Microscopic derivation of the Fr\"ohlich Hamiltonian for the Bose polaron in the mean-field limit

We consider the quantum mechanical many-body problem of a single particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the Hartree limit and rigorously show that, at sufficiently low energies, the problem is well described by the Fr\"ohlich polaron model, in which the interaction of the impurity with the environment is described as a linear coupling to a scalar quantum field.


The polaron
When considering a quantum particle of mass M moving through a large uniform system, it is natural to model the interaction of the particle with the environment using a linear coupling to an appropriate scalar boson field. For a translation invariant system, this corresponds to the formal (class of) Hamiltonian(s) where R denotes the position of the impurity particle, and k labels the momentum modes of the field. Moreover, P = −i∇ R is the particle's momentum operator in the canonical representation, and a † k , a k are the usual field mode creation and annihilation operators. They satisfy the canonical commutation relations [a k , a † k ′ ] = δ k,k ′ , [a k , a k ′ ] = 0, which corresponds to a scalar bosonic field. The g k are coefficients quantifying the coupling of the particle's motion to the field, with * denoting the complex conjugate, and e k is the free field dispersion relation. The natural domain of this Hamiltonian lies in the Hilbert space H ⊗ F(K), where H is the Hilbert space of the particle and K is the Hilbert space of a single field mode, with F(K) denoting the symmetric Fock space over K.
The Hamiltonian (1.1) is commonly referred to as the Fröhlich Hamiltonian, as it was introduced by Fröhlich in 1937 [1] in order to describe electronic motion in polar crystals. The polaron in this context refers to the picture of an electron dressed with the emerging optical phonons dragged along as it moves. Later, this concept was extended to include other phenomena related to mobile impurities coupled to excitations of the background, giving rise to interesting effects in many materials [2,3] which are still the subject of ongoing research [4,5].
In this work, we are interested in a rigorous justification of the use of Hamiltonians of the type (1.1) as an effective description of a full quantum mechanical many-body problem. In the case of the original Fröhlich model this task seems too ambitious (see, however, [6], where the classical approximation to the original polaron problem, the Pekar functional, is rigorously derived from a specific model of an electron moving through a quantum crystal). The applicability of the polaron picture is not limited to electrons in crystal lattices, however. In fact, recent progress in experiments with ultracold atoms opened the possibility of studying impurity atoms immersed in an environment consisting of many bosonic atoms at low temperatures, displaying Bose-Einstein condensation. As discussed below, at sufficiently low energies the excitations of the bosonic bath correspond to quantized acoustic phonons, and hence the Bose polaron corresponds to the impurity atom dressed with these phonons. We refer to [7] for a review of recent theoretical progress concerning the application of Fröhlich Hamiltonians to these systems. As the mathematical description of cold Bose systems, and in particular the structure of their excitation spectra at low energies, have recently been studied rigorously in numerous works [8][9][10][11][12][13], we find it natural to provide a rigorous microscopic derivation of (1.1) based on these results.

The N + 1 Bose gas
We consider a system of N bosons of mass 1/2 and one additional particle (of an unspecified type of statistics) of mass M , all confined to move on the unit torus in d dimensions, T d . Assumption 1.1 (Assumptions on the potentials). We assume that 1. the bosons interact among themselves via a two-body potential v, which is bounded, Borel measurable, even and of positive type, i.e., all its Fourier coefficients v p are non-negative.
2. the additional impurity particle interacts with the bosons via a two-body potential w, which is bounded, Borel measurable and even.
Note that no assumption is made on the Fourier coefficients w p of w. Nevertheless w being even implies w p = w −p ∈ R. Without loss of generality, we may in addition assume that v and w are non-negative, since they can be shifted by a constant otherwise.
The positions of the bosons are labeled by {x i } N i=1 , x i ∈ T d and the position of the impurity by R ∈ T d . The Hamiltonian of this system reads where we introduced some coupling (λ,µ) and scaling (η,ν) parameters to be chosen. It acts on L 2 (T d ) ⊗ H N with H N being the Hilbert space of square-integrable symmetric functions on T dN . Here, △ y denotes the d−dimensional Laplacian in the coordinate y acting on functions on the unit torus. The coupling parameters λ and µ determine the strength of the potentials v and w (for the functional forms of v and w being fixed), whereas η and ν determine the respective ranges (relative to the system size). They can be adjusted to consider various scaling regimes. The usual thermodynamic limit corresponds to the choice η ∼ ν ∼ N 1/d and λ ∼ µ ∼ N 2/d . In contrast, we consider here the mean-field, or Hartree, limit, where the interactions are weak and extend over the entire system. In particular, we choose λ = (N − 1) −1 , µ = N −1/2 , and η = ν = 1. For systems without impurity, this was the scaling for which the first rigorous results on the excitation spectrum were obtained [8,10,13,14], and our analysis is based on them. The choice µ = N −1/2 for the impurity-boson coupling turns out to be a natural in the analysis, compatible with the methods from [8,10] we use, as explained below. Therefore, from now on we consider the Hamiltonian with v and w 1-periodic functions satisfying Assumption 1.1.

Motivation of the Fröhlich Hamiltonian
With v p and w p denoting the Fourier coefficients of v and w, respectively, the second-quantized version of (1.4) We defined the Hartree ground state energy which captures the effect of interactions between particles in the p = 0 mode. The sums run over (2πZ) d with p = 0 excluded. Here, a p denotes the usual annihilation operator The second-quantized Hamiltonian (1.4) acts on L 2 (T d ) ⊗ F, with F the bosonic Fock space F over L 2 (T d ), i.e., F := ∞ i=0 H i (with H 0 = C). Actually, it preserves L 2 (T d ) ⊗ H N . For the system without impurity, it was predicted by Bogoliubov [15] that for sufficiently low energies, the excitation spectrum of H N should be composed of elementary excitations, which are physically interpreted as quantized (acoustic) free phonons. This serves as the basis for the microscopic explanation of the emergence of superfluid behavior in low-temperature bosonic systems. From the formal perspective, it provides a specific example of the appearance of an effective quantum field theoretical description of a many-body system. The low-energy effective theory is predicted to be that of the Hamiltonian and β p = γ p α p . These algebraic relations are realized via a suitable unitary (Bogoliubov) transformation. From (1.7) we deduce that, for low energies, the excitation spectrum is expected to be composed of free bosonic quasi-particles with dispersion relation e p . In the mean-field scaling λ = (N − 1) −1 considered here, one can prove [8] that e p = p 4 + 2v p p 2 . Additionally, it can be shown that in this scaling the ground state energy equals 1 2 The method employed by Bogoliubov leading to H B consists of the following steps: 1. the operators a 0 , a † 0 are replaced by the number √ N 2. all the terms of higher order than quadratic in creation and annihilation operators that remain in the Hamiltonian are dropped.
This procedure is physically motivated by the expectation that for sufficiently small energies there is Bose-Einstein condensation in the system, that is, the p = 0 mode is occupied by an overwhelming fraction of particles. Whereas this has not been proven for a generic bosonic system with general interactions, the validity of the Bogoliubov approximation has been rigorously verified (in the case w ≡ 0) for a variety of assumptions on v. The first such result refers precisely to our conditions on v and, as already mentioned, the mean-field scaling λ = (N − 1) −1 , which corresponds to a very weak and long-ranged potential. If one applies the Bogoliubov approximation to the Hamiltonian (1.4) with impurity, one expects that the system is, for small energies, effectively described by the Fröhlich Hamiltonian By expressing the a p 's in terms of the operators b p , b † −p , we see that it equals which belongs to the class of Hamiltonians defined in (1.1). The Hamiltonian H F acts on L 2 (T d ) ⊗ F + , where F + is the Fock space over the complement of the unit function in L 2 (T d ), describing solely the p = 0 modes of the field. In order to obtain (1.10) via a Bogoliubov approximation, we supplemented this procedure by additionally dropping, in the impurity-boson interaction, all the terms that are of higher order than linear in the creation and annihilation operators (after first replacing the a 0 and its adjoint by √ N ), whereas we kept the quadratic terms in the boson-boson interaction. One of elements of our analysis below is the justification of this additional step while checking that the other steps, known to be rigorously justifiable in the mean-field case in the absence of an impurity, are still applicable. It is important, however, to realize that in some instances, especially when the impurity-boson interaction is strong, additional terms not present in the Fröhlich Hamiltonian (1.10) cannot be neglected [16,17].

Main results
The interpretation of our main results, as stated below, is that the Fröhlich Hamiltonian (1.10) may indeed be seen as an effective low-energy, large N theory for the original model described by H N in (1.3). Our analysis consists of a rigorous justification of the extended Bogoliubov approximation, based on suitable operator inequalities. It leads to two main Theorems, the first of which concerns the excitation spectrum of H N .

Theorem 1: convergence of eigenvalues
Let us denote by e i (A) the i−th eigenvalue resp. the i−th min-max value of an operator A, starting at i = 0. Our first Theorem states that as long as one considers the energy levels of H N lying in a not too large window above the ground state, their values are provided by the corresponding eigenvalues of the Fröhlich Hamiltonian if N is sufficiently large. In particular, we provide explicit bounds on the size of that window as compared with N .
for some constant C v,w > 0, independent of the parameters ξ, N .
In the special case of the ground state energy we have The interaction with the impurity thus gives rise to a N 1/2 contribution to the ground state energy and, more importantly, leads to an O(1) contribution to the excitation spectrum via the last term in (1.9). This can be understood as follows. In the impurity-free case, the effect of the emergence of phonons is reflected as a O(1) correction to the ground state and low-lying excitation energies, in the mean-field (Hartree) limit considered here. There are only finitely many (even for very large N ) phonons that emerge in the system. In the Fröhlich description, the system is seen as a condensate background slightly perturbed by the impurity, which itself interacts with the phonons. Therefore, the phonon-impurity interaction should as well give rise to an O(1) correction. Since it scales as a 0 ∼ √ N , we see that µ ∼ 1/ √ N is consistent with these considerations.

Remark 1.2.
The error bounds are of the form ξ(ξ/N ) 1/2 . Therefore, as long as the total excitation energy satisfies ξ ≪ N , the error made by using the Fröhlich Hamiltonian instead of the original one when computing the energy levels is small compared to the total excitation energy. The size of this energy window is presumably optimal. In fact, if the condition ξ ≪ N is not fulfilled one cannot expect the onset of BEC anymore, which is an essential assumption in the Bogoliubov approximation. It is noteworthy that precisely the same error scaling was obtained in [8] for the pure bosonic system. The effects of the inclusion of the impurity thus manifest themselves only in the value of the constant C v,w . Remark 1.3. By a direct inspection of the proof, one sees that the result can easily be generalized to the case of multiple impurities (as long as their number is fixed, i.e., independent of N ).

Remark 1.4.
We emphasize that the proof is valid in the mean-field scaling λ = (N − 1) −1 and µ = N −1/2 and under Assumption 1.1 on v and w. These restrictions on the potentials could possibly be weakened; we stick to them in this paper in order to avoid unnecessary technical complications.
Physically, our choice of scaling corresponds to very weak and long-ranged potentials. Extending the results to the case of more realistic, short-ranged potentials remains a challenge. In fact, the w ≡ 0 cases with either λ = N 2/d , η = N 1/d (equivalent to the thermodynamic limit) or λ = N 2 , η = N in d = 3 (the Gross-Pitaevskii limit) were rigorously analyzed only very recently. The results for the thermodynamic limit concern the ground state energy only [9,[18][19][20], whereas in the Gross-Pitaevskii scaling regime the emergence of the Bogoliubov spectrum for low energies was shown as well [21].
where n 0 is the condensate density and ζ = (2g BB n 0 ) −1/2 is the healing length; the parameters g IB and g BB are the coupling constants describing the impurity-boson and boson-boson interactions, respectively. Additionally, ǫ p = c 2 p 2 (1 + (ζp) 2 ) with c = 1/ζ = √ 2g BB n 0 denoting the speed of sound in the bosonic bath. This Hamiltonian displays an evident ultraviolet divergence, recently analyzed in [22]. By naively replacing v p and w p in (1.10) with the respective coupling constants g BB and g IB , one arrives at H B-F with unit condensate density. We conjecture that (1.13), resp. some renormalized version of it, arises in place of H F in scaling regimes corresponding to more realistic interactions of shorter range than the Hartree limit considered here. Remark 1.6. Our proof makes use of selected methods from [8] and [10], and suitably extends these to allow for the inclusion of an impurity. In the case w ≡ 0, we reproduce the results of [8], but by utilizing techniques from [10] we are able to substantially simplify the proof.

Theorem 2: convergence of eigenvectors
In order to compare the two operators H N and H F , which act on different Hilbert spaces, we utilize an operator introduced by Lewin, Nam, Serfaty and Solovej in [10], which maps H N to (a subspace of) F + . We give here a quick review of their construction, as it is important to formulate our second result.

The LNSS transform
. . , N }, are uniquely determined by the above considerations. The space F ≤N + is naturally seen to be a proper subset of the Fock space over the orthogonal complement of v 0 ∈ H. Moreover, U N is unitary. Performing this construction for H = L 2 (T d ) with, for instance, the plane wave basis and with v 0 ≡ 1 we arrive at a unitary transformation U N : H N → F ≤N + ⊂ F + with F + being the Fock space over the orthogonal complement of the unit function on T d . This space has a clear physical interpretation of being the space of excitations from the condensate, and the fully condensed state plays the role of the vacuum. It is due to the algebraic properties of U N , however, that it becomes helpful in the analysis, as it can be seen to rigorously realize the Bogoliubov substitution a 0 , a † 0 → √ N . More precisely, with Q denoting the projection onto the orthogonal complement of the unit function in L 2 (T d ), one can check that for all Ψ ∈ H N and consequently that for k, l = 0 (the last two follow from the first, in fact). Equipped with the operator U N , which we shall frequently denote by U for short, we now state our second main result concerning the eigenvectors.
in operator norm. Here U N as to be understood as a partial isometry, i.e., U † N is extended by 0 to all of F + . Remark 1.10. Another interesting problem concerns the dynamics of the impurity and the use of the Fröhlich Hamiltonian as its generator. This question has been recently studied from a physics perspective [17,23]. From a mathematical point of view, there exist results concerning the dynamics of a tracer particle immersed in a Bose gas [24,25], which concern a different scaling limit than the one considered here and do not utilize the Fröhlich description. The convergence (1.19) can also be reformulated as convergence of the corresponding group of time evolutions, and hence can be used to determine also the dynamics of small excitations of the condensate. In the absence of an impurity, more general results are known where the condensate itself is excited and evolves according to the time-dependent Hartree equation (see, e.g., [26,27]).
The remainder of this paper contains the proofs of Theorems 1.1 and 1.2. Throughout the text, the symbol C denotes a positive constant whose exact value may change at different appearances. Moreover, unless stated otherwise, all states on the relevant Hilbert spaces are normalized. Finally, all operators that are defined as acting on functions of the Bose gas coordinates or the field modes only are actually everywhere understood as their tensor products with the unit operator on L 2 (T d ), the latter being the Hilbert space of the impurity particle.

Auxiliary considerations
In this Section we introduce four preparatory Lemmas that will be needed in the proofs of Theorems 1.1 and 1.2. For their statement, we need to introduce some notation. We shall often denote the terms on the right side of (1.3), from left to right, by P 2 /2M, T, V and W . Let P denote the projection onto the unit wave function in L 2 (T d ), and Q = 1 − P. We define the excitation number operator as an operator on H N , whose second quantized form in the plane wave basis equals 2) The first Lemma explores the consequences of the mean-field structure of H N . In particular, the ground state energy of H N is, to leading order in N , equal to E H (N ), and the excitation number operator is uniformly bounded in N for states of fixed excitation energy.

Remark 2.1.
Below, we will make use of a direct consequence of this Lemma, namely for any state Ψ such that (Ψ, H N Ψ) ≤ e 0 (H N ) + ξ with ξ > 0.
Proof. The upper bound on the ground state energy is obtained by taking the constant wave function in L 2 (T d ) ⊗ H N as trial function. We write H N = P 2 2M + 1 2 T + V + ( 1 2 T + W ); by a standard argument using the positivity of the Fourier coefficients of v we have ≥ 0. Next, we use Temple's inequality [28]. Consider a Hamiltonian H = H 0 + W with a self-adjoint operator W ≥ 0 and a Hamiltonian H 0 ≥ 0 with ground state energy satisfying e 0 (H 0 ) = 0. Denoting by e 0 , e 1 the first two eigenvalues of H, we have clearly (H − e 0 )(H − e 1 ) ≥ 0. We evaluate this at the ground state of H 0 , Ψ 0 . We get (W − e 0 )(W − e 1 ) 0 ≡ (Ψ 0 , (W − e 0 )(W − e 1 )Ψ 0 ) ≥ 0 and rewrite this, since e 1 > 0, as Using the positivity of W and e 1 ≥ e 1 (H 0 ) we finally get and Ψ 0 -the unit function on T d , we have, with w 2 (0) = w 2 . This leads to (2.10) Using that N + ≤ (2π) −2 T , we see that the desired result holds.
The second Lemma concerns the fluctuations of the condensate in the ground state, which are seen to be strongly suppressed due to the mean field scaling. where the constants depend only on v and w but not on N . This fact will be of importance below.
Proof. Because N + ≤ 1 2π 2 ( 1 2 T ) and N + commutes with T , we find it convenient to give a bound on the operator 1 2 N + T , as the latter can be directly linked to H N . Writing and we estimate the relevant terms. By the Cauchy-Schwarz inequality, Note that (S + S 1 )Ψ is permutation symmetric in the Bose gas coordinates, so that (Ψ, N + (S + S 1 )Ψ) = N (Ψ, Q 1 (S + S 1 )Ψ), where Q 1 = 1 − P 1 . Moreover, S is independent of x 1 hence it commutes with Q 1 . Using the inequality (2.6) (with N replaced with N − 1) as well as Temple's inequality (2.9) and the upper bound on e 0 (H N ) in (2.3), we see that Since S commutes with Q 1 we thus have (2.17) The part of N + S 1 not containing −△ 1 /2 We introduce the short-hand v 12 to denote v(x 1 − x 2 ). We write, following [8] (Ψ, Observe that (Ψ, Q 1 P 2 v 12 P 2 Ψ) = (Ψ, Q 1 P 2 v 12 P 2 Q 1 Ψ) + (Ψ, Q 1 P 2 v 12 P 2 P 1 Ψ), where the last term vanishes and the remaining one is positive. For the first term, we use (Ψ, as P 2 ≤ 1 and (Ψ, Q 1 Ψ) = (Ψ, Q 2 Ψ) due to the permutation symmetry. The remaining part of S 1 is bounded as We thus have where the N -independent constant α equals α = 1 2 w ∞ + v ∞ + δE ′ . As N + ≤ gT , with g = (2π) 2 being the energy gap of the Laplacian on the torus, this implies for any ǫ, λ, κ > 0. By choosing ǫ = λ = κ = g 4 , we arrive at the desired result.
The third and fourth Lemmas concern H F . They will be of importance when proving the upper bound on the difference of eigenvalues in Theorem 1.1. Hamiltonian (1.9). Then there exist positive constants C 5 , C 6 , C 7 such that the inequalities

the diagonal part of the Fröhlich
Proof. Clearly, as v p ≥ 0, one can take C 5 = g −1 = (2π) −2 . The off-diagonal part of H F consists of the purely bosonic (v-dependent) and a w-dependent part. The latter,W , can be bounded byW for any ǫ > 0. To see this, simply complete the square for a single mode using the inequality (ηa † p + η −1 w p e ipR )(ηa p + η −1 w p e −ipR ) ≥ 0, then choose η 2 = ǫ(p 2 + v p ) and sum over the modes. It is hence enough to show that the bosonic off-diagonal part, given by for any ǫ > 0. Now take ǫ = λ(p 2 + v p ) for some λ > 0 and define µ := Consequently, Proof. We will show that N + H F 0 ≤ D ′ 1 (H F ) 2 + D ′ 2 , which implies the desired result by the previous lemma. As [N + , H F 0 ] = 0, we have withW and V OD defined as in the proof of Lemma 2.3. Using the canonical commutation relations [a p , a † q ] = δ p,q , we compute vp 2 a p a −p and finally An analogous computation forW yields By completing the square similarly as in Lemma 2.3, we havẽ for any λ > 0. We obtain for any λ > 0, γ > 0. By Lemma 2.3 and Proceeding similarly with V OD using (2.26) and the fact that p =0 a † p H F 0 a p = H F 0 (N + − 1), we have for someC > 0 and a constant µ < 1. By Lemma 2.3 and the Cauchy-Schwarz inequality, the last two terms of the above are bounded by C(H F ) 2 + C. Finally, using again the Cauchy-Schwarz inequality, we can bound for any ǫ > 0. Invoking Lemma 2.3 again, we obtain for any ǫ > 0 and λ > 0, By choosing ǫ and λ small enough, we arrive at the desired result.

Comparing H N and H F
The estimates provided in the previous section concern the relation of the number of excitations operator N + (or its square) to the Hamiltonians H N and H F independently. Now, making use of the LNSS transformation U introduced in Sec.
The proof of the proposition is divided into two main steps. In step 1, we take care of the higher-order terms in the creation and annihilation operators that appear in the second quantization of H N , but are absent in H F . Let viewed as an operator on L 2 (T d ) ⊗ H N .

Lemma 3.1.
For any ǫ > 0, one has the operator inequalities: and Proof. Using the Cauchy-Schwarz inequality and positivity of v, we have By translation invariance Q ⊗ PvP ⊗ P = 0. Moreover, the boundedness of v enables us to bound Therefore, we have the bounds v ≥ P ⊗ PvP ⊗ P + P ⊗ PvQ ⊗ Q + Q ⊗ QvP ⊗ P and v ≤ P ⊗ PvP ⊗ P + P ⊗ PvQ ⊗ Q + Q ⊗ QvP ⊗ P Similarly, treating w(x − R) as a one-body multiplication operator parametrized by R, we have Taking into account that one easily arrives, after computing the relevant second quantization representations of the operators appearing in the bounds (3.8)- (3.10), at the desired result. Since this is essentially the same computation as in [8,Sec. 5], we omit the details.
The operator inequalities in Lemma 3.1 quantify the effect of dropping the higher order terms in the creation and annihilation operators appearing in the original Hamiltonian. As a second step, we now estimate the effect of the Bogoliubov substitution a 0 , a † 0 → √ N ∈ R by using the unitary transform U N , which replaces the a 0 , a † 0 by an operator √ N − N + acting on

Lemma 3.2.
We have the following inequality for all Φ ∈ F ≤N + : where the positive constants α ′ , β ′ do not depend on N .
Proof. By using the algebraic properties (1.15)-(1.17) of U we see that the expressions to estimate are the following. First, using (1.15), which gives an expression of the type claimed in the Proposition for ǫ −1 = N 2 /(N − 1). In the above, we summed the Cauchy-Schwarz inequality for A = a † e −ipR and B = w p (1 − (N − N + )/N ), and used the bound Similarly, from (1.17), we arrive at the second term to estimate: for ǫ −1 = N − 1. We used (3.14) for A = a † p a † −p and Similarly, By combining these inequalities, we obtain the desired bound.
The main result of this section, Proposition 3.1, is a direct consequence of the last two Lemmas.

Proof of Theorem 1
For brevity we denote

Lower bound
Let

Upper bound
For the upper bound, we use Fock space localization. It is quantified by the following result [10,29].
for all M ∈ N. Here f M denotes the operator and analogously for g M .
For the proof, see [10,Appendix B]. Proposition 4.1 can be used to quantify the error made by constraining the states on Fock space to contain only up to M particles. The proof is based on an IMS-type argument, which yields that this error scales as M −2 . From the Proposition, we deduce Lemma 4.1. There exist non-negative constants C, K such that Proof. We apply Proposition 4.1 for A = H F − e 0 (H F ). From Lemma 2.3 it follows e 0 (H F ) ≥ −C 7 /C 6 and further that jP , which leads to the right hand side of the claimed inequality, with σ = 2. Using which is a contradiction for large N and small ξ/N .
Let us now take Y ⊂ F + to be the spectral subspace of H F corresponding to energies E ≤ e i (H F ), and let 1 ≤ ξ ≤ N . The bound (4.1) together with the upper bound of Lemma 2.1 implies that e i (H F ) ≤ Cξ, and hence also (Φ, (H F ) k Φ) ≤ Cξ k for k = 1, 2 for any Φ ∈ Y . By Lemma 4.1 and Proposition 3.1 (with the choice ǫ = ξ/N ) we have By taking the expectation value in any normalized Φ ∈ Y , we obtain, by Lemmas 2.3 and 2.4 and the simple inequalities N k Since for some C > 0, which is the desired bound.

Existence of eigenvectors
We shall now conclude the existence of eigenvectors of H N and H F by showing that these operators have compact resolvents. By the definition of compactness and the spectral theorem one easily sees that if A ≥ B > 0, then the compactness of B −1 implies the compactness of A −1 . Since the particles are confined to the unit torus, for any ǫ > 0 the operators T + ǫ and P 2 + ǫ are strictly positive and have purely discrete spectra with eigenvalues accumulating at infinity; therefore, they have a compact inverse. The same observation applies to the operator since lim |p|→∞ e p = ∞ and inf p e p > 0. Since H N ≥ T + P 2 2M , we conclude that H N has compact resolvent, which, by the spectral theorem, implies that the spectrum of H N is discrete and eigenvectors exist. On the other hand, by completing the square, as in Lemma 2.3, it is easy to see that for appropriate constants c, d > 0. The existence of eigenvectors of H F , along with the fact that its spectrum is discrete, follows now from precisely the same reasoning as above. This proves the first part of Theorem 2. We first show the convergence for ground states. By writing U † Ψ 0 = a N + b N , a N ∈ P 0 F + and b N ∈ P 0 F ⊥ + , we have (1 −P k )U Ψ i 2 , (5.6) which can be rewritten as

Convergence of eigenvectors
Note that the last term goes to zero as N → ∞ by (5.3). Take now l to be the largest integer such that e l (H F ) < e k (H F ). The dimension of the eigenspace corresponding to e k (H F ) therefore equals k − l. We have the simple identity (U Ψ i ,P k U Ψ i ) (5.8) (note the presence of both tilded and untilded operators). For the first two terms, we can use (5.7) for a lower bound. Moreover, since the Ψ i are orthonormal, we have k i=0 (U Ψ i ,P l U Ψ i ) ≤ TrP l = l + 1. The last term in (5.8) is trivially bounded from below by −(l + 1). We thus conclude that where the quantities C N > 0, D N > 0 can be read off from (5.7) and vanish as N → ∞, because of (5.3). Therefore, k i=l+1 (U Ψ i , P k U Ψ i ) → k − l, but as each individual term in the sum is ≤ 1, we must have lim(U Ψ i P k U Ψ i ) = 1 for every eigenstate of H ′ N with energy e k (H ′ N ). This is precisely the convergence result stated in Theorem 2, whose proof is now complete.