Abstract
We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.
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1 Introduction and main results
The polaron is quasi-particle that models an electron moving through an ionic crystal while interacting with its self-induced polarization field. The polarization field can be either described as a quantum field by the Fröhlich model [1] (called quantum polaron) or as a classical field by the Landau-Pekar equations [2,3,4] (called classical polaron). The Landau-Pekar equations describe the polaron as a pair \((\psi , \varphi ) \in H^1 (\mathbb {R}^3) \times L^2_{\sqrt{\epsilon }} ( \mathbb {R}^3)\) where \(\psi \) denotes the \(L^2\)-normalized wave function of the electron and \(\varphi \) the classical field which is for a positive function \(\varepsilon >0\) an element of
The Landau-Pekar equations are given by the coupled system of differential equations
where \(\alpha >0\) denotes the coupling constant, \(m>0\) the electron’s mass,
and where \(\varrho _\psi : =\vert \psi \vert ^2\),
The strong coupling limit is linked with a classical field approximation: For \(\alpha \rightarrow \infty \), the classical Landau-Pekar equations can be derived from the quantum dynamics generated by the Fröhlich model [5,6,7,8,9,10].
1.1 Effective mass problem for the Landau-Pekar equations
The dynamics of the polaron is closely related to the outstanding problem of its effective mass: Due to the interaction with the self-induced polarization field, the electron slows down. In physics this phenomenon is described by the emergence of a quasi-particle, the polaron, with an increased effective mass \(m_\textrm{eff}>0\).
Based on the classical polaron, Landau and Pekar [2,3,4] formulated a famous quantitative prediction for the effective mass in the strong coupling limit. Their heuristic ideas (described in more detail in [11]) rely on the existence of traveling waves of the Landau-Pekar equations, i.e. solutions of (1.2) with initial data \((\psi _\textrm{v}, \varphi _\textrm{v}) \in H^1( \mathbb {R}^3) \times L^2_{\sqrt{\varepsilon }} ( \mathbb {R}^3)\), \(\Vert \psi _\textrm{v}\Vert _2 =1\) satisfying
with phase \(e_\textrm{v}\in \mathbb {R}\) and velocity \(\textrm{v}\in \mathbb {R}^3\). Traveling waves were, however, conjectured to not exists for \(\textrm{v}\not = 0\) for the Landau-Pekar equations [11] due to a vanishing speed of sound.
Related to that, the corresponding energy functional to (1.2) does not dominate the total momentum. Thus a computation of the energy as function of conserved total momentum yields a constant function and therefore an infinite mass. Contrarily for the quantum Fröhlich model such an energy-momentum expansion allows to approach the quantum polaron’s effective mass (see [12,13,14] resp. [15,16,17] for recent progress based on different techniques).
However for the classical polaron, i.e. the Landau-Pekar equations, neither traveling wave solutions nor an energy-momentum expansion serve for a mathematical rigorous definition of the effective mass. To overcome these problems [11] provides a definition of the effective mass that is based on a novel energy-velocity expansion and verifies the quantitative prediction by Landau and Pekar for the classical polaron.
The goal of this paper is to verify Landau and Pekar’s heuristic approach for the effective mass, originally formulated for the non-regularized classical polaron, mathematical rigorously for a regularized classical polaron model, namely the regularized Landau-Pekar equations.
More precisely, we choose instead of (1.4) the functions \(\varepsilon ,v\) to be sufficiently regular (see Assumptions 1.1, 1.2 below) and show that subsonic traveling wave solutions with non-vanishing velocity \(\textrm{v}\not = 0\) do exist (Theorem 1) and serve for a definition of the effective mass (Theorem 2). Moreover, the resulting formula agrees with a definition of the effective mass through an energy-momentum expansion (Theorem 3) and, furthermore, with results obtained for the quantum regularized Fröhlich model [18].
1.2 Regularized Landau-Pekar equations
The regularized Landau-Pekar equations describe more generally a particle moving through an excitable medium. We impose the following assumptions on the functions \(\varepsilon ,v\) and
Assumption 1.1
(Regularity) Let \(\varepsilon , v\) be radial with \(\varepsilon >0\) and such that \(g\in H^2 (\mathbb {R}^3)\) and \(\vert \widehat{g} (k) \vert \ge (1+ \vert k \vert )^{-9/2}\) for all \(k \in \mathbb {R}^3\).
Furthermore we consider underlying media of positive critical velocity \(\textrm{v}_\textrm{crit}>0\) formulated in the assumption below. The critical velocity is often referred to as speed of sound of the medium.
Assumption 1.2
Let \(\varepsilon >0\) satisfy \(\inf _{k \in \mathbb {R}^3} \frac{\varepsilon (k)}{\vert k \vert }:= \textrm{v}_\textrm{crit}\) for a constant \(\textrm{v}_\textrm{crit}>0\).
We remark that Assumptions 1.1 and 1.2 exclude the (non-regularized) Landau-Pekar equations with \(\varepsilon ,v\) given by (1.4) that, in particular, have vanishing speed of sound with the above definition.
The dynamical equations (1.2) for \(\varepsilon ,v\) satisfying Assumption 1.1, 1.2 are well defined (see Lemma 2.1 below). Moreover the energy functional
where \(h_{\varphi }\) is given by (1.3) is preserved along the dynamics. For the regularized Landau-Pekar equations we show that there exists subsonic traveling wave solutions with \(0< \vert \textrm{v}\vert < \textrm{v}_\textrm{crit}\).
Theorem 1
Let \(\varepsilon , v\) satisfy Assumptions 1.1 and 1.2 and \(\vert \textrm{v}\vert < \textrm{v}_\textrm{crit}\). Then there exists a traveling wave solution of the form (1.5) with \(\textrm{v}\not = 0\).
Theorem 1 follows from Proposition 2.3 and is proven in Sect. 4.
We can not treat the case of supersonic traveling waves \(\vert \textrm{v}\vert > \textrm{v}_\textrm{crit}\). However we conjecture that supersonic traveling waves do not exist. This conjecture is based on the observation that for \(\vert \textrm{v}\vert > \textrm{v}_\textrm{crit}\), the energy functional does not dominate the total momentum, similarly as for the non-regularized model discussed before.
1.2.1 Effective mass problem for the regularized Landau-Pekar equations
We provide two definitions for the effective mass. The first definition (Theorem 2) is based on an energy-velocity expansion of traveling wave solutions and inspired by ideas of Landau and Pekar. The second (Theorem 3) is based on an energy-momentum expansion for low energy states. Both definitions lead to the same formula for the effective mass and, in particular, verify the physicists’ predictions.
1.2.2 Traveling waves approach
We derive an energy-velocity expansion of low-energy traveling wave solutions (1.5) with small velocities. To be more precise we consider states \(( \psi , \varphi ) \in H^1( \mathbb {R}^3) \times L_{\sqrt{\varepsilon }}^2 ( \mathbb {R}^3)\)
- (i):
-
with small energy, i.e. satisfying
$$\begin{aligned} \mathcal {G}_\alpha ( \psi , \varphi ) \le e_\alpha + \kappa \; \text {for sufficiently small} \quad \kappa >0 \quad \text {(independent of }\; \alpha ) \end{aligned}$$(1.8) - (ii)\(_\textrm{v}\):
-
and which are traveling wave solutions of velocity \(\textrm{v}\), i.e. let \(\textrm{v}< \textrm{v}_\textrm{crit}\) (uniformly in \(\alpha \)), then \(( \psi _\textrm{v}, \varphi _\textrm{v})\) solves (1.5) with velocity \(\textrm{v}\) and with phase \(e_\textrm{v}\ge - e_\alpha + \textrm{v}^2/4\).
The definition of the effective mass through traveling waves is based on their energy-velocity expansion, i.e. for states of the set
we study the energy expansion
around the ground state energy \(e_\alpha \).
Theorem 2
Let \(\varepsilon , v\) satisfy Assumptions 1.1 and 1.2. Assume that for the pair of ground states \((\psi _\alpha , \varphi _\alpha )\) of \(\mathcal {G}_\alpha \) given by (1.7) where \(\varphi _\alpha = - \sqrt{\alpha } \sigma _{\psi _\alpha }\), the minimizer \(\psi _\alpha \) is unique up to translations and changes of phase. There exists \(\alpha _0 >0\) such that for all \(\alpha \ge \alpha _0\) and \(\alpha \textrm{v}\le 1\), we have
The energy expansion of Theorem 2 (i.e. (1.11)) is proven in Sect. 5.
The coefficients of the energy expansion are well defined as
for any
We define the effective mass as the second order coefficient of the expansion of \(E_\textrm{v}^\textrm{TW}\) around the ground state energy. It follows from the ground state’s approximation (see Proposition 2.2 below) that \(\widehat{\varrho }_{\psi _\alpha } (k) \rightarrow 1\) point-wise in the limit \(\alpha \rightarrow \infty \). Therefore in the strong coupling limit the leading order in \(\alpha \) of the effective mass is given by
and agrees with the findings of for the quantum Fröhlich model [18].
1.2.3 Approach through energy-momentum-expansion
For the second approach we are interested in the infimum of \(\mathcal {G}_\alpha \) w.r.t. to the set of states \(( \psi , \varphi ) \in H^1 ( \mathbb {R}^3) \times L^2_{\sqrt{\varepsilon }} ( \mathbb {R}^3 )\) with small energy (i.e. satisfy (i)) and
- (ii)\(_\textrm{p}\):
-
with mean momentum \(\textrm{p}\in \mathbb {R}^3\), i.e.
$$\begin{aligned} \langle \widehat{\psi } \vert p \vert \widehat{\psi } \rangle + \langle \varphi |p|\varphi \rangle = \textrm{p}\; . \end{aligned}$$(1.15)
Thus we consider states of the set
The definition of the effective mass then relies on an expansion of
stated in the following theorem.
Theorem 3
Let \(\varepsilon , v\) satisfy Assumptions 1.1 and 1.2. Assume that for the pair of ground states \((\psi _\alpha , \varphi _\alpha )\) of \(\mathcal {G}_\alpha \) given by (1.7) where \(\varphi _\alpha = - \sqrt{\alpha } \sigma _{\psi _\alpha }\), the minimizer \(\psi _\alpha \) is unique up to translations and changes of phase.
-
(a)
There exists \(\alpha _0 >0 \) such that for all \(\alpha \ge \alpha _0\) and \( \alpha ^{-1/4} \textrm{p}\le 1 \)
$$\begin{aligned} E_\textrm{p}=&e_\alpha + \left( m +\frac{ 2 (2\pi )^3 \alpha }{3} \Vert k v \varepsilon ^{-3/2} \; \widehat{\varrho }_{\psi _\alpha } \Vert _2^2\right) ^{-1} \frac{\textrm{p}^2}{2} + O( \alpha ^{-5/4} \textrm{p}^3 ) \; . \end{aligned}$$(1.17) -
(b)
There exists \(\alpha _0 >0\) such that for all \(\alpha \ge \alpha _0\) and \(\alpha ^{-1/2}\textrm{p}\le 1\), a pair of minimizers \((\psi _\textrm{p},\varphi _\textrm{p}) \) of \(E_\textrm{p}\) is a traveling wave solution \(( \psi _{\textrm{v}'}, \varphi _{\textrm{v}'} )\) to (4.1) with velocity \(\textrm{v}' = m_\textrm{eff}^{-1} \textrm{p}+ O( \alpha ^{-3/2} \textrm{p}^2)\) and
$$\begin{aligned} E_\textrm{p}= \mathcal {G}_\alpha ( \psi _{\textrm{v}'}, \varphi _{\textrm{v}'} ) + O ( \alpha ^{-2} \textrm{p}^3 ) \; . \end{aligned}$$(1.18)
Theorem 3(a), (b) are proven in Sect. 5.
We define the effective mass as the coefficient of the second order contribution of the energy-momentum expansion and thus in leading order in \(\alpha \) given in the strong coupling limit by
which agrees with the effective mass \(m_\textrm{eff}^\textrm{TW}\) defined in (1.14) and findings from the quantum Fröhlich model [18].
In particular Theorem 3(b) shows that any minimizer of \(E_\textrm{p}\) is given by a traveling wave solution of velocity \(\textrm{v}' = m_\textrm{eff}^{-1} \textrm{p}\), thus, by approximate elements of the set \(\mathcal {I}_\textrm{v}\) considered in Theorem 2.
We remark that traveling wave solutions for non-linear Schrödinger equations with non-vanishing speed of sound were studied in various other settings (see for example [19] for the Gross-Pitaevksi and [20] for pseudo-relativistic Hartree equation). We note that [19] considers a variational approach to traveling waves in the spirit of Theorem 3(b).
1.3 Structure of the paper
In Sect. 2 we collect properties and approximations of the ground state, ground state energy (Sect. 2.1, Proposition 2.1 resp. Proposition 2.2) and traveling wave solutions (Sect. 2.2, Proposition 2.3) that will be important to prove our main theorems. In Sect. 3 we prove Propositions 2.1, 2.2 on the ground state’s properties. For this we first show the existence of ground states for all \(\alpha >0\) in Sect. 3.1, then the approximation of the ground (state) by the harmonic oscillator in Sect. 3.2 and finally the positivity of the Hessian for large \(\alpha > \alpha _0\) yielding coercivity estimates. We combine those results in Sect. 3.4 to finally prove Propositions 2.1, 2.2. In Sect. 4 we prove afterwards Proposition 2.3 (yielding in Theorem 1) on the properties of traveling waves. In Sect. 5 we finally prove Theorems 2 and 3 on the two definitions of the effective mass bases on the results before.
2 Properties of the ground state and traveling waves
2.1 Properties of the ground state
For the regularized polaron’s ground state
the infimum can be taken first w.r.t. to the phonon field yielding by a completion of the square to the choice
with \(\varrho _\psi = \vert \psi \vert ^2\). The resulting energy functional for \(\psi \in H^1 ( \mathbb {R}^3)\) is
Thus if \(\psi _\alpha \) is an element of the manifold of minimizers \(\mathcal {M}_{\mathcal {E}_\alpha }\) of the energy functional \(\mathcal {E}_\alpha \) defined by
then the pair \(( \psi _\alpha , \varphi _\alpha )\) with \(\varphi _\alpha \) given by (2.2) is an element of the manifold of minimizers \( \mathcal {M}_{\mathcal {G}_\alpha }\) of the energy functional \(\mathcal {G}_\alpha \)
The energy functional \(\mathcal {E}_\alpha \) is symmetric with respect to translations and changes of the phase of the wave function. Thus for any minimizer \(\psi _\alpha \) of \(\mathcal {E}_\alpha \) (i.e. \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\)) it follows \(\Theta ( \psi _\alpha ) \subseteq \mathcal {M}_{\mathcal {E}_\alpha }\) where
For the (non-regularized) Pekar functional corresponding to (1.4), the existence of a unique pair of ground states \(( \psi _\textrm{Pekar}, \varphi _\textrm{Pekar})\) up to phases and translations was proven [21] for all \(\alpha >0\). For the regularized model we prove the existence of a ground state for all \(\alpha >0\).
Proposition 2.1
(Existence) Let \(\varepsilon , v\) satisfy Assumption 1.1. For all \(\alpha >0\) there exists a pair of minimizers \( ( \psi _\alpha , \varphi _\alpha ) \in \mathcal {M}_{\mathcal {G}_\alpha }\) with \(\varphi _\alpha = -\sqrt{\alpha } \sigma _{\psi _\alpha }\) and \(0< \psi _\alpha \in C^\infty ( \mathbb {R}^3)\) satisfying the Euler-Lagrange equation
We remark that the ground state’s uniqueness for the regularized model is in general not known. For technical reasons, we can not prove uniqueness up to translations and phases for large \(\alpha > \alpha _0\). For that, a refined approximation of the ground state than the one in Proposition 2.2 is needed to conclude ground state’s uniqueness up to translations and phase by the local coercivity estimates in Corollary 3.1 for large \(\alpha > \alpha _0\).
Note that the first part of Theorems 2 and 3 immediately follow from Proposition 2.1 that is proven in Sect. 3.4.
The ground state \(\psi _\alpha \)’s properties for large coupling constants \(\alpha >\alpha _0\) results from the asymptotic behavior of the energy functional \(\mathcal {G}_\alpha \). In fact in the strong coupling limit \(\alpha \rightarrow \infty \) the ground state energy \(e_\alpha = \mathcal {G}_\alpha (\psi _\alpha )\) is well described through the harmonic oscillator
Furthermore its well known ground state
approximates the true ground state of \(\mathcal {G}_\alpha \) as the following Lemma shows.
Proposition 2.2
(Approximation of the ground state) Let \(\varepsilon ,v\) satisfy Assumption 1.1. There exists \(\alpha _0 >0 \) and constants \(C_1,C_2 >0\) (independent of \(\alpha \)) such that
Furthermore let \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\). There exists \(C_3>0\) (independent of \(\alpha \)) such that for all \(\alpha \ge \alpha _0\)
Proposition 2.2 is proven in Sect. 3.4.
Here we introduced the norm
(and similarly for the \(H^1\)-norm) quantifying the distance of an element \(\psi _\alpha \) of the manifold of minimizers to the harmonic oscillator’s ground state.
We remark that the rate of convergence of (2.11) depends for technical reasons on Assumption 1.1 namely the regularity of the function g.
2.2 Traveling waves
The dynamical equations corresponding to the energy functional \(\mathcal {G}_\alpha \) in (1.7) are given for \((\psi _t, \varphi _t ) \in H^1 (\mathbb {R}^3) \times L^2_{\sqrt{\varepsilon }} ( \mathbb {R}^3)\) by the system of coupled partial differential equations (1.2). The dynamical equations are well-posed as the following Lemma shows.
Lemma 2.1
Let \(\varepsilon , v\) satisfy Assumption 1.1. For any \(\left( \psi _0, \varphi _0 \right) \in H^1 ( \mathbb {R}^3) \times L_{\sqrt{\varepsilon }}^2 ( \mathbb {R}^3)\) there exists a unique global solution of (1.2). Furthermore,
and there exists \(C>0\) such that for \((\psi _0, \varphi _0)\) with \(\mathcal {G}( \psi _0, \varphi _0) \le C \alpha \) we have for all \(t \in \mathbb {R}\)
The proof of the Lemma follows similarly to [5, Lemma 2.1] considering the non-regularized Landau-Pekar equations. The arguments presented in [5] apply for the regularized case, too, so that we refer for the proof of Lemma 2.1 to [5, Lemma 2.1].
A traveling wave of velocity \(\textrm{v}\in \mathbb {R}^3\) is a solution of (1.2) with initial data \((\psi _\textrm{v}, \varphi _\textrm{v}) \in H^1( \mathbb {R}^3) \times L^2_{\sqrt{\varepsilon }} ( \mathbb {R}^3)\), \(\Vert \psi _\textrm{v}\Vert _2 =1\) satisfying (1.5). The existence of subsonic traveling waves is given by the following Proposition.
Proposition 2.3
Let \(\varepsilon , v\) satisfy Assumptions 1.1 and 1.2. Furthermore assume that \(\vert \textrm{v}\vert < \textrm{v}_\textrm{crit}\).
-
(a)
There exists a traveling wave solution of the form (1.5).
-
(b)
Furthermore assume that \(\mathcal {G}( \psi _\textrm{v}, \varphi _\textrm{v}) - e_\alpha \le \kappa \) for sufficiently small \(\kappa >0\) (independent of \(\alpha \)) and \(e_\textrm{v}\ge - e_\alpha + \textrm{v}^2/4\). Assume that the ground state \(\psi _\alpha \) of \(\mathcal {E}_\alpha \) is unique up to translations and rotations. Then there exists \(\alpha _0 >0\) and a constant \(C>0\) (independent of \(\alpha , \textrm{v}\)) such that
$$\begin{aligned} {{\,\textrm{dist}\,}}_{L^2} \left( \Theta ( \psi _\alpha ) , \; \psi _\textrm{v}\right) \le C \vert \textrm{v}\vert \; \end{aligned}$$(2.15)for all \(\alpha \ge \alpha _0\) and \(\vert \textrm{v}\vert \le 1\).
Note that Theorem 1 follows immediately from Proposition 2.3(a). The proof of Proposition 2.3 is given in Sect. 2.2.
Furthermore note that a similar approximation as in (2.15) holds for the field \(\varphi _\textrm{v}\), too. For this we remark that instead of minimizing w.r.t. to the field \(\varphi \) in (2.1) first (as explained in Sect. 2.1) we can take the infimum w.r.t. to the wave function \(\psi \) first, too yielding the functional
We remark that by the energy functional’s symmetries for any \(\varphi _\alpha \in \mathcal {M}_{\mathcal {F}_\alpha }\) and
it follows that \(\Omega ( \varphi _\alpha ) \subseteq \mathcal {M}_{\mathcal {F}_\alpha }\). Then under the same assumption as in Proposition 2.3 there exists \(C>0\) (independent of \(\alpha \)) such that
We remark that Proposition 2.3(a) shows that subsonic traveling waves exist for all \(\alpha >0\). However the approximations (2.15), (2.18) of the second part of the Theorem holds for sufficiently large \(\alpha \ge \alpha _0\) only. The restriction to sufficiently large \(\alpha >0\) in part (b) ensures the validity of the global coercivity estimates (see Corollary 3.2) that are proven for sufficiently large \(\alpha > \alpha _0\) only. Furthermore we notice that for \(\textrm{v}=0\) the pair of ground states \(( \psi _\alpha , \varphi _\alpha )\) provide a traveling wave solution with \(e_\textrm{v}= e_\alpha \). In particular the assumption on the phase from part (b), made for technical reasons only, is satisfied for \(\textrm{v}=0\).
3 Properties of the energy functional \(\mathcal {E}_\alpha \)
In this section, we prove Propositions 2.1 and 2.2 on the properties of the energy functional \(\mathcal {G}_\alpha \).
The proof of Proposition 2.1 relies on a comparison of properties of \( \mathcal {E}_\alpha \) with the properties of \(\mathrm{h_{osc}}\) in the limit \(\alpha \rightarrow \infty \). Then existence and uniqueness (up to translations and changes of the phase) for pairs of minimizers \(( \psi _\alpha , \varphi _\alpha )\) of \(\mathcal {G}_\alpha \) follow with the choice \(\varphi _\alpha = - \sqrt{\alpha } \sigma _{\psi _\alpha }\).
First, in Lemma 3.1, we prove the existence of a minimizer \(\psi _\alpha \) for all \(\alpha \). Next we show the ground state (energy) is well approximated through the harmonic oscillator (Lemma 3.2). This approximations allows to show that the Hessian modulo its zero modes of \(\mathcal {E}_\alpha \) is asymptotically for \(\alpha \rightarrow \infty \) characterized by the harmonic oscillator, and thus positive (Lemma 3.3). This fact has several consequences: We infer first local (Corollary 3.1) and later global coercivity estimates (Corollary 3.2) for sufficiently large \(\alpha \ge \alpha _0\). For the latter we assume that the ground state \(\psi _\alpha \) of \(\mathcal {M}_{\mathcal {E}_\alpha }\) is unique up to translations and phase. Furthermore we obtain that the ground state energy \(e_\alpha \) is separated from the first excited eigenvalue by a gap of order \(\sqrt{\alpha }\) (Corollary 3.3).
We remark that the strategy for the proofs in this section follow [6, Section 3] considering the non-regularized Pekar functional on a Torus of length L. For sufficiently large L the uniqueness of the ground state and coercivity estimates are proven based on a comparison with the non-regular Pekar functional defined on the full space for which these properties are well known.
3.1 Existence
First we show the existence of minimizers of \(\mathcal {E}_\alpha \) for all \(\alpha >0\) in the subsequent Lemma.
Lemma 3.1
Let \(\varepsilon , v\) satisfy Assumption 1.1. For all \(\alpha >0\), there exists a minimizer \(0< \psi _\alpha \in C^\infty ( \mathbb {R}^3)\) of the functional \(\mathcal {E}_\alpha \) satisfying the Euler-Lagrange equation
Proof
Since \(h = g* g\), we have \(\Vert h \Vert _{\infty } \le C \Vert g \Vert _2^2 \) and
so that by Assumption 1.1 there exists \(C>0\) such that
From (3.3) we infer on one hand that \(e_\alpha \ge -C \alpha \) for all \(\alpha \). On the other hand, in order to prove the existence of a minimizer, we remark that (3.3) shows that any minimizing sequence \(\left( \psi _n \right) _{n \in \mathbb {N}}\) is bounded in \(H^1\), uniformly in \(n \in \mathbb {N}\). For this reason the sequence by [22, Lemma 6] resp. [23, Theorem 8.10], there exists a sequence \(\left( y_n \right) _{n \in \mathbb {N}} \in \mathbb {R}^3\) such that the translated sequence \(\left( \psi _{n}^{y_n} \right) _{n \in \mathbb {N}}\) has a sub-sequence \(\left( \psi _{n_j} \right) _{n_j \in \mathbb {N}}:= \left( \psi _{n_j}^{y_{n_j}} \right) _{n_j \in \mathbb {N}}\) that converges weakly in \(H^1( \mathbb {R}^3)\) to a non-zero function. It follows from the Sobolev inequality that this sub-sequence converges strongly in \(L^p\) for \(2 \le p \le 6\) to a non-zero limit. The limiting function \(\psi _\alpha \in H^1\) is again \(L^2\)-normalized and, moreover, satisfies
by semi-lower continuity of the \(H^1\)-norm. Since
and \(\Vert \left( h* \vert \psi _1 \vert ^2 \right) \psi _2 \Vert _{2} \le C \Vert \psi _1 \Vert _{2}^2 \Vert \psi _2 \Vert _{2} \) for any \(\psi _1,\psi _2 \in L^2\), we find
as \(n \rightarrow \infty \). Therefore,
and with \(\mathcal {E}_ \alpha ( \psi _\alpha ) \ge \liminf _{{n_j} \rightarrow \infty } \mathcal {E}_\alpha ( \psi _{n_j}) = e_\alpha \) (as \(( \psi _{n_j} )_{{n_j} \in \mathbb {N}}\) is a minimizing sequence), we conclude that \(\mathcal {E}_{\alpha } ( \psi _\alpha ) = e_\alpha \) and, thus, \(\psi _\alpha \) is a minimizer. By invariance of \(\mathcal {E}_\alpha \) w.r.t. to translations and phase, any element of \(\Theta (\psi _\alpha )\) defined in (2.6) is a minimizer, too. The positivity and regularity properties of \(\psi _\alpha \) follows by standard bootstrap arguments (see for example [6, Lemma 3.3]). \(\square \)
3.2 Approximation
Next we prove that in the strong coupling limit the spectrum of \(\mathcal {E}_\alpha \) is well approximated by the harmonic oscillator \(\mathrm{h_{osc}}\)’s spectrum. The idea is to use the Taylor expansion of the potential given by Assumption 1.1 through
to show its asymptotic quadratic behavior. The ground state energy of \(\mathrm{h_{osc}}\) (as defined in (2.8)) is well known
and separated from the rest of the spectrum by a gap of order \(\sqrt{\alpha }\). For this we compare the Hessian of \(\mathcal {E}_\alpha \) with
that is known to be positive, and thus, yielding coercivity estimates of the form
Furthermore we compare the deviation of the ground state of \(\psi _\alpha \) with the one of the harmonic oscillator that is known to be \(\Vert x^2 \psi _\textrm{osc}\Vert _2 = C \alpha ^{-1/2}\) for some \(C>0\).
Lemma 3.2
Let \(\varepsilon , v\) satisfy Assumption 1.1.
-
(a)
Then there exists \(C_1,C_2,C_3>0\) (independent of \(\alpha \)) such that
$$\begin{aligned} C_1 \alpha ^{1/3} \le e_\alpha + \alpha \Vert g\Vert _2^2 - \frac{\sqrt{3 \alpha } \Vert \nabla g\Vert _2^2}{2 \sqrt{m}} \le C_2 \; \end{aligned}$$(3.12)and
$$\begin{aligned} \vert \mu _{\psi _\alpha } - \mu _{\psi _\textrm{osc}} \vert \le C_3 \alpha ^{5/12} \; \end{aligned}$$(3.13)where we introduced the notation \(\mu _{\psi _\textrm{osc}} = \langle \psi _\textrm{osc}|- \Delta + 2 \alpha (h * \vert \psi _\textrm{osc}\vert ^2)|\psi _\textrm{osc}\rangle \). Let \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\) such that
$$\begin{aligned} {{\,\textrm{dist}\,}}_{L^2} \left( \Theta ( \psi _\alpha ), \psi _\textrm{osc} \right) = \Vert \psi _\textrm{osc}- \psi _\alpha \Vert _2 \; . \end{aligned}$$(3.14)Then there exists \(\alpha _0 >0\) and \(C>0\) (independent of \(\alpha \)) such that for all \(\alpha \ge \alpha _0\) we have
$$\begin{aligned} \Vert x^2 \psi _\alpha \Vert _2 \le C \alpha ^{-1/2} \end{aligned}$$(3.15) -
(b)
Let \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\). Then there exists \(\alpha _0 >0\) and \(C>0\) (independent of \(\alpha \)) such that for all \(\alpha \ge \alpha _0\) we have
$$\begin{aligned} {{\,\textrm{dist}\,}}_{L^2} \left( \Theta ( \psi _\alpha ), \psi _\textrm{osc} \right) \le C \alpha ^{-1/20} \; . \end{aligned}$$(3.16) -
(c)
Let \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\) such that
$$\begin{aligned} {{\,\textrm{dist}\,}}_{L^2} ( \Theta ( \psi _\alpha ), \psi _\textrm{osc}) = \Vert \psi _\alpha - \psi _\textrm{osc}\Vert _2 \; . \end{aligned}$$(3.17)Then there exists \(\alpha _0 >0 \) and \(C>0\) (independent of \(\alpha \)) such that for sufficiently large \(\alpha \ge \alpha _0\) we have
$$\begin{aligned} \Vert x^2 \left( \psi _\alpha - \psi _\textrm{osc}\right) \Vert _2 \le C \alpha ^{-11/20} \; . \end{aligned}$$(3.18) -
(d)
Under the same assumptions as in part (c) there exists \(\alpha _0 >0 \) and \(C_1,C_2,C_3 >0\) (independent of \(\alpha \)) such that for sufficiently large \(\alpha \ge \alpha _0\) we have
$$\begin{aligned} C_1 \alpha ^{1/4} \le \Vert \nabla \psi _\alpha \Vert _2 \le C_2 \alpha ^{1/4} \; \end{aligned}$$(3.19)and furthermore
$$\begin{aligned} \Vert \nabla \left( \psi _\alpha - \psi _\textrm{osc}\right) \Vert _2 \le C_3 \alpha ^{9/40} \; . \end{aligned}$$(3.20)
Proof
First we remark that in the following proof we denote with \(C>0\) a constant independent of \(\alpha \).
Proof of (a): For the upper bound of the ground state \(e_\alpha \) we pick the harmonic oscillator’s ground state \(\psi _\textrm{osc}\) defined in (2.9) as trial state. Its energy serves as an upper bound for the ground state energy
and can be explicitly computed with by the potential’s (3.8) Taylor expansion, Assumption 1.1 and \(\cos ( x) \ge 1 - \frac{x^2}{2}\)
for a constant \(C >0\) (independent of \(\alpha \)).
For the lower bound we use the IMS localization technique to show that it suffices to consider the problem on a ball of radius R, where we can use the potential’s Taylor expansion. To this end, let \(\psi _\alpha \) denote a minimizer realizing
and \(\chi \in C^\infty ( \mathbb {R}^3)\) a function with support on the ball \(B_1\) with radius one such that \(\Vert \chi \Vert =1\) and \(\chi (0) =1\). We define the rescaled function \(\chi ^R\) with \(\Vert \chi ^R \Vert _2 =1 \) supported on \(B_R\) and denote with \(\chi ^{R,z} = \chi ^R ( \cdot - z)\) its shift. The idea is to choose R dependent on \(\alpha \). However for simplicity we neglect the dependence of R on \(\alpha \) in the notation. We observe that the \(L^2\)-normalized function
satisfies
and thus, by completing the square and standard techniques of IMS localization
Since \(\chi ^{R,z} \psi _\alpha \Vert _2^2 dz \) denotes a probability measure, there exists \(z \in B_R\) such that
By scaling, we furthermore find \(\Vert \nabla \chi ^{R} \Vert _2^2 \le C R^{-2}\) yielding
We use (3.28) to prove both, the energy’s lower bound and the approximation of the ground state. We start with the lower bound on the ground state energy first. For this, we observe that (3.28) implies
i.e. it suffices to compute the energy \(\mathcal {E}_\alpha \) for function \(\psi ^{R,z}_\alpha \) supported on \(B_R\), where we can use the Taylor expansion of \(h\)
We observe that (3.27) is invariant w.r.t. translations and changes of phase of \(\psi _\alpha \) and \(\psi ^{R,z}\) and thus, we can furthermore restrict to \(\psi _\alpha ^{R,z}\) such that
for which we find
where \(\mathrm{h_{osc}}\) denotes the harmonic oscillator \(\textrm{h}_\textrm{osc} = - \frac{\Delta }{2\,m} + \frac{m\omega ^2}{2} x^2\). By definition, \(\psi ^{R,z}_\alpha \) is \(L^2\)-normalized and thus a competitor for the ground state of \(\mathrm{h_{osc}}\), i.e.
Optimizing w.r.t. to the parameter R (yielding \(R= \alpha ^{-1/6}\)), we arrive at
proving part (a).
Properties of \(\psi _\alpha , \psi _\alpha ^{R,z}\): As a preliminary step to prove the remaining parts of this Proposition we prove useful properties of the ground state \(\psi _\alpha \) and \(\psi _\alpha ^{R,z}\) (constructed in (3.24), satisfying (3.27) and by translational invariance of the problem (3.31)). We observe that \(\cos ( x ) \le 1\) (and thus \(h \le \Vert g \Vert _2^2\)) implies
With(3.28) and the ground state energy’s approximation (part (a)) we obtain
and in the same way
With the upper bound (3.34) we furthermore deduce from (3.35) resp. the Euler-Lagrange equation
for all \(\alpha \ge \alpha _0\). However for \(\psi _{\alpha }^{R,z}\) we observe first that (3.28) resp. (3.32) together with the harmonic oscillator’s coercivity property (3.11) show for \(R= \alpha ^{-1/6}\)
With the upper bound on the energy (3.22) we find
In particular for sufficiently large \(\alpha \ge \alpha _0\) we have
Approximation of Lagrange multipliers: We prove the approximation of the Lagrange multiplier \(\mu _{\psi _\alpha }\) with \(\mu _{\psi _\textrm{osc}}\) using the previous results. In particular by translational invariance of the problem we choose \(\psi _\alpha \) such that \(\psi _\alpha ^{R,z}\) (as constructed in (3.24)) satisfies (3.27) and (3.31)). We write
yielding with (3.27) and (3.37) to
Since
we find with (3.27), \(\Vert v / \varepsilon ^{1/2} ( \varrho _{\psi _\alpha ^{R,z}} -\varrho _{\psi _\textrm{osc}}) \Vert _2 \le C \Vert \psi _\textrm{osc}- \psi _\alpha ^{R,z}\Vert _2\) and (3.40)
Thus we obtain from (3.43) for sufficiently large \(\alpha >\alpha _0\)
Scaling of the ground state: To show the ground state’s scaling (3.15) for \(\psi _\alpha \) satisfying (3.14) we observe that by the Euler-Lagrange equation we have
The idea is now to use the properties of the resolvent \(\left( - \Delta - \mu _{\psi _\alpha } \right) ^{-1}\) (that is well defined sind from (3.46) we have \(\mu _{\psi _\alpha } \ge - C \alpha \) for sufficiently large \(\alpha \ge \alpha _0\)) to prove the desired bound. The resolvent’s Green’s function is given in terms of the inverse Fourier transform \(\mathcal {F}^{-1}\) of
and can by functional calculus explicitly computed. In fact we have
which leads with (3.48) to
and we arrive with Fubini’s theorem at
for \(\vert z \vert \not =0\). We plug this identity into (3.47) and find using assumption (3.14), and the notation \(F: = 2 \alpha \left( h * \vert \psi _\alpha \vert ^2 \right) \psi _\alpha \) (i.e. F denotes the right hand side of (3.47))
Thus the weighted \(L^2\)-norm, we aim to find an upper bound for, becomes
that we can estimate by Cauchy Schwarz with
We split the integral into several regions to find the desired bound: First we consider the case \(\vert x \vert > 2 \vert y \vert \) for which we have \(\vert x - y\vert \ge \vert x \vert - \vert y \vert \ge \frac{\vert x \vert }{2}\). By substitution we can therefore estimate the integral in this region by
where we used \(- \mu _{\psi _\alpha } \ge C \alpha \) for sufficiently large \(\alpha \ge \alpha _0\). Now let \(\vert x \vert <2 \vert y \vert \) and \(\vert x-y\vert > \frac{\vert y \vert }{2}\), then we have \(\frac{\vert x \vert }{\vert x-y\vert } \le 4\) and it follows
with similar arguments as before. Finally for \(\vert x \vert < 2 \vert y \vert \) and \(\vert x-y\vert \le \frac{\vert y \vert }{2}\) it follows \(\vert x \vert \ge \vert y \vert - \vert x - y \vert \ge \vert y \vert /2\). Hence
where we used that \( \vert y \vert ^{6} \; e^{- C \sqrt{-4\,m \mu _{\psi _\alpha }} \vert y\vert } \le \frac{\widetilde{C}}{ \alpha ^{3}}\). Summarizing the estimates we conclude by
where F denotes the r.h.s. of (3.47). Since \(\Vert h \Vert _\infty \le C\),\(\psi _\alpha \) is a \(L^2\)-normalized function we find that
for sufficiently large \(\alpha \ge \alpha _0\). In particular for \(n=3\) we find \(\Vert x^3 \psi _\alpha \Vert _2 \le C \alpha ^{-3/4}\) and thus, in particular,
Proof of (b): In order to prove the ground state \(\psi _\alpha \)’s approximation we observe that by the previous discussion it is enough to consider the problem on the ball \(B_{R_1} (0) \) with \(R_1 = \alpha ^{-1/5}\). In fact (3.15) shows
We consider \(\psi _\alpha \) and \(\psi _\alpha ^{R,z}\) (constructed in (3.24)), satisfying (3.27) and by translational invariance of the problem (3.31)). With these notations we in particular have from (3.40)
and thus it remains to show \(\psi _\alpha ^{R,z}\) is close to \(\psi _\alpha \). To this end we control the \(L^2\)-norm of the difference \(\varrho _{\psi _\alpha } - \varrho _{\psi ^{R,z}_\alpha }\) first and deduce as a second step from this estimates for \(L^2\)-norm of \(\psi _\alpha ^{R,z} -\psi _\alpha \). For this we write the \(L^2\)-norm of \(\varrho _{\psi _\alpha } - \varrho _{\psi ^{R,z}_\alpha }\) in momentum space. We shall show that for low momenta, we control the norm with Assumption 1.1 and (3.27) while for high momenta the \(L^2\)-norm is small by regularity properties of \(\varrho _{\psi _\alpha },\varrho _{\psi ^{R,z}_\alpha }\). For the latter one we observe that from (3.38) we have
and thus there exists \(C>0\) such that
To obtain a similar bound for \(\widehat{\varrho }_{\psi _\alpha ^{R,z}}\) we first have to derive a bound for the \(H^2\)-norm of \(\psi _\alpha ^{R,z}\). For this we remark that by definition (3.24) we have
With the Taylor expansion of \(\chi ^{R,z}\) around z we find that \(\Vert \chi ^{R,z} \psi _\alpha \Vert _2 \ge \Vert \psi _\alpha \Vert _2 + R^{-1} \Vert x ( \nabla \chi ^{R,z} ( w) ) \psi _\alpha \Vert _2 \) for \(w \in B_R (z)\) and thus with the ground state’s scaling properties (3.15) we have \(\Vert \chi ^{R,z} \psi _\alpha \Vert _2 \ge C \) for sufficiently large \(\alpha \ge \alpha _0\). Hence we find with (3.19), (3.38) and the scaling of \(\chi ^{R,z}\) at \(\Vert \Delta \psi ^{R,z}_\alpha \Vert _2 \le C \sqrt{\alpha }\) and thus
From (3.64) and (3.66) we find that for high momenta, i.e. all \(k \in B_{R_2}^c = \lbrace k \in \mathbb {R}^3: \vert k \vert > R_2 \rbrace \) we have
We remark that we choose \(R_2:= R_2 ( \alpha ) = \alpha ^{1/5}\), however, for simplicity (as for R), we neglect the dependence of \(\alpha \) in its notation. For low momenta, i.e. \(k \in B_{R_2}\) we use that by Assumption (1.1) we have \(\vert \widehat{g} (k) \vert \ge ( 1 + \vert k \vert )^{-9/2}\) and thus
and we find with (3.27) (assuming \(R_2 \le \alpha ^{1/5}\))
Optimizing with respect to \(R_2\) leads to \(R_2 = \alpha ^{1/5}\) and thus for sufficiently large \(\alpha \ge \alpha _0\) to
In particular it follows as \(\psi _\alpha ^{R,z}\) and \(\psi _\alpha \) have the same phase
We recall that we need to estimate the \(L^2\)-norm difference of \(\psi _\alpha , \psi _\alpha ^{R,z}\) on the ball \(B_{R_1}\), i.e. we have by Cauchy Schwarz’s inequality with \(R_1= \alpha ^{-1/5}\)
and we arrive with (3.61) at
Proof of part (c): In order to prove \(H^1\)-norm convergence of \(\psi _\alpha \) to \(\psi _\textrm{osc}\) we observe that from the Euler-Lagrange equations of \(\psi _\alpha \) resp. \(\psi _\textrm{osc}\), Assumption 1.1 and the scaling properties of \(\psi _\textrm{osc}\)
for a constant \(C>0\) independent of \(\alpha \). We recall that both \(\psi _\alpha \) and \(\psi _\textrm{osc}\) are \(L^2\)-normalized functions. Hence introducing the notation \(\widetilde{h} = h - \Vert g \Vert _2^2\) and
for any \(\psi \in H^1 ( \mathbb {R}^3)\) we arrive at
From Assumption 1.1 we have \(\vert \widetilde{h} (x) \vert \le C x^2\) and thus together with the ground state’s scaling properties (part (c)) we find \(\vert \widetilde{\mu }_{\psi _\alpha } \vert \le C \alpha ^{1/2}\), \(\vert \widetilde{\mu }_{\psi _\textrm{osc}} \vert \le C \alpha ^{1/2}\) and
from part (part (a)). Therefore we find with the approximation of the ground state (part (b)) and the ground state’s scaling properties (part (c)) that
finally yielding
and the desired lower and upper bound in (3.19).
Proof of (d): To prove convergence of \(\psi _\alpha \) to \(\psi _\textrm{osc}\) for \(\psi _\alpha \) satisfying (3.14) in the weighted \(L^2\)-norm, too, we proceed similarly as in part (d). From the Euler-Lagrange equation of \(\psi _\alpha \) we get
By assumption resp. part (a) we have
and furthermore we have from (3.58) (using assumption (3.14)) the estimate
where F denotes the r.h.s. of (3.80). It follows from the approximation of the Lagrange multiplier (part (a)) and the ground state (part (b)) that \(\Vert F \Vert _2 \le C \alpha ^{-1/20}\) and we finally arrive at the desired bound of part (c). \(\square \)
3.3 Properties of the Hessian
The Hessian \(\mathcal {H}_{\alpha }\) of \(\mathcal {E}_\alpha \) is defined for any \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\) by
We can explicitly compute the Hessian and find
where \(Q_\alpha = 1 - P_\alpha = 1 - \left| \psi _\alpha \right\rangle \left\langle \psi _\alpha \right| \) and
3.3.1 Positivity of the Hessian
We compare the Hessian’s components
with \(\mathcal {H}_\textrm{osc}\) (defined in (3.10)) known to satisfy \(\mathcal {H}_\textrm{osc} \ge C \sqrt{\alpha }\) for a constant \(C>0\) independent of \(\alpha \). We remark that by definition the Hessian \(\mathcal {H}_\alpha \) is defined modulo its zero modes namely the ground state \(\psi _\alpha \) for the first component resp. \(\psi _\alpha \) and its partial derivatives \(\partial _j \psi _\alpha \) for \(i=1,2,3\) for the second component.
Lemma 3.3
Let \(\varepsilon , v\) satisfy Assumption 1.1. Then, there exists \(\alpha _0>0\) and \(C>0\) (independent of \(\alpha \)) such that
Proof
We present the proof of the Hessian’s component \(\mathcal {H}_\alpha ^{(2)}\). The statement for \(\mathcal {H}^{(1)}\) then follows with similar arguments.
For any \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\) and \(f \in H^1 ( \mathbb {R}^3)\) we define the projection \(Q_\alpha ' = 1- P_\alpha ', P_\alpha ' = \left| \psi _\alpha \right\rangle \left\langle \psi _\alpha \right| + \sum _{j=1}^3 \left| \partial _j \psi _\alpha \right\rangle \left\langle \partial _j \psi _\alpha \right| / \Vert \partial _j \psi _\alpha \Vert _2^2 \) and furthermore the \(L^2\)-normalized function
Thus in the following we consider the expectation value
where we introduced the notation
with \(\widetilde{h} = h - \Vert g \Vert _2^2\) and used that the zero-th order term of the expansion of \(h\) in the above expectation value vanishes as \(Q'_\alpha f\) is orthogonal to \(\psi _\alpha \). By translational invariance of the problem we restrict to \(\psi _\alpha \) such that \({{\,\textrm{dist}\,}}_{L^2} ( \Theta (\psi _\alpha ), \psi _\textrm{osc}) = \Vert \psi _\textrm{osc}- \psi _\alpha \Vert _2\) (so that Lemma 3.2(c), (d) apply). We recall that we want to compare \(\mathcal {H}_\alpha ^{(i)}\) with \(\mathcal {H}_\textrm{osc}\) that is of order \(\sqrt{\alpha }\). Thus any term we can show to be \(o( \sqrt{\alpha })\) will be dominated in the end by \(\mathcal {H}_\textrm{osc}\) and thus will be considered to be sub-leading for sufficiently large \(\alpha \ge \alpha _0\).
We shall first show that \(g_\alpha \) is approximately orthogonal to the harmonic oscillator’s ground and first excited state \(P_\textrm{osc}' = \left| \psi _\textrm{osc}\right\rangle \left\langle \psi _\textrm{osc}\right| + \sum _{j=1}^3 \left| \partial _j \psi _\textrm{osc}\right\rangle \left\langle \partial _j \psi _\textrm{osc}\right| / \Vert \partial _j \psi _\textrm{osc} \Vert _2^2 \), i.e. that it is enough to consider
This follows from the observation that the difference is given by
On the one hand, since
and similarly denoting \(P_\textrm{osc}^{(j)} =: \left| \partial _j \psi _\textrm{osc}\right\rangle \left\langle \partial _j \psi _\textrm{osc}\right| / \Vert \partial _j \psi _\textrm{osc} \Vert _2^2 \) and \(P_{\alpha }^{(j)} =: \left| \partial _j \psi _\alpha \right\rangle \left\langle \partial _j \psi _\alpha \right| / \Vert \partial _j \psi _\alpha \Vert _2^2\),
from Lemma 3.2(d) for sufficiently large \(\alpha \ge \alpha _0\) and \(\Vert \nabla \psi _\textrm{osc}\Vert \ge c \alpha ^{1/4}\) for some \(c>0\). Thus we arrive at
On the other hand we have
so that with Lemma 3.2(d) we have \( \Vert \textrm{H}_{\alpha } P'_\textrm{osc} f \Vert _2 \le C \alpha ^{1/2}\) and we arrive at
As a next step, we shall replace \(Q_\textrm{osc} g_\alpha \) with the \(L^2\)-normalized function
For this we first observe that
and thus, we need to control the normalization constants’ ratio. For this we use (3.95) and find that
for a constant \(C>0\) independent in \(\alpha \). In particular we obtain
for sufficiently large \(\alpha \ge \alpha _0\). Thus from (3.97) and (3.98) we get
Next we show that the operator \(\widetilde{X}_{\psi _\alpha }\) contributes sub-leading (i.e. \(o ( \sqrt{\alpha })\)) only. For this we write
With \(\vert \widetilde{h} (x) \vert \le C x^2\) we find from Lemma 3.2(c) and \(\Vert g_\textrm{osc} \Vert _2 =1\) that
We recall that \(g_\textrm{osc}\) is orthogonal to \(\psi _\textrm{osc}\) and its partial derivatives. In particular, as \(\nabla \psi _\textrm{osc}= x \psi _\textrm{osc}\), the function \(g_\textrm{osc}\) is orthogonal to \(x \psi _\textrm{osc}\), too. Therefore not only the zero-th but also the first-order term of the Taylor expansion of h in \(\widetilde{X}_{\psi _\textrm{osc}}\) evaluated in \(g_\textrm{osc}\) vanishes, i.e.
where \(\widetilde{\widetilde{X}}_{\psi _\textrm{osc}}(x,y) = \alpha \psi _\textrm{osc}(x) ( h (x-y) - \Vert g \Vert _2^2 -\Vert \nabla g \Vert _2^2 (x-y)^2 ) \psi _\textrm{osc}(y)\). Since \(\vert \widetilde{h}(x) -\Vert \nabla g \Vert _2^2 x^2\vert \le C x^4\) by Assumption 1.1 we find with the harmonic oscillators scaling properties that \(\langle g_\textrm{osc}|\widetilde{\widetilde{X}}_{\psi _\textrm{osc}}|g_\textrm{osc}\rangle \ge - C \), and thus from (3.102)
for sufficiently large \(\alpha \ge \alpha _0\). Now it remains to compare the r.h.s. with the harmonic oscillator. For this we split the operator \({\textrm{H}}_\alpha \) into one part that is localized on a ball \(B_R\) with \(R= \alpha ^{-1/6}\) (that we shall show is bounded from below by \(\mathcal {H}_\textrm{osc}\) that is \(O( \sqrt{\alpha })\)) and a part outside \(B_R^c\) (that we will show is bounded from below by a positive constant of \(O( \alpha )\), i.e. trivially satisfying the claim for sufficiently large \(\alpha \ge \alpha _0\)).
For the localization we consider a partition of unity \(0 \le \eta ^1, \eta ^2 \le 1\) with \(\eta ^i \in C_0^\infty ( \mathbb {R}^3)\) and
and define the rescaled version \(\eta _R^i (x) = \eta ^i (x/R )\) and the \(L^2\)-normalized function
With standard arguments of IMS localization we find
By scaling the last summand is of order \(R^{-2} = \alpha ^{1/3}\) yielding
and it remains to estimate the expectation value \(\langle f_R^i|{\textrm{H}}_\alpha |f_R^i\rangle \) for \(i=1,2\).
We start with the expectation value w.r.t \(g_{R}^2\) supported on \(B_R^c\). Since \(\cos (k \cdot x ) \le 1 \), we find
To show that the remaining term contributes sub-leading (i.e. \(o(\alpha )\)) only, we need to control the \(L^\infty \)- norm of \(V_{\sqrt{\alpha }\varphi _\alpha }\) on \(B_R^{c}\), i.e.
for \(\vert x \vert \ge \alpha ^{-1/6}\). We split the integral in \(B_{\widetilde{R}}\) and \(B_{\widetilde{R}}^c\) where now we choose \(\widetilde{R} = \alpha ^{-1/5}\). For \(y \in B_{\widetilde{R}}^c\) we find by the scaling properties of \(\psi _\alpha \) that
and we arrive with \(\Vert h \Vert _{L^\infty ( \mathbb {R}^3)}\le C\) for \(\vert x \vert \ge \alpha ^{-1/6} \) at
Now let \( \vert y\vert \le \alpha ^{-1/5}\) and \(\vert x \vert \ge \alpha ^{-1/6}\). Then we have \(\vert x - y \vert \ge \vert x \vert - \vert y \vert \ge C \alpha ^{-1/6}\) for sufficiently large \(\alpha \ge \alpha _0\). Thus, with \( \Vert h \Vert _{L^\infty ( \mathbb {R}^3)} \le C \)
for \(\vert x \vert \ge \alpha ^{-1/6}\). With Cauchy Schwarz inequality and \(\Vert \psi _{\alpha } \Vert _4 \le C \Vert \psi _{\alpha } \Vert _{H^1} \le C \alpha ^{1/4}\) (from Lemma 3.2(d)) we find
Hence we deduce from (3.111)
for constant \(C_1,C_2\) independent of \(\alpha \) and sufficiently large \(\alpha \ge \alpha _0\).
We recall that the goal for the expectation value
with \(f_R^1\) supported on \(B_R\) is a comparison with the Hessian of the harmonic oscillator \(\mathcal {H}_\textrm{osc}\) that is \(O( \sqrt{\alpha })\). For this we observe that \(f_R^1\) is almost orthogonal to \(\psi _\textrm{osc}\) and its partial derivatives as
by Lemma 3.2(c)) and similarly for the partial derivatives. Thus (with similar arguments as in the beginning of this proof (see Eq. (3.91) and subsequent)) instead of \(g_R^1\) we consider in the following the \(L^2\)-normalized function
paying a price sub-leading in \(\alpha \) (i.e. \(o( \sqrt{\alpha })\) and given by
We use the Taylor expansion of \(h\), \(\langle \psi _\alpha |x|\psi _\alpha \rangle =0\) and Lemma 3.2(d) and find
and thus (since \(\left\langle \psi _\alpha \right| \textrm{h}_\textrm{osc}\left| \psi _\alpha \right\rangle \ge \left\langle \psi _\textrm{osc}\right| \textrm{h}_\textrm{osc}\left| \psi _\textrm{osc}\right\rangle \))
By construction \(\widetilde{g}_R^1\) is a \(L^2\)-normalized function and orthogonal to the harmonic oscillator’s ground state and its partial derivatives. Thus \(\widetilde{g}_R^1\) is a competitor for a minimizer of the harmonic oscillator’s Hessian and we conclude that
Since \(\mathcal {H}^{(2)}\) is a convex combination of (3.117) and (3.124), we find
and conclude that there exists \(\alpha \ge \alpha _0\) such that \(\mathcal {H}^{(2)} \ge C \alpha ^{1/2}\) for all \(\alpha \ge \alpha _0\). \(\square \)
The Hessian’s positivity in the strong coupling limit \(\alpha \rightarrow \infty \) leads to local coercivity estimates summarized in the following Corollary.
Corollary 3.1
There exists \(\alpha _0 >0\) and \(\kappa , C >0\) (independent of \(\alpha \)) such that for all \(\alpha > \alpha _0\) and \(\psi _\alpha \in \mathcal {M}_{\mathcal {E}_\alpha }\), any \(L^2\)-normalized \(\psi \in H^1 ( \mathbb {R}^3)\) and \(\varphi \in L^2 ( \mathbb {R}^3)\) with
we have
The proof is based on an expansion of \(\mathcal {G}_\alpha \) around the ground state energy \(e_\alpha \). In the following, we provide an expansion of \(\mathcal {G}( \psi , \varphi )\) which will be useful for later proofs. For this,let \(\delta _1 = \psi - \psi _\alpha \) and \(\delta _2 = \varphi - \varphi _\alpha \)
We observe that the sum of the second and third term vanish by the definition of the potential (1.3) and \(\varphi _\alpha = - \sqrt{\alpha }\sigma _{\psi _\alpha }\). For the last two terms of the r.h.s., we complete the square
where \(X_{\psi _\alpha }\) is defined in (3.85) so that we arrive at
The Euler-Lagrange equation of \(\psi _\alpha \) together with the notation (3.85), (3.85) and the observation that by \(L^2\)-normalization of \(\psi _\alpha \) and \(\psi \)
and therefore
we find with (3.85)
Proof of Corollary 3.1
In order to prove (3.127) first, we remark that it suffices to consider \(\psi \in H^1 ( \mathbb {R}^3)\) such that
hold. In particular we assume w.l.o.g. that \(\theta =0\) and \(y = 0\).
Furthermore, completing the square we get
and thus it suffices to consider the case \(\delta _2 = \sqrt{\alpha } ( \sigma _\psi - \sigma _{\psi _\alpha })\). It follows from (3.134) and (3.135)
We recover back the Hessian of \(\mathcal {E}_\alpha \) which by Lemma 3.3 is positive for sufficiently large \(\alpha \ge \alpha _0\). Moreover, it follows from Lemma 3.3 that there exists \(C_1>0\) (independent of \(\alpha \)) such that with \(\Vert \delta _1 \Vert _2 \le \delta \alpha ^{-1/2}\) for sufficiently small \(\delta >0\) by assumption, we have
Moreover, since \(\cos ( k \cdot x) \le 1 \) by Lemma 3.2 there exists a constants \(\kappa _1,\kappa _2 >0\) such that
Interpolating between (3.138) and (3.139), there exists a constant \(C_2 >0\) such that for \(\alpha \ge \alpha _0\)
By translational and rotational invariance of the energy, we conclude
Second we prove (3.128): Completing the square leads to
so that we find from (3.138) that there exists \(y \in \mathbb {R}^3\) and \(\kappa _1 >0\) such that
By regularity of \(\varepsilon , v\), there exists \(\kappa _3 >0\) such that
and we find by completing the square
for a constant \(\kappa _4 >0\). We conclude that
\(\square \)
3.3.2 Global coercivity estimates
The Hessian’s positivity shows the validity of global coercivity estimates of the energy. For this, we additionally have to assume that the ground state \(\psi _\alpha \) is unique up to translations and phase, i.e. that \(\mathcal {M}_{\mathcal {E}_\alpha } = \Theta ( \psi _\alpha )\). We remark that to prove the ground states uniqueness up to translations and phase by the local coercivity estimates Corollary 3.1, one needs an improved approximation of the ground state than in comparison to Proposition 2.2.
Corollary 3.2
Assume that the ground state \(\psi _\alpha \) of \(\mathcal {E}_\alpha \) is unique up to translations and rotations. There exists universal constants \(\alpha _0 \ge 0\) and \(C >0\) (independent of \(\alpha \)) such that for any \(L^2\)-normalized \(\psi \in H^1 ( \mathbb {R}^3)\) and \(\varphi \in L^2 ( \mathbb {R}^3)\) with we have for all \(\alpha \ge \alpha _0\)
The proof follows the arguments presented in [24, Lemma 2.6].
Proof
We first prove the global bound (3.147). Then the second bound (3.148) follows similarly to the proof of Corollary 3.1.
In order to prove (3.147) we remark (similarly to the proof of Corollary 3.1) that is suffices to consider \(\psi \in H^1 ( \mathbb {R}^3)\) such that
W.l.o.g. we assume \(y=0 \) and \(\theta =0\). By contradiction we assume that there does not exist a universal constant \(C>0\) such that (3.147) holds. Then there exists a sequence of functions \(\psi _n \in L^2( \mathbb {R}^3)\) with \(\Vert \psi _n \Vert _{L^2( \mathbb {R}^3)} =1 \) such that
It follows that \(\mathcal {E}_\alpha ( \psi _n ) \ge \frac{1}{2} \Vert \nabla \psi _n \Vert _2^2 - C \alpha \). Therefore \(\psi _n\) is uniformly bounded in \(H^1\) and moreover a minimizing sequence. With similar arguments as in the proof of Lemma 3.1\(\psi _n\) converges to an element of the set of minimizes \(\Theta (\psi _\alpha )\) given by (3.149) through \(\psi _\alpha \). This is a contradiction since Corollary 3.1 shows that locally coercivity estimates hold true. \(\square \)
Another consequence of the Hessian’s approximate behavior is the following property.
Corollary 3.3
There exists \(\alpha _0\) and a constant \(C>0\) (independent of \(\alpha \)) such that for all \(\alpha \ge \alpha _0\), we have \({\textrm{H}}_\alpha - e_\alpha > C \sqrt{\alpha }\).
Proof
The existence of a spectral gap of \({\textrm{H}}_\alpha \) of order \(\sqrt{\alpha }\) follows immediately from the global coervitiy estimates in Corollary 3.2. \(\square \)
3.4 Proof of Propositions 2.1, 2.2
In this section we prove Proposition 2.1, 2.2 based on the results proven before.
Proof of Proposition 2.1
The proposition follows immediately from Lemma 3.1. \(\square \)
Proof of Proposition 2.2
The proposition follows from Lemma 3.2 and Corollary 3.2. \(\square \)
4 Proof for traveling waves
In this section, we prove Proposition 2.3 on existence of subsonic traveling waves of the regularized Landau-Pekar equations.
For this, we remark that it follows from the regularized polaron’s dynamics that the traveling wave (1.5) satisfies
Proof of Proposition 2.3
Proof of (a): First, we prove the existence of traveling waves for sufficiently small velocities. Traveling wave solutions of (1.5) are stationary points of the action functional \(\mathcal {I}_\textrm{v}\) given by
In the following we show that there exists a minimizer \((\psi _\textrm{v}, \varphi _\textrm{v})\) of \(\mathcal {J}_\textrm{v}\), and thus a traveling wave solution. Since
and for arbitrary \(\delta >0\)
the action functional is bounded from below by
We remark that in the last step we used the \(L^2\)-normalization of \(\psi \). By Assumption 1.2, we have \(\varepsilon (k)\ge \textrm{v}_\textrm{crit}\vert k \vert \) and thus,
For \(\vert \textrm{v}\vert \le \textrm{v}_\textrm{crit}- \delta \), it follows that any minimizing sequence \((\psi _n,\varphi _n)_{n \in \mathbb {N}}\) of \(\mathcal {I}_\textrm{v}\) is uniformly bounded in \(H^1( \mathbb {R}^3) \times L^2_{\sqrt{\vert \cdot \vert }}(\mathbb {R}^3)\). Any minimizing sequence, thus, uniformly bounded and weakly converging in \(H^1 ( \mathbb {R}^3) \times L_{\sqrt{\varepsilon }} ( \mathbb {R}^3)\) to a limiting functional that is possibly zero (by translational invariance of the action). With similar arguments as after Eq. (3.3), there exists a sequence \(( \psi _n^{y_n}, e^{i p_n }\varphi _n)_{n \in \mathbb {N}}\) that converges strongly in \(L^2(\mathbb {R}^3) \times L^2_{\sqrt{\varepsilon }}( \mathbb {R}^3)\) to a pair of non-zero limiting functions. By semi-lower continuity of the \(H^1\)- and the \(L_{\sqrt{\varepsilon }}\)-norm, and
we conclude that the action functional \( \mathcal {J}_\textrm{v}\) attains its infimum for \(( \psi _\textrm{v}, \varphi _\textrm{v})\) which is a non-zero traveling wave solution (4.1).
Proof of scaling properties: As a preliminary step towards proving Proposition 2.3(b) we shall first prove that
i.e. that the traveling wave satisfies similar scaling properties as the harmonic oscillator. We proceed similarly as in the proof of Lemma 3.2(b). For this let \(\textrm{H}_0:= - \Delta /(2\,m) + i \textrm{v}\cdot \nabla \). Then the traveling wave equation (4.1) implies
Since \(\textrm{H}_0 \ge - \textrm{v}^2/4\) and \(e_\textrm{v}\ge - e_\alpha + \textrm{v}^2/4 \), the resolvent \(\left( \textrm{H}_0 + e_\textrm{v}\right) ^{-1}\) is well defined and we can write
The resolvent’s Green’s function is given in terms of the inverse Fourier transform \(\mathcal {F}^{-1}\) of
and can (by self-adjointness of \(\textrm{H}_0\) and functional calculus) explicitly computed. In fact by functional calculus we have
which leads with (4.11) to
and we arrive with Fubini’s theorem at
for \(\vert z \vert \not =0\). We plug this identity into (4.10) and find that
Thus the weighted \(L^2\)-norm, we aim to find an upper bound for, becomes
that we can estimate by Cauchy Schwarz with
With
and \(\Vert \varphi _\textrm{v}\Vert _{L_{\sqrt{\varepsilon }}} \le C \sqrt{\alpha }\) from Lemma 2.1, we can use a similar splitting of the above integral as in the proof of Lemma 3.2(b) to then conclude by \(e_\textrm{v}- \textrm{v}^2/4 \ge - e_\alpha \ge C \alpha \) with (4.8).
Proof of (b): We observe that for \(\textrm{v}= 0\) a traveling wave solution is given by \(\psi _{\textrm{v}=0} = \psi _\alpha ^y, \varphi _{\textrm{v}= 0} = \varphi _\alpha ^y\) with \(e_{\textrm{v}} = \mu _{\psi _\alpha ^y}\) for any \(y \in \mathbb {R}\). To prove (2.15) it follows (similarly to the proof of Corollary 3.1) that it suffices to consider the decomposition
with \(\left\langle \mathrm{Re\,}\delta _1\right| \left| \nabla \psi _\alpha ^y\right\rangle = 0\), \(\left\langle \mathrm{Im\,}\delta _1\right| \left| \psi _\alpha ^y\right\rangle =0\). In particular it follows from Corollary 3.2 and condition (i) that
for sufficiently small \(\kappa _1, \kappa _2, \kappa _3 >0\) (independent of \(\alpha \)). In the following we assume w.l.o.g. that \(y=0\).
By definition of \({\textrm{H}}_\alpha \) (see (3.85)) and the decomposition (4.19), we can write the traveling wave equations (4.1) as
and it follows that the phase \(\mu _\textrm{v}\) is given through the identity
Plugging this identity back into (4.21), we get
where we introduced the notation \(Q_\textrm{v}= 1- \left| \psi _\textrm{v}\right\rangle \left\langle \psi _\textrm{v}\right| \) and the operator
With the decomposition’s properties (4.20) we find that
Thus by Corollary 3.3 there exists \(\kappa _4 >0\) such that by assumption \(Q_{\textrm{v}}({\textrm{H}}_\alpha + A )Q_{\textrm{v}} \ge \kappa _4 \sqrt{\alpha } \), i.e. we can write
The second term of the r.h.s. leads with Proposition 3.2 to the desired bound. For the first term we observe that by definition of the potential and radialilty of v
where \(\delta _2^s\) denotes the symmetric part of \(\delta _2\), i.e. \(\delta _2^s(k) = \delta _2^s( -k )\). We observe that splitting \(\delta _2\) into its symmetric \(\delta _2^{s} (k) = \delta _2^{s} (-k)\) and anti-symmetric \(\delta _2^{a} (k) = - \delta _2^{a} (-k)\) we have from the traveling wave Eq. (4.22)
Here we used that \(\varepsilon ^{-2} ( k \cdot \textrm{v})^2 \le \textrm{v}^2/ \textrm{v}_\textrm{crit}^2 <1\) by Assumption 1.2. Hence
We observe that due to the projection \(Q_\textrm{v}\) the first term of the Taylor expansion of \(\cos ( k \cdot )\) vanishes. Thus with \(\varepsilon ^{-2} ( k \cdot \textrm{v})^2 \le \textrm{v}^2/ \textrm{v}_\textrm{crit}^2 <1\) and \(\left\langle \mathrm{Re\,}\psi _\alpha \right| \left| \psi _\alpha \right\rangle = - \Vert \delta _1 \Vert _2^2\) we arrive at
From the scaling properties of the traveling wave (4.8) and the ground state (see Corollary 3.3) we conclude
yielding for \(\vert \textrm{v}\vert \le 1\)
and (2.18) follows from (4.22) resp. (4.23)
\(\square \)
5 Proofs for definitions effective mass
In this section, we prove Theorems 2 and 3 on the definition of the effective mass.
We remark that the proofs presented in this Section follow ideas from [11] where the non-regularized Landau-Pekar equations have been considered. For the non-regularized Landau-Pekar equations traveling waves are conjectured to not exists. However assuming their existence an energy expansion in the vein of the proof of Theorem 2 was sketched. Furthermore a different approach for a definition of the mass through an energy-velocity expansion was presented. The proof of Theorem 3 given below uses ideas presented there.
5.1 Effective mass through traveling waves
We consider the definition of the effective mass through subsonic traveling waves first (whose existence follow from Proposition 2.3).
Proof of Theorem 2
Let \(\alpha \ge \alpha _0\) large enough and \(\vert \textrm{v}\vert < \textrm{v}_\textrm{crit}\). Then, by Proposition 2.3 and 2.1, there exists \(y,z \in \mathbb {R}^3\) and \(\theta \in (0, 2 \pi ]\) such that
with \(\left\langle \mathrm{Im\,}\psi _\textrm{v}\right| \left| e^{i \theta } \psi _\alpha \right\rangle = 0\), \(\left\langle \mathrm{Re\,}\nabla \psi _\textrm{v}\right| \left| e^{i \theta }\psi _\alpha ^y\right\rangle = 0\) and \(\left\langle e^{i z \cdot }\varphi _\alpha \right| \left| \nabla \varphi _\textrm{v}\right\rangle =0\) and \(\psi _\alpha \) is uniquely given (up to translations and changes of phase). W.l.o.g. we assume in the following \(y=0, \theta =0\). Then, (3.145) shows (with similar arguments as used in [11]) that we it suffices to consider \(z =0\), too. Thus, we decompose the traveling wave as
with \( \Vert \xi _\textrm{v}\Vert _2 \le C\) and \(\Vert \eta _\textrm{v}\Vert _{L^2_{\sqrt{\varepsilon }}} \le C \sqrt{\alpha }\). Note that Proposition 2.3 moreover shows that \(e_\textrm{v}= \mu _{\psi _\alpha } + O( \alpha ^{3/4}\textrm{v}^2) \) and the linearisation of the traveling wave Eq. (4.1) read
In particular, it follows
Combining (5.7) and (5.6), we find
As \(\textrm{H}_{\alpha }\) is invertible on the span of \(\partial _1 \psi _\alpha \), we find from (3.131) with (5.4) and (5.5) for all \( \vert \textrm{v}\vert \le C \alpha ^{-1} \)
\(\square \)
5.2 Effective mass through energy-momentum expansion
Here, we consider the definition of the effective mass through the energy-momentum expansion explained in Sect. 5.
Proof of Theorem 3
We first pick a trial state to show an upper bound on \(\inf _{\mathcal {I}_\textrm{p}}\mathcal {G}(\psi , \varphi )\) which we use later for the lower bound. For this, we choose \(\alpha _0 >0\) sufficiently large, such that by Proposition 2.1 (b), there exists a unique (up to translations and phases) pair of minimizers \((\psi _\alpha , \varphi _\alpha )\) of \(\mathcal {G}_\alpha \).
Proof of the upper bound of (a): It is easy to check that the trial states
with the choice
and \((1-\mu _\textrm{p})^2 + \lambda _\textrm{p}^2 =1 \) (i.e. \(\mu _p = O( \alpha ^{-2}\textrm{p}^2)\)) satisfy constraint (1.15) (using that \(\textrm{H}_{\alpha }^{-1} \nabla \psi _\alpha = m x \psi _\alpha \)) and that for large \(\alpha \ge \alpha _0\)
With these observations, we plug the trail states into the expansion of \(\mathcal {G}_\alpha \) in (3.134) and find
Proof of the lower bound of (a): For \( \textrm{p}\le \alpha ^{1/4}\), it follows from the upper bound and Corollary 3.2 that for any element \(( \psi , \varphi ) \in \mathcal {I}_\textrm{p}\), there exists \(y \in \mathbb {R}^3\) and \(\theta , \omega \in (0, 2 \pi ]\) such that
W.l.o.g. we assume \(y=0, \theta =0\) and, with (3.145) we consider furthermore \(\omega =0\). It follows from the expansion (3.134) of the energy \(\mathcal {G}_\alpha \)
Completing the square, we find with (5.11)
For the lower bound, we can neglect the first two lines and obtain
For the first line, we use the constraint (ii)\(_{\textrm{p}}\), (5.14) together with Assumption 1.2 and the trivial bound \(\Vert \nabla \psi _0 \Vert \le C \sqrt{\alpha }\).The second line, we compute explicitly and obtain
Combining now the upper (5.18) and the lower bound (5.13), we arrive at Theorem 3.
Proof of (b): The corresponding Lagrange functional to the minimization problem is given by
Thus any minimizer satisfies the traveling wave equations with velocity \(\lambda \), i.e.
By definition of the set \(\mathcal {I}_\textrm{p}\) in (1.16), Proposition 2.3 shows that we can decompose
with \(\Vert \delta _1 \Vert _2 \le C \alpha ^{-1/4}\) and \( \Vert \varepsilon ^{1/2} \delta _2 \Vert _2 \le C \alpha ^{1/4}\). The coercivity estimates from Corollary 3.2 together with the upper bound of part (a) then show
Together with the traveling waves equation it follows from constraint (ii)\(_\textrm{p}\) for all \(\textrm{p}\le 1\)
Furthermore from the first part of the theorem and Theorem 2, we conclude
where \(( \psi _{\textrm{v}'}, \varphi _{\textrm{v}'} )\) denotes a traveling wave with velocity \(\textrm{v}' = m_\textrm{eff}^{-1} \textrm{p}+ O( \alpha ^{-1} \textrm{p}^2)\). \(\square \)
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Acknowledgements
S.R. would like to thank Robert Seiringer for fruitful discussions, Krzysztof Myśliwy for helpful remarks and the reviewer for careful reading and useful comments. Funding from the European Union’s Horizon 2020 research and innovation program under the ERC grant agreement No. 694227 is gratefully acknowledged.
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Rademacher, S. Traveling waves and effective mass for the regularized Landau-Pekar equations. Calc. Var. 63, 121 (2024). https://doi.org/10.1007/s00526-024-02735-3
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DOI: https://doi.org/10.1007/s00526-024-02735-3