\lambda _L=2(\lambda '-\lambda _L)\) in the second integral and fix \(\nu =\varepsilon /8\), obtaining
where in the second step we used that \(\{\Vert (-\Delta _L+1)^{-1/2}\Pi ^L_{\nabla }\varphi \Vert _2 \leqq (\varepsilon +\nu )/3\}\cap S_{\nu } \subset T_{\varepsilon /2}\) by the triangle inequality if \(\nu =\varepsilon /8\), and (4.1.28) to estimate the Gaussian factor on \(S_{\nu /2}^c\). Therefore, as long as \(\sqrt{N}\leqq C_L^1 \alpha \varepsilon \) for a sufficiently small \(C_L^1\), we conclude that
Plugging estimates (4.1.17), (4.1.19), (4.1.23), and (4.1.31) into (4.1.13), we infer, for \(\sqrt{N}\leqq C_L^1 \alpha \varepsilon \), that for a sufficiently large \(C_L^2\)
We now proceed to choose a real orthonormal basis for \({{\,\mathrm{ran}\,}}\Pi \) which is convenient to bound the r.h.s. of (4.1.32). Let \(\{g_j\}_{j\in {\mathbb {N}}}\) be an orthonormal basis of eigenfunctions of \(K_L\) with corresponding eigenvalue \(k_j\), ordered such that \(k_{j+1}\geqq k_j\). By Proposition 3.18 we have \(k_j=1\) for \(j=1,2,3\) and \(k_j<1\) for \(j>3\). Moreover, \(\Pi ^L_{\nabla }\) coincides with the projection onto \({{\,\mathrm{span}\,}}\{g_1,g_2,g_3\}\). We pick \(\Pi '\) to be the projection onto \({{\,\mathrm{span}\,}}\{g_4,\dots , g_{N}\}\) if \(\varphi _L\) is spanned by \(\{g_1,\dots , g_N\}\) and onto \({{\,\mathrm{span}\,}}\{g_4,\dots , g_{N-1}, \varphi _L\}\) otherwise. With this choice the eigenvalues \(\mu _i\) of \(\Pi (1\!\!1-K_L)\Pi \) appearing in the Gaussian factor G are equal to
with 
. In any case
In order to estimate \(\sum _{j=N}^{\infty } (1-(1-k_j)^{1/2})\), we note that Lemma 3.16 implies that \(k_j\lesssim _L (l_j+1)^{-2}\), where \(l_j\) denotes the ordered eigenvalues of \(-\Delta _L\). Since \(l_j\sim j^{2/3}\) for \(j\gg 1\), we have
This allows us to conclude that
as long as \(\sqrt{N}\leqq C_L^1 \alpha \varepsilon \). The error term is minimized, under this constraint, for \(\varepsilon \sim \alpha ^{-8/11}\) and \(N\sim \alpha ^2 \varepsilon ^2 \sim \alpha ^{6/11}\), which yields
as claimed in (2.3.2).
4.2 The Cutoff Hamiltonian
As a first step to derive the lower bound in (2.3.2), we show that it is possible to apply an ultraviolet cutoff of size \(\Lambda \) to \({\mathbb {H}}_L\) at an expense of order \(\Lambda ^{-5/2}\) (this is proven in Proposition 4.5 in Section 4.2.3). Our approach follows closely the one in [9]. It relies on an application of a triple Lieb–Yamazaki bound (extending the method of [19]) which we carry out in Section 4.2.1, and on a consequent use (in Section 4.2.2) of a Gross transformation [13, 23].
We shall in the following, for any real-valued \(f\in L^2({\mathbb {T}}^3_L)\), denote
We recall that (see (2.1.4)) the interaction term in the Fröhlich Hamiltonian is given by
where \(v_L\) was defined in (2.1.3) and a and \(a^{\dagger }\) satisfy the rescaled commutation relations (2.1.5). We shall apply an ultraviolet cutoff of size \(\Lambda \) in k-space, which amounts to substituting the interaction term with
where
To quantify the expense of such a cutoff we clearly need to bound
where
4.2.1 Triple Lieb–Yamazaki Bounds
Let us introduce the notation \(p=(p_1,p_2,p_3)=-i\nabla _x\) for the electron momentum operator. Note that on any function of the form \(f(x,y)=f(y-x)\), such as \(w^x_{L,\Lambda }\) for example, the operator p simply acts as multiplication by k in k-space and agrees, up to a sign, with \(-i\nabla _y\).
The purpose of this section is to prove the following Proposition.
Proposition 4.1
Let \(w_{L,\Lambda }\) be defined as in (4.2.7) and \(\Lambda >1\). Then
as quadratic forms on \(L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}(L^2({\mathbb {T}}^3_L))\).
We first need the following Lemma.
Lemma 4.2
Let \(w_{L,\Lambda }\) be defined as in (4.2.7) and \(\Lambda >1\). Then for any \(j,l,m\in \{1,2,3\}\)
as quadratic forms on \(L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}(L^2({\mathbb {T}}^3_L))\).
Proof
For any \(j,l,m\in \{1,2,3\}\), (4.2.9) follows from \(a^{\dagger }(g)a(g)\leqq \Vert g\Vert _2^2 {\mathbb {N}}\) for \(g\in L^2({\mathbb {T}}^3_L),\) and then proceeding along the same lines of the proof of (4.2.10). To prove (4.2.10) we estimate
If we denote \(f_{j,l}^x:=(-\partial _j \partial _l (-\Delta _L)^{-2} w_{L,\Lambda })^x\), in order to show (4.2.11) it suffices to prove that
where the bracket notation refers to the second factor in the tensor product, i.e., the left side is a rank-one projection on the second factor parametrized by x, which acts via multiplication on the first factor. For any \(\Psi \in L^2({\mathbb {T}}^3_L)\otimes L^2({\mathbb {T}}^3_L)\) with Fourier coefficients \(\Psi _{q,k}\), we have
which shows our claim. We only need to justify the last step, i.e., that the supremum appearing in (4.2.14) is bounded by \(C\Lambda ^{-5}\). We have
This concludes the proof. \(\square \)
We are now able to prove Proposition 4.1.
Proof of Proposition 4.1
Following the approach by Lieb and Yamazaki in [19], we have
Applying this three times, we obtain
Similarly,
Therefore, if we define
we have
Using that \(B_{jkl}^{\dagger }=-B_{jkl}\) and that \(B_{jkl}\) is invariant under exchange of indices, we arrive at
By the Cauchy–Schwarz inequality, we have for any \(\lambda >0\)
Moreover, using (4.2.9) and the rescaled commutation relations (2.1.5) satisfied by a and \(a^{\dagger }\), we have
Using (4.2.22) and (4.2.23) and picking \(\lambda =C^{1/2}\Lambda ^{-5/2}\) we conclude that
We now define
Using (4.2.10), (4.2.11) and the Cauchy-Schwarz inequality, we have for any \(\lambda >0\)
Moreover,
Picking \(\lambda =\Lambda ^{-5/2}+\alpha ^{-1}\Lambda ^{-3/2}\), we therefore conclude that
Applying (4.2.24) and (4.2.28) in (4.2.21), we finally obtain
as claimed. \(\square \)
4.2.2 Gross Transformation
The bound (4.2.8), derived in Proposition 4.1, is not immediately useful as it stands. In order to relate the r.h.s. of (4.2.8) to the square of the Fröhlich Hamiltonian \({\mathbb {H}}_L\) in (2.1.4), we shall apply a Gross transformation [13, 23].
For a real-valued \(f\in H^1({\mathbb {T}}^3_L)\), recalling that \(f^x(\,\cdot \,)=f(\,\cdot \,-x)\), we consider the following unitary transformation on \(L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}\)
where U is understood to act as a ‘multiplication’ with respect to the x variable. For any \(g\in L^2({\mathbb {T}}^3_L)\), we have
and therefore
Moreover,
This implies that
Therefore, we also have
We denote
and we shall pick
for some \(K>0\). Recalling (4.2.5), this implies that
For simplicity we suppress the dependence on K in the notation for f and g, but we will keep track of the parameter K by denoting the operator U related to this choice of f (depending on \(\alpha \) and K) via (4.2.30) by \(U^K_{\alpha }\). We shall need the following estimates for norms involving f and g. We have
We now state and prove the main result of this subsection, the proof of which follows the approach used in [12] for the analogous statement on \({\mathbb {R}}^{3}\), and in [9] for the analogous statement on a domain with Dirichlet boundary conditions.
Proposition 4.3
For any \(\varepsilon >0\) there exist \(K_{\varepsilon }>0\) and \(C_{\varepsilon }>0\) such that, for all \(\alpha \gtrsim 1\) and any \(\Psi \in L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}\) in the domain of \(|p|^2+{\mathbb {N}}\)
Proof
We shall use the following standard (given the rescaled commutation relations satisfied by a and \(a^{\dagger }\)) properties, which hold for any \(\Psi \in {\mathcal {F}}\), any \(f\in L^2({\mathbb {T}}^3_L)\) and any function \(h: [0,\infty )\rightarrow {\mathbb {R}}\)
It is then straightforward, with the aid of the estimates (4.2.39), (4.2.40), (4.2.41) and (4.2.42), to show, for any \(\Psi \in L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}\), any \(\delta >0\) and any \(K>0\), that
It remains to bound the term
As in (4.2.48), we can easily bound
By (4.2.36) and (4.2.38) and recalling (4.2.5) and (4.2.7), we have
With the same arguments used in the proof of Lemma 4.2 we obtain
and therefore, using (4.2.40) to bound \(\Vert a(f^x) \Psi \Vert \) , we arrive at
Combining (4.2.46)–(4.2.48), (4.2.50), (4.2.53), (4.2.40) and (4.2.41) with (4.2.35), we obtain, for any \(K\geqq 1\)
which allows us to conclude the proof by picking \(K_{\varepsilon }\sim \varepsilon ^{-2}\), \(\delta \sim \varepsilon \) and \(C_{\varepsilon }\sim \varepsilon ^{-3}\).
\(\square \)
Remark 4.4
Proposition 4.3 has as an important consequence the fact that the ground state energy of \({\mathbb {H}}_L\) is uniformly bounded for \(\alpha \gtrsim 1\).
4.2.3 Final Estimates for Cut-off Hamiltonian
With Propositions 4.1 and 4.3 at hand, we are finally ready to prove the main result of this section. Note that all the estimates performed in this section are actually independent of L.
Proposition 4.5
Let
where \(v_{L,\Lambda }\) is defined in (4.2.5). Then, for any \(\Lambda \gtrsim 1\) and \(\alpha \gtrsim 1\),
Note that for the error term introduced in (4.2.57) to be negligible compared to \(\alpha ^{-2}\) it suffices to pick \(\Lambda \gg \alpha ^{4/5}\).
Proof
We begin by recalling that Proposition 4.1 implies that
Applying the unitary Gross transformation \(U^K_{\alpha }\) introduced in the previous subsection (with f defined in (4.2.37) and K large enough for Proposition 4.3 to hold for some \(0<\varepsilon <1\)) to both sides of the previous inequality and recalling (4.2.31), we obtain
Proposition 4.3 implies that
where C is a positive constant (independent of \(\alpha \) for \(\alpha \gtrsim 1\)). Recalling the definitions of f and \(w_{L,\Lambda }\) we also have
and this allows us to conclude, in combination with (4.2.59) and (4.2.60), that
Hence
By Remark 4.4, to compute the ground state energy of \({\mathbb {H}}_L\) it is clearly sufficient to restrict to the spectral subspace relative to \(|{\mathbb {H}}_L|\leqq C\) for some suitable C, which then yields (4.2.57). This concludes the proof and the section. \(\square \)
4.3 Final Lower Bound
In this section we show the validity of the lower bound in (2.3.2), thus completing the proof of Theorem 2.5. With Proposition 4.5 at hand, we have good estimates on the cost of substituting \({\mathbb {H}}_L\) with \({\mathbb {H}}_L^{\Lambda }\) and, in particular, we know that the difference between the ground state energies of the two is negligible for \(\Lambda \gg \alpha ^{4/5}\). We are thus left with the task of giving a lower bound on \({{\,\mathrm{inf\,spec}\,}}{\mathbb {H}}_L^{\Lambda }\).
While the previous steps in the lower bound follow closely the analogous strategy in [9], the translation invariance of our model leads to substantial complications in the subsequent steps, and the analysis given in this subsection is the main novel part of our proof. In contrast to the case considered in [9], the set of minimizers \({\mathcal {M}}^{{\mathcal {F}}}_L=\Omega _L(\varphi _L)\) is a three-dimensional manifold, and in order to decouple the resulting zero-modes of the Hessian of the Pekar functional we find it necessary introduce a suitable diffeomorphism that ’flattens’ the manifold of minimizers and the region close to it. Special attention also has to be paid on the metric in which this closeness is measured, necessitating the introduction of the family of norms in (3.3.47).
We emphasize that the non-uniformity in L also results from the subsequent analysis, where the compactness of resolvent of \(-\Delta _L\) enters in an essential way.
Let \(\Pi \) denote the projection
For later use we note that
The Fock space \({\mathcal {F}}(L^2({\mathbb {T}}^3_L))\) naturally factorizes into the tensor product \({\mathcal {F}}(\Pi L^2({\mathbb {T}}^3_L))\otimes {\mathcal {F}}((1\!\!1-\Pi ) L^2({\mathbb {T}}^3_L))\) and \({\mathbb {H}}_L^{\Lambda }\) is of the form \({\mathbb {A}}\otimes 1\!\!1+ 1\!\!1\otimes {\mathbb {N}}^>\), with \({\mathbb {A}}\) acting on \(L^2({\mathbb {T}}^3_L)\otimes {\mathcal {F}}(\Pi L^2({\mathbb {T}}^3_L))\) and \({\mathbb {N}}^>\) being the number operator on \({\mathcal {F}}((1\!\!1-\Pi ) L^2({\mathbb {T}}^3_L))\). In particular, \({{\,\mathrm{inf\,spec}\,}}{\mathbb {H}}_L^{\Lambda }={{\,\mathrm{inf\,spec}\,}}{\mathbb {A}}\).
As in Section 4.1, we can, for any \(L^2\)-orthonormal basis of real-valued functions \(\{f_n\}\) of \({{\,\mathrm{ran}\,}}\Pi \), identify \({\mathcal {F}}(\Pi L^2({\mathbb {T}}^3_L))\) with \(L^2({\mathbb {R}}^N)\) through the Q-space representation (see [25]). In particular, any real-valued \(\varphi \in {{\,\mathrm{ran}\,}}\Pi \) corresponds to a point \(\lambda \in {\mathbb {R}}^N\) via
Note that, compared to Section 4.1, we are using a different choice of \(\Pi \) here for the decomposition \(L^2({\mathbb {T}}^3_L)={{\,\mathrm{ran}\,}}\Pi \oplus ({{\,\mathrm{ran}\,}}\Pi )^{\perp }\).
In the representation given by (4.3.3), the operator \({\mathbb {A}}\) is given by
on \(L^2({\mathbb {T}}^3_L)\otimes L^2({\mathbb {R}}^N)\). For a lower bound, we can replace \(h_{\varphi }=-\Delta _L+V_{\varphi }\) with the infimum of its spectrum \(e(\varphi )\), obtaining
where \({\mathbb {K}}\) is the operator on \(L^2({\mathbb {R}}^N)\) defined as
where \({\mathcal {F}}_L\), which is understood as a multiplication operator in (4.3.6), can be seen as a function of \(\varphi \in {{\,\mathrm{span}\,}}_{{\mathbb {R}}}\{f_j\}_{j=1}^N\) or \(\lambda \in {\mathbb {R}}^N\) through the identification (4.3.3).
Using IMS localization we shall split \({\mathbb {R}}^N\) into two regions, one localized around the surface of minimizers of \({\mathcal {F}}_L\), i.e., \({\mathcal {M}}^{{\mathcal {F}}}_L=\Omega _L(\varphi _L)\), and the other localized away from it. On each of these regions we can bound \({\mathcal {F}}_L\) from below with the estimates contained in Proposition 3.15 and in Corollary 2.4, respectively. Because of the prefactor \(\alpha ^{-4}\) in front of \(-\Delta _{\lambda }\) the outer region turns out to be negligible compared to the inner one (at least if we define the inner and outer region with respect to an appropriate norm). At the same time, employing an appropriate diffeomorphism, the inner region can be treated as if \(\Omega _L(\varphi _L)\) was a a flat torus, leading to a system of harmonic oscillators whose ground state energy can be calculated explicitly.
We start by specifying the norm with respect to which we measure closeness to \(\Omega _L(\varphi _L)\). Recall the definition of the \(W_T\)-norms given in (3.3.47). Note that for \(T\geqq \Lambda \) the \(L^2\)-norm coincides with the \(W_T\)-norm on \({{\,\mathrm{ran}\,}}\Pi \), which makes \(0<T< \Lambda \) the relevant regime for our discussion. In fact, we shall pick
where \(T\gg 1\) is needed for the inner region to yield the right contribution, and \(T\ll \Lambda ^{2/3}\) ensures that the outer region contribution is negligible.
We proceed by introducing an IMS type localization with respect to \(\Vert \cdot \Vert _{W_T}\). Let \(\chi :{\mathbb {R}}_+\rightarrow [0,1]\) be a smooth function such that \(\chi (t)=1\) for \(t\leqq 1/2\) and \(\chi (t)=0\) for \(t\geqq 1\). Let \(\varepsilon >0\) and let \(j_1\) and \(j_2\) denote the multiplication operators on \(L^2({\mathbb {R}}^N)\)
Then
where \({\mathbb {E}}\) is the IMS localization error given by
which is estimated in the following lemma.
Lemma 4.6
Proof
To bound \({\mathbb {E}}\) we apply Lemma 3.17, which states that for \(\varepsilon \) sufficiently small, for any \(\varphi \in \text {supp} j_1\), there exists a unique \(y_{\varphi }\in {\mathbb {T}}^3_L\) such that
Likewise, for any \(n\in \{1,\dots ,N\}\) and any h sufficiently small there exists a unique \(y_{n,h}\in {\mathbb {T}}^3_L\) such that
It is easy to see, using again Lemma 3.17, that \(\lim _{h\rightarrow 0}y^{h,n}= y^{\varphi }\) for any n. Therefore, using that \({{\,\mathrm{dist}\,}}_{W_T}(\varphi +hf_n,\Omega _L(\varphi _L))\leqq \Vert \varphi -\varphi _L^{y_{\varphi }}\Vert _{W_T}\) and \({{\,\mathrm{dist}\,}}_{W_T}(\varphi ,\Omega _L(\varphi _L))\leqq \Vert \varphi -\varphi _L^{y_{h,n}}\Vert _{W_T}\), we arrive at
which shows that
Using that \(|\chi '|, \left| \left[ (1-\chi ^2)^{1/2}\right] '\right| \lesssim 1\!\!1_{[1/2,1]}\), for \(k=1,2\) we obtain
Summing over n, using that \(\Vert W_T\Vert \leqq 1\) and that \(\{f_n\}\) is an orthonormal system, we arrive at (4.3.11). \(\square \)
Thus, the localization error is negligible as long as \(\varepsilon \gg \alpha ^{-1}\). Hence, we are left with the task of providing lower bounds for \(j_1 {\mathbb {K}} j_1\) and \(j_2 {\mathbb {K}} j_2\) under the constraint \(\varepsilon \gg \alpha ^{-1}\). We carry out these estimates in the next two Sections 4.3.1 and 4.3.2. Finally, in Section 4.3.3, we combine these bounds to prove the lower bound in (2.3.2).
4.3.1 Bounds on \(j_1 {\mathbb {K}} j_1\)
Let us look closer at the intersection of the \(\varepsilon \)-neighborhood of \(\Omega _L(\varphi _L)\) with respect to the \(W_T\)-norm with \({{\,\mathrm{ran}\,}}\Pi \), i.e., the set
In the following we shall show that this set is, for \(\varepsilon \) small enough, a tubular neighborhood of \(\Pi \Omega _L(\varphi _L)\), which can be mapped via a suitable diffeomorphism (given in Definition 4.7) to a tubular neighborhood of a flat torus.
Since \(\varphi \in {{\,\mathrm{ran}\,}}\Pi \) and \(\Pi \) commutes both with \(W_T\) and with the transformation \(g\mapsto g^y\) for any \(y\in {\mathbb {T}}^3_L\), we have
This implies that \(\left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\) is non-empty if and only if
Since \(\varphi _L\in C^{\infty }({\mathbb {T}}^3_L)\), \(r_{T,\varepsilon }>0\) as long as
for some \(h>0\) and \(\Lambda \) sufficiently large. In particular, (4.3.20) is satisfied with \(h=5/4\) for \(\alpha \) large enough since, as discussed above, we need to pick \(\varepsilon \gg \alpha ^{-1}\) and \(\Lambda \gg \alpha ^{4/5}\) for the IMS and the cutoff errors to be negligible.
Lemma 3.17 implies that any \(\varphi \in \left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\), for \(\varepsilon \leqq \varepsilon '_L\) (independently of T and N), admits a unique \(W_T\)-projection \(\varphi _L^{y_{\varphi }}\) onto \(\Omega _L(\varphi _L)\) and
Since \(W_T\) and \(\Pi \) commute, \(\Omega _L(\varphi _L)\) is ‘parallel’ to \({{\,\mathrm{ran}\,}}\Pi \) with respect to \(\Vert \cdot \Vert _{W_T}\), i.e., \({{\,\mathrm{dist}\,}}_{W_T}({{\,\mathrm{ran}\,}}\Pi , \varphi _L^y)\) is independent of y and the \(W_T\)-projection of \(\varphi _L^y\) onto \(\Pi \) is simply \(\Pi (\varphi _L^y)=(\Pi \varphi _L)^y\). Therefore, for \(\varepsilon \leqq \varepsilon '_L\), any \(\varphi \in \left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\) admits a unique \(W_T\)-projection \((\Pi \varphi _L)^{y_\varphi }\) onto \(\Omega _L(\Pi \varphi _L)\) and (4.3.21) induces a unique decomposition of the form
where \(\eta _{\varphi }=\Pi v_{\varphi }\) (note that \((1\!\!1-\Pi )v_{\varphi }=-(1\!\!1-\Pi )\varphi _L\)). This allows to introduce the following diffeomorphism, which is a central object in our discussion. It maps \(\left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\) onto a tubular neighborhood of a flat torus. We shall call this diffeomorphism Gross coordinates, as it is inspired by an approach introduced in [14].
Definition 4.7
(Gross coordinates) For
we define the Gross coordinates map u as
where \(y_{\varphi }\) and \(\eta _{\varphi }\) are defined through the decomposition (4.3.22).
By the discussion above it is clear that u is well-defined and invertible, for \(\varepsilon \leqq \varepsilon '_L\) (defined in Lemma 3.17), with inverse \(u^{-1}\) given by
We emphasize that the whole aim of the discussion above is to show that u is well-defined, since once that has been shown the invertibility of u and the form of \(u^{-1}\) are obvious. In other words, the map \(u^{-1}\) as defined in (4.3.25) is trivially-well defined, but it is injective and surjective with inverse u only thanks to the existence and uniqueness of the decomposition (4.3.22).
To show that u is a smooth diffeomorphism, we prefer to work with its inverse \(u^{-1}\), which we proceed to write down more explicitly. For this purpose, we pick a real \(L^2\)-orthonormal basis \(\{f_k\}_{k=1}^N\) of \({{\,\mathrm{ran}\,}}\Pi \), such that \(f_1\), \(f_2\) and \(f_3\) are an orthonormal basis of \({{\,\mathrm{span}\,}}\{\Pi W_T \partial _j \varphi _L\}_{j=1}^3\) and \(f_4=\frac{\Pi \varphi _L}{\Vert \Pi \varphi _L\Vert _2}\). Note that \({{\,\mathrm{span}\,}}\{\Pi W_T \partial _j \varphi _L\}_{j=1}^3\) is three dimensional, as remarked after (3.3.45), at least for N and T large enough, and that \(f_4\) is indeed orthogonal to \(f_1\), \(f_2\) and \(f_3\) since in k-space \(W_T\) and \(\Pi \) are even multiplication operators while the partial derivatives are odd multiplication operators. We denote the projection onto \({{\,\mathrm{span}\,}}\{\Pi W_T \partial _j \varphi _L\}_{j=1}^3\) by
Having fixed a real orthonormal \(L^2\)-basis, we can identify any real-valued function in \({{\,\mathrm{ran}\,}}\Pi \) (and hence also any function in \(\left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\)) with a point \((\lambda _1,\dots ,\lambda _N)\) via (4.3.3). In these coordinates, the orthogonal transformation that acts on functions in \({{\,\mathrm{ran}\,}}\Pi \) as the translation by y, i.e., \(\varphi \mapsto \varphi ^y\), reads
and we can write \(B^{T,\Lambda }_{\varepsilon }\) in (4.3.23) as
In this basis, we can write \(u^{-1}\) explicitly as
The following Lemma uses this explicit expression for \(u^{-1}\) and shows that it is a smooth diffeomorphism (therefore showing that the Gross coordinates map u is as well).
Lemma 4.8
Let \(u^{-1}\) be the map defined in (4.3.29). There exists \(\varepsilon ^1_L\leqq \varepsilon '_L\) (independent of T and N) and \(N_L>0\) such that for any \(\varepsilon \leqq \varepsilon _L^1\), any \(T>0\) and any \(N>N_L\) the map \(u^{-1}\) is a \(C^{\infty }\)-diffeomorphism from \({\mathbb {T}}^3_L\times B_{\varepsilon }^{T,\Lambda }\) onto \(\left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\). Moreover, for \(\varepsilon \leqq \varepsilon ^1_L\), \(|\det Du^{-1}|\) and all its derivatives are uniformly bounded independently of T and N.
Proof
We introduce the notation \(J(y,\eta )=D u^{-1}(y,\eta )\) and \(d(y,\eta ):=| \det J(y,\eta )|\). Note that R(y) in (4.3.27) satisfies \(R(-y)=R(y)^{-1}=R(y)^t\) since \(\{f_j^y\}_{j=1}^N\) is an orthonormal basis of \({{\,\mathrm{ran}\,}}\Pi \) for any y. Hence, for \(j=1,\dots ,N\) we have
This yields the smoothness of \(u^{-1}\) in \(\eta \) and in y (noting that \(\{f_j\}_{j=1}^N\subset {{\,\mathrm{ran}\,}}\Pi \) is a set of smooth functions for any N). We proceed to compute J. We have, for \(4\leqq k \leqq N\),
and
for \(1\leqq k \leqq 3\). Therefore
where \(v_k(\eta ):= -\partial _k u^{-1}(0,\eta )=-\partial _k \left( \Pi \varphi _L+\sum _{l=4}^N\eta _l f_l\right) \). Since R(y) is orthogonal, we see that \(d=|\det J_0|\) (implying, in particular, that d is independent of y).
Observe that
where \(A_0\) is the \(3\times 3\) matrix given by
and \(A_1\) is the \((N-3)\times 3\) matrix defined by
Since \(J_0\) is the identity in the bottom-right \((N-3)\times (N-3)\) corner and 0 in the top-right \(3\times (N-3)\) corner, \(d=|\det A_0|\). On \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla ,T}\) the operators \(\partial _k\) with \(k=1,2,3\) and \(W_T^{-1}\) are uniformly bounded in N and T. Recall also that \(\Vert \eta \Vert _{W_T}\leqq \varepsilon ^1_L\). Hence, for some constant \(C_L\) independent of N and T, and for any \(j,k\in \{1,2,3\}\), we have
Moreover, for any \(j, k \in \{1,2,3\}\) and any \(l, l_1, l_2\in \{4,\dots ,N\}\), we also have
Clearly, (4.3.37) and (4.3.38) together with the fact that \(d=|\det A_0|\) show that d and all its derivatives are uniformly bounded in N and T. To show that there exists \(\varepsilon _L^1\) and \(N_L\) such that \(d\geqq C_L>0\) for all \(\varepsilon \leqq \varepsilon _L^1\), \(T>0\) and \(N>N_L\), we show that the image of the 3-dimensional unit sphere under \(A_0\) is uniformly bounded away from 0, which clearly yields our claim. For this purpose, we observe that the k-th column of \(A_0\) is given by \(\Pi ^L_{\nabla ,T}\left[ -\partial _k \left( \Pi \varphi _L+\sum _{l=4}^N\eta _l f_l\right) \right] \) and therefore, for any unit vector \(a=(a_1,a_2,a_3)\in {\mathbb {R}}^3\),
where we denote \(\sum _{k=1}^3 a_k \partial _k = \partial _{a}\). To bound the norm of \(A_0 a\) from below, it is then sufficient to test \(\partial _a u^{-1}(0,\eta )\) against one normalized element of \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla ,T}\), say \(\frac{\Pi W_T \partial _a \varphi _L}{\Vert \Pi W_T \partial _a \varphi _L\Vert _2}\). We obtain
where we used that \(\Vert \eta \Vert _{W_T}\leqq \varepsilon \), \(0\leqq W_T\leqq 1\!\!1\) and \(\Pi \leqq 1\!\!1\), and \((\, \cdot \,)_+\) denotes the positive part. As remarked after (3.3.45), \(\partial _a \varphi _L =(-\Delta _L)^{-1/2} \partial _a |\psi _L|^2\ne 0\) and since \(\varphi _L\in C^{\infty }\), \(\partial _a \varphi _L\) and \(\partial _a^2 \varphi _L\) are uniformly bounded in a. We can thus find \(N_L>0\) and \(\varepsilon _L^1\) such that the r.h.s. of (4.3.40) is bounded from below by some constant \(C_L>0\) uniformly for \(T>0\), \(N>N_L\) and \(\varepsilon \leqq \varepsilon _L^1\). This shows that \(A_0\) (and hence J) is invertible at every point and that \(d\geqq C_L>0\) uniformly in \(T>0\), \(N>N_L\) and \(\varepsilon \leqq \varepsilon ^1_L\), as claimed. This concludes the proof. \(\square \)
Since u is a diffeomorphism, we can introduce a unitary operator that lifts \(u^{-1}\) to \(L^2\), defined by
Recall that \(j_1\) is supported in \(\left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\), hence we can apply U to \(j_1 {\mathbb {K}} j_1\), obtaining an operator that acts on functions on \({\mathbb {T}}^3_L\times {\mathbb {R}}^{N-3}\) that are supported in \({\mathbb {T}}^3_L\times B^{T,\Lambda }_{\varepsilon }\). In particular,
where the subscript indicates that the operator has to be understood as the corresponding quadratic form with form domain \(H^1_0({\mathbb {T}}^3_L\times B^{T,\Lambda }_{\varepsilon })\) (i.e., with Dirichlet boundary conditions on the boundary of \(B^{T,\Lambda }_{\varepsilon }\)). We are hence left with the task of giving a lower bound on \({{\,\mathrm{inf\,spec}\,}}_{H^1_0\left( {\mathbb {T}}^3_L\times B^{T,\Lambda }_{\varepsilon }\right) }[U^*{\mathbb {K}}U]\), which will be done in the remainder of this subsection.
Recalling the definition of \({\mathbb {K}}\) given in (4.3.6), we proceed to find a convenient lower bound for \(U^* {\mathcal {F}}_LU\). Any \((\Pi \varphi _L)^{y_{\varphi }}+(w_{\varphi })^{y_{\varphi }}=\varphi \in \left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\) satisfies (3.3.22) with \(\varphi _L^{y_{\varphi }}\) in place of \(\varphi _L\), and we can therefore expand \({\mathcal {F}}_L(\varphi )\) using Proposition 3.15, obtaining
Since \(K_L\) and \(J_L\) are trace class operators,
holds for \(\Lambda \) sufficiently large and \(\varepsilon \) sufficiently small. Moreover, since \(\varphi _L\in C^{\infty }({\mathbb {T}}^3_L)\),
for arbitrary \(h>0\) and uniformly in T. This implies that, for any \(\varphi =(\Pi \varphi _L)^{y_{\varphi }}+(w_{\varphi })^{y_{\varphi }} \in \left[ \Pi \Omega _L(\varphi _L)\right] _{\varepsilon ,T}\), any \(\Lambda \) sufficiently large, any \(\varepsilon \) sufficiently small and an arbitrary h
Therefore, if we define the \([(N-3)\times (N-3)]\)-matrix M with coefficients
then, by (4.3.46), the multiplication operator \(U^* {\mathcal {F}}_LU\) satisfies
It is easy to see that M is a positive matrix, at least for \(\varepsilon \) sufficiently small and T and \(\Lambda \) sufficiently large. Indeed, the positivity of M is equivalent to the positivity of \((1\!\!1-K_L-\varepsilon C_LJ_L)\) on \({{\,\mathrm{ran}\,}}(\Pi -\Pi ^L_{\nabla ,T})\) and, by Proposition 3.18, \((1\!\!1-K_L-\varepsilon C_LJ_L)\) is positive on any vector space with trivial intersection with \({{\,\mathrm{ran}\,}}\Pi ^L_{\nabla }\). Clearly, since \(\Pi ^L_{\nabla ,T}\rightarrow \Pi ^L_{\nabla }\) as \(T\rightarrow \infty \), the bound
holds, uniformly in T, \(\Lambda \) and for \(\varepsilon \) sufficiently small.
We now proceed to bound \(- U^* \Delta _{\lambda } U\) from below.
Lemma 4.9
Let U be the unitary transformation defined in (4.3.41). There exists \(C_L>0\), independent of N, T and \(\varepsilon \), such that, for \(\varepsilon \leqq \varepsilon ^1_L\), \(T>0\) and \(N>N_L\)
Proof
Since (4.3.33) shows that \(J(y,\eta )=R(y)J_0(\eta )\) with R(y) orthogonal, we have
with \(d(y,\eta )=| \det J(y,\eta )|\) and \(\nabla \) denoting the gradient with respect to \((y,\eta )\in {\mathbb {R}}^N\). Recalling the expression (4.3.34) for \(J_0\), we find
Since \(D(1\!\!1-\Pi ^L_{\nabla ,T})=(1\!\!1-\Pi ^L_{\nabla ,T})D^t=0\), we have
With (4.3.51) and (4.3.53), we thus obtain
Lemma 4.8 guarantees that d and all its derivatives are bounded, and d is bounded away from 0 uniformly in \(N>N_L\), \(T>0\) and \(\varepsilon \leqq \varepsilon _L^1\), leading to (4.3.50). \(\square \)
In combination, (4.3.48), (4.3.50) and the positivity of M imply that
Note that since we are taking \(\Lambda \gg \alpha ^{4/5}\), \(\varepsilon \ll 1\) and \(h>0\) was arbitrary, picking \(h=5\) allows to absorb the error term \(O(\varepsilon \Lambda ^{-h})\) in the error term \(O(\alpha ^{-4})\). Recalling the definition of M given in (4.3.47), we have
With \(\{t_j\}_{j=1}^{N-3}\) an orthonormal basis of \({{\,\mathrm{ran}\,}}(\Pi -\Pi ^L_{\nabla ,T})\) of eigenfunctions of \((\Pi -\Pi ^L_{\nabla ,T})(1\!\!1-K_L-\varepsilon C_LJ_L)(\Pi -\Pi ^L_{\nabla ,T})\), we can write
for some \(\{\xi _j\}_{j=1}^{N-3}\) satisfying
for T and \(\Lambda \) large enough and \(\varepsilon \) small enough, where we used (4.3.49) for the lower bound. Using the concavity of the square root and the trace class property of \(J_L\), we conclude that
Since \(\varphi _L\in C^{\infty }\) and recalling (4.3.7), for an arbitrary \(h>0\) we can bound
which also implies the same estimate for the trace-norm of the difference of \(\Pi ^L_{\nabla }\) and \(\Pi ^L_{\nabla ,T}\), both operators being of rank 3. Recalling that \(\Pi ^L_{\nabla }\) projects onto \(\ker (1\!\!1-K_L)\), we finally obtain
The error term \(O(T^{-h})\) forces \(T\rightarrow \infty \) as \(\alpha \rightarrow \infty \), but allows T to grow with an arbitrarily small power of \(\alpha \). By picking h to be sufficiently large we can absorb it in the error term \(O(\varepsilon )\).
We obtain the final lower bound
4.3.2 Bounds on \(j_2 {\mathbb {K}} j_2\)
We recall Corollary 2.4, which implies that, for any \(\varphi \in L^2_{{\mathbb {R}}}({\mathbb {T}}^3_L)\),
where B acts in k-space as the multiplication by
Note that \(B-\eta W_T>0\) for \(\eta >0\) small enough (independently of T). Moreover, for any \(\varphi \) in the support of \(j_2\) and any \(y\in {\mathbb {T}}^3_L\),
Therefore, on the support of \(j_2\), we have
By the Cauchy–Schwarz inequality, using that all the operators involved commute, we have
for any \(\gamma >0\). Note that the right hand side is independent of y. Since \(\varphi _L\in C^{\infty }({\mathbb {T}}^3_L)\), the Fourier coefficients of \(\varphi _L\) satisfy
for any \(t>0\). Using the positivity of \(B-\eta W_T\) we can bound
Therefore we conclude, using the positivity of \(1\!\!1-W_{\gamma _{\beta }}^{1/2}\) and of \(B-\eta W_T\), that
We need to estimate the behavior in \(N=\mathrm{rank}\Pi \), T and \(\gamma \) of the trace appearing in the last equation, which equals
The contribution to the sum from \(|k|\leqq \max \{\gamma ,T\}\) can be bounded by \(C(L\max \{\gamma ,T\})^3\). For \(|k|> \max \{\gamma ,T\}\), \(W_\gamma (k) = W_T(k) = (1+|k|^2)^{-1}\), and the coefficient under the square root in the last line of (4.3.72) behaves asymptotically for large momenta as \(1 - |k|^{-1}\). Hence, recalling (4.3.2), we conclude that
Because of (4.3.7), the first term on the right hand side is negligible compared to the second if we choose \(\gamma \) to equal \(\alpha \) to some small enough power. Because t was arbitrary, we thus arrive at
Therefore, if
for a sufficiently large constant \(C_L\), we conclude that for sufficiently large \(\alpha \) and \(\Lambda \)
4.3.3 Proof of Theorem 2.5, lower bound
By combining the results (4.3.63) and (4.3.76) of the previous two subsections with (4.3.9) and (4.3.11), we obtain
under the constraint (4.3.75). With Proposition 4.5 we can thus conclude that
To minimize the error terms under the constraint (4.3.75), we pick \(\varepsilon \sim \alpha ^{-1/7}\) and \(\Lambda \sim \alpha ^{6/7}\), which yields the claimed estimate
This concludes the proof of the lower bound, and hence the proof of Theorem 2.5.
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Acknowledgements
Funding from the European Union’s Horizon 2020 research and innovation programme under the ERC grant agreement No 694227 is gratefully acknowledged. We would also like to thank Rupert Frank for many helpful discussions, especially related to the Gross coordinate transformation defined in Def. 4.7.
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Feliciangeli, D., Seiringer, R. The Strongly Coupled Polaron on the Torus: Quantum Corrections to the Pekar Asymptotics. Arch Rational Mech Anal 242, 1835–1906 (2021). https://doi.org/10.1007/s00205-021-01715-7
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DOI: https://doi.org/10.1007/s00205-021-01715-7