Quantum corrections to the Pekar asymptotics of a strongly coupled polaron

We consider the Fr\"ohlich polaron model in the strong coupling limit. It is well known that to leading order the ground state energy is given by the (classical) Pekar energy. In this work, we establish the subleading correction, describing quantum fluctuation about the classical limit. Our proof applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a set of finite volume, with linear size determined by the natural length scale of the Pekar problem.


Introduction
The polaron model was introduced by Fröhlich [9] as a model of an electron interacting with the quantized optical modes of a polar crystal. It represents a simple and well-studied model of non-relativistic quantum field theory, and we refer to [1,7,10,21,28] for properties, results and further references.
In the strong coupling limit α Ñ 8, the model allows for an exact solution, in the sense that the ground state energy asymptotically equals the one given by the Pekar approximation [25], which amounts to a classical approximation to the quantum field theory. This was first shown by Donsker and Varadhan [4] using a path integral formulation of the problem. (See also [22,23] for recent work on the construction of the Pekar process [28].) Later the result was improved by Lieb and Thomas [19] who provided a quantitative bound on the difference.
We are interested here in the subleading correction to the classical (Pekar) approximation. It was predicted by Allcock [2] that this correction results from quantum fluctuations about the classical limit, and is Opα´2q smaller than the main term. It can be calculated by evaluating the ground state energy of a system of (infinitely many) harmonic oscillators with frequencies determined by the Hessian of the Pekar functional. Allcock's result is verified rigorously in this paper, by giving upper and lower bounds on the ground state energy of the Fröhlich polaron model that establish this subleading correction. Our analysis applies to a model of a confined polaron, where both the electron and the polarization field are restricted to a finite volume (with linear size of the natural length scale set by the Pekar problem).
The confinement breaks translation invariance, which removes zero modes otherwise present in the Hessian of the Pekar functional, and avoids having to localize the electron on the Pekar scale, which simplifies the problem. The singular ultraviolet behavior is unaffected by the confinement, however, and represents one of the main technical challenges. A key ingredient in our analysis is a multiple use of the commutator method of Lieb and Yamazaki [20], combined with Nelson's Gross transformation [13,24]. with a parameter α ą 0. The field energy is given by the number operator N " ř j a : pϕ j qapϕ j q for some orthonormal basis tϕ j u in L 2 pΩq, with spectrum σpNq " α´2t0, 1, 2, . . .u. We are interested in the ground state energy of H as α Ñ 8.

Model and Main Results
We note that the expression (2.1) is somewhat formal, since v x R L 2 pΩq and hence a : pv x q is not densely defined. The operator H can be defined with the aid of its corresponding quadratic form, however. It is in fact well known that H defines a self-adjoint operator on a suitable domain, see [12] or Section 6 below.
Remark 2.2. Typically the polaron model is considered without confinement, i.e., for Ω " R 3 , in which case the electron-phonon coupling function equals p´∆ R 3 q´1 {2 px, yq " p2πq´3 ş R 3 e ik¨px´yq |k|´1dk " p2π 2 q´1|x´y|´2. For the proof of our main theorem the compactness of p´∆ Ω q´1 will be important, hence we need to consider bounded sets Ω here.
Our proof works under the more general assumption that the set of minimizers of E is discrete (up to the phase degeneracy). The case where minimizers form a continuous manifold requires additional ideas, however.
(2.6) Recall that, for Ω " R 3 , uniqueness of minimizers of E P (up to translations and phase factor) is known [17] (see also [29]). We expect Assumption 1 to hold if Ω is convex, for instance. The proof in [17] can be adapted to show uniqueness in case Ω is a ball [5].
Assumption 2. There exists a κ ą 0 such that E P pψq ě E P pψ P q`κ ż Ωˇ∇`ψ´ψ P˘ˇ2 @ψ P H 1 0 pΩq, ψ ě 0, }ψ} 2 " 1 . (2.7) The bound (2.7) follows from an a priori weaker spectral assumption on the absence of non-trivial zero modes of the Hessian of E P at its minimizer ψ P , by a simple compactness argument. For completeness, we spell out the details of this argument in Appendix A. The analogue of this spectral assumption in the case Ω " R 3 is known (up to zero-modes resulting from the translation invariance) [16,30]. Using the method in [16], one can prove Assumption 2 in case Ω is a ball [5].
(2.9) An explicit computation gives where ψ P acts as a multiplication operator, µ P " inf specp´∆ Ω`Vϕ P q " e P´} ϕ P } 2 2 , and Q P is the projection orthogonal to ψ P , i.e., orthogonal to the kernel of´∆ ΩV ϕ P´µ P . It is not difficult to see that Assumption 2 implies that H P is non-degenerate, i.e., strictly positive (compare with Proposition 3.2 in Section 3.2 below).
Finally, we need a regularity assumption on the domain Ω.
For a proper definition of the meaning of C 3,δ boundary, see Appendix B. Assumption 3 allows to estimate derivatives of the integral kernel of certain functions of the Dirichlet Laplacian (see Appendix C). The required estimates certainly hold under less restrictive assumptions on Ω, and we expect our main result to hold also in case Ω is a cube, for instance. We shall not try to investigate the minimal regularity assumptions, however, and shall henceforth work with Assumption 3.

Main Result.
Recall the definition (2.5) for the Pekar energy e P , as well as (2.10) for the Hessian H P of F P in (2.8) at the unique minimizer ϕ P . Our main result is as follows. hold for some constant C ą 0 and α large enough.
The trace in (2.11) and (2.12) is over L 2 pΩq. We shall see below that 1´?H P is actually trace class. Note also that H P ă 1, hence the coefficient of α´2 in (2.11) is strictly negative.
In the case Ω " R 3 , the correctness of the leading term e P was shown in [4,19]. The proof in [19] gives an error bound of the order α´1 {5 . In the confined case considered here, we improve this error bound to Opα´2q, and actually compute the next order correction. We conjecture that the formula (2.11) also holds true in case Ω " R 3 , as predicted in the physics literature [2,14].
The α´2 correction to the ground state energy in (2.11) can be interpreted as arising from quantum fluctuations around the classical limit described by the Pekar functional.
In fact, the trace originates from the ground state energy of a Hamiltonian describing a system of (infinitely) many harmonic oscillators.
The remainder of the paper is devoted to the proof of Theorem 2.3. We start with a brief outline to guide the reader.

2.4.
Outline of the Proof. In Section 3 we study the Pekar functional (2.8). We shall compute its Hessian at the unique minimizer ϕ P , and use it to estimate the functional in a small neighborhood of its minimizer. We shall also derive a useful quadratic lower bound that is valid globally, i.e., not just close to the minimizer.
In Section 4 we shall derive an upper bound on the ground state energy of H that has the desired asymptotic form as α Ñ 8. We shall construct an appropriate trial state and utilize the estimate of the Pekar functional close to its minimizer from the previous section.
Sections 5 and 6 contain auxiliary results that are essential for the lower bound, in particular to allow for an ultraviolet regularization of the problem. In Section 5 the commutator method of Lieb and Yamazaki [20] is applied multiple (in fact, three) times in order to estimate the effect of an ultraviolet cutoff in the coupling function v x in terms of the number operator N and the electron kinetic energy´∆ Ω . In fact, the relevant operator that needs to be bounded is N 1{2 p´∆ Ω q 3{2 , which cannot be controlled in terms of H 2 , however. The necessary bound does hold after a unitary Gross transformation, which shall be explained in Section 6. This will be sufficient for our purpose.
In Section 7 we shall give a lower bound on the ground state energy of H of the desired asymptotic form. We shall use the results of Sections 5 and 6 to implement an ultraviolet cutoff, which effectively reduces to problem to finitely many modes. We shall then use an IMS localization in Fock space and the bounds in Section 3 to conclude the desired lower bound.
In Appendix A we shall give an equivalent formulation of Assumption 2 in terms of spectral properties of the Hessian of E P . In further appendices we shall derive bounds on derivatives of the integral kernel of certain functions of the Dirichlet Laplacian ∆ Ω that we need in our proof. These bounds are derived in Appendix C utilizing a theorem in Appendix B on bounds on solutions of Poisson's equation.
Throughout the proof, we shall use the symbol a À b if a ď Cb for some constant C ą 0. Recall that for ϕ P L 2 R pR 3 q we set V ϕ "´2p´∆ Ω q´1 {2 ϕ. In this section we work under Assumption 1 which states that F P pϕq has a unique minimizer ϕ P . We have epϕq`}ϕ} 2 2 ě epϕ P q`}ϕ P } 2 2 and our goal in this section is to obtain upper and lower bounds on the difference.

The Pekar Functional
Recall that ψ P denotes the unique non-negative minimizer of E P , which is the ground state of H ϕ P . We have For later use, we record that ψ P is a bounded function.
Let P " |ψ P yxψ P | and Q " 1´P . We introduce the following non-negative operators and where ψ P acts as a multiplication operator. We shall see that K " 1´H P , where H P denotes the Hessian of F P pϕq at ϕ " ϕ P , introduced in (2.9) above. It is easy to see that L is trace class, since p´∆ Ω q´1 {2 ψp´∆ Ω q´1 {2 is Hilbert-Schmidt for any multiplication operator ψ P L 2 pΩq. In fact, since p´∆ Ω q´1 {2 ď ? 2p´∆ Ωè 1 q´1 {2 (with e 1 " inf specp´∆ Ω q ą 0) and p´∆ Ω`e1 q´1 {2 px, yq ď p´∆ R 3`e 1 q´1 {2 px, yq for any x, y P R 3 by (C.2), the Cauchy-Schwarz inequality implies that To show that also K is trace class, we shall first prove the following lemma, which implies, in particular, that V ϕ is operator-bounded relative to´∆ Ω if ϕ P L 2 pΩq.
Proof. For any ψ P L 2 pΩq, we can write where on the right side ψ is understood as a multiplication operator. By arguing as in (3.7) one readily checks that }p´∆ Ω q´1ψ} À }ψ} 2 , which implies the result. (In fact, this bound even holds with the Hilbert-Schmidt norm on the left side.) Lemma 3.2 readily implies that p´∆ Ω q 1{2 Q H ϕ P´e pϕ P q p´∆ Ω q 1{2 is bounded, hence the trace class property of K follows from the one of L.
Our main result in this section is the following.
for ε ą 0 small enough. TheňˇF P pϕq´F P pϕ P q´xϕ´ϕ P |1´K|ϕ´ϕ P yˇˇÀ ε xϕ´ϕ P |L|ϕ´ϕ P y . (3.11) Note that this result implies, in particular, that 0 ď K ď 1. It identifies H P " 1´K as the Hessian of F P pϕq " epϕq`}ϕ} 2 2 at the minimizer ϕ P . Our assumption on the strict positivity of the Hessian thus translates, in view of the compactness of K, to the statement }K} ă 1.
Proof. By choosing ε ą 0 small enough and arguing as in the proof of Lemma 3.2 we can ensure that the family of operators´∆ Ω`Vϕ pxq has a unique eigenvalue close to epϕ P q and this eigenvalue is epϕq. The rest of the spectrum of H ϕ is uniformly bounded away from epϕ P q. Hence we can write for a fixed (i.e., ϕ-independent) contour C that encircles epϕ P q.
We also have that ∆ Ω pz´H ϕ P q´1 is uniformly bounded for z P C. This follows from the fact that V ϕ P is infinitesimally bounded relative to´∆ Ω , which can easily be checked following the proof of Lemma 3.2. Hence for small ε, by Lemma 3.2 and our assumption (3.10). We can thus use the resolvent identity in the form 14) The first term on the right side is analytic in z for all z inside the contour C, and hence gives zero after integration when inserted in (3.12). The second term is rank one, and Fubini's theorem implies that we can interchange the trace and the integral after inserting this term in (3.12). We thus obtain For simplicity, let us introduce the notation Note that due to (3.13) these operators are smaller than 1 in norm, uniformly in z P C. We shall use the identity 1 1´A We insert the various terms into (3.15) and do the contour integration. The term 1 then yields epϕ P q. The term A yields using (3.3). A standard calculation shows that the term ApA`Bq leads to xψ P |V ϕ´ϕ P Q epϕ P q´H ϕ P V ϕ´ϕ P |ψ P y "´xϕ´ϕ P |K|ϕ´ϕ P y . Furthermore, since Q|ψ P y " 0, the term Bp1´Bq´1 yields zero. We conclude that To bound the first term on the right side of (3.20), note that We claim that which implies that (3.21) is bounded, in absolute value, as as desired. To prove (3.22) we use the fact that}p´∆ Ω q 1{2 pz´H ϕ P q´1p´∆ Ω q 1{2 } is uniformly bounded to reduce the problem to showing }p´∆ Ω q´1 {2 Ap1´Aq´1p´∆ Ω q 1{2 } À ε. Since S´1Ap1´Aq´1S " S´1ASp1´S´1ASq with S " p´∆ Ω q 1{2 , it suffices to show that }p´∆ Ω q´1 {2 Ap´∆ Ω q 1{2 } À ε, which follows from }p´∆ Ω q´1 {2 V ϕ p´∆ Ω q´1 {2 } ď }V ϕ p´∆ Ω q´1} and Lemma 3.2. For the last term in (3.20), we simply bounďˇˇˇB The same bounds as above easily lead to the conclusion that also this term is bounded by the right side of (3.23). This concludes the proof of Proposition 3.1.

3.2.
A Uniform Quadratic Lower Bound. Inequality (3.11) gives a bound on F P for ϕ near the minimizer ϕ P . We shall also need the following rougher global bound.
We start with the following lemma.
Proof. With f pxq " |ψpxq|`|ψ P pxq| and gpxq " |ψpxq|´|ψ P pxq|, the Schwarz inequality and the symmetry and positivity of the integral kernel of p´∆ Ω q´1 {2 imply that For fixed x, we can use the Hardy inequality and the fact that Since ş Ω f 2 ď 4, the result follows. Proof of Proposition 3.2. From our assumption (2.7) on the Hessian of the Pekar functional E P and Lemma 3.3, it follows that Minimizing with respect to ψ and using (3.3) leads to the desired lower bound.

Proof of Theorem 2.3: Upper bound
In this section we construct a trial state to derive an upper bound on the polaron ground state energy. We think of Fock space as the space L 2 pR 8 q (called Q-space in [26]) with Gaussian measure given by Ś ně1 a 2{παe´2 α 2 λ 2 n dλ n , where the λ n are the coefficients of ϕ in some fixed basis tϕ n u ně1 of L 2 R pΩq. In this representation, the number operator N acts as N " ř ně1 p´p4α 4 q´1B λ 2 n`α´2 λ n B λn q, and the vacuum vector is simply represented by the constant function.
Our trial state Ψ will depend only on finitely many variables tλ n u N n"1 , hence the problem effectively reduces to one on L 2 pR N q. We find it more convenient to work with the flat Lebesgue measure on this space, instead of the Gaussian measure mentioned above, which is legitimate for N ă 8. In this representation, the trial state is chosen as follows: where ‚ ε ą 0 is a small parameter that will be chosen to go to zero as α Ñ 8. ‚ 0 ď χ ď 1 is a smooth cut-off function with χptq " 1 for t ď 1{2 and χptq " 0 for t ě 1 ‚ Π is a finite rank projection on L 2 R pΩq, with range containing ϕ P . ‚ ψ ϕ is the unique non-negative, normalized ground state of H ϕ "´∆ Ω`Vϕ ‚ K " 1´H P , explicitly given in (3.5).
Using the eigenvalue equation for ψ ϕ , the energy of our trial state Ψ is given as Since our state corresponds to the vacuum for all modes outside the range of Π, we have For an upper bound on epΠϕq, we use Proposition 3.1. This leads to xΨ |H| Ψy ď e P xΨ|Ψy`@ΨˇˇN´}Πϕ} 2 2`x Πϕ´ϕ P |1´K`εCL|Πϕ´ϕ P yˇˇΨ D (4.4) for a suitable constant C ą 0.
Let N " dim ran Π and let tϕ n u N n"1 be an orthonormal basis of the range of Π. Then, with Πϕ " ř N n"1 λ n ϕ n , Utilizing the fact that the Gaussian factor in Ψ satisfies´1 we can integrate by parts and rewrite the right side of (4.4) aŝ We claim that L is bounded by p´∆ Ω q´1. This follows immediately from the boundedness of ψ P shown in Lemma 3.1. Alternatively, one can use that ψp´∆ Ω q´1ψ is a bounded operator for ψ P L 3 pΩq by Sobolev's inequality. Hence we can use the rough bound A À ε 3 xΨ|Ψy . (4.10) Moreover, by a simple Cauchy-Schwarz inequality, B ď 2pB 1`B2 q with and To bound B 1 , we use standard first order perturbation theory for eigenvectors to compute where Q Πϕ " 1´|ψ Πϕ yxψ Πϕ |. In particular, where we again interpret ψ Πϕ as multiplication operator on the right side. It is not difficult to see that p´∆ Ω q 1{2 Q Πϕ H Πϕ´e pΠϕq p´∆ Ω q 1{2 is uniformly bounded on the support of χ (compare with the proof of Proposition 3.1). Using this fact and (3.7), we see that (4.14) is uniformly bounded, independently of N. Hence B 1 À α´4xΨ|Ψy.
For B 2 , we have where we have used the fact that ϕ P is in the range of Π. We have to compare this with the norm of Ψ, which is bounded from below by Since }K} ă 1 by assumption, p´∆ Ω q´1 ď νp1´ΠKΠq 1{2 for some constant ν ą 0 independent of N. Hence we can bound the characteristic function of S ε from above by expp´1 4ν α 2 ε 2 qˆexppα 2 @ Πϕ´ϕ Pˇp 1´ΠKΠq 1{2ˇΠ ϕ´ϕ P D q. Therefore, In particular, as long as αε ě const.
? N with a sufficiently large constant, we have xΨ|Ψy Á detp1´ΠKΠq´1 {4 , and hence In summary, we have shown that as long as αε ě const. ? N and ε is small enough. We shall choose Π to be the projection onto the span of g 1 , . . . , g N´1 , ϕ P , where we denote by tg j u j an orthonormal basis of eigenfunctions of K, ordered in a way that the corresponding eigenvalues k j " xg j |Kg j y form a decreasing sequence 1 . Then and hence Since Ω is assumed to be a smooth and bounded domain, we have the Weyl asymptotics e j " j 2{3 for j " 1 (see, e.g., [27, Sec. XIII.15]), which implies that In order to minimize the error term, we shall choose ε " α´8 {11 and N " α 2 ε 2 " α 6{11 , which leads to the bound for large enough α. This concludes the proof of the upper bound in Theorem 2.3.

Multiple Lieb-Yamazaki Bound
In [20] Lieb and Yamazaki used the fact that the interaction between the particle and the field can be written as a commutator, together with a Cauchy-Schwarz inequality, to get a uniform lower bound on the ground state energy of H (for Ω " R 3 ) for large α. In fact, their method shows that the introduction of an ultraviolet cutoff Λ in the interaction affects the ground state energy at most by OpΛ´1 {2 q. We shall apply their method multiple (in fact, three) times, which will allow us to conclude that the effect of the cutoff is in fact at most OpΛ´5 {2 q (up to logarithmic corrections). It will be essential to use the Gross transformation explained in the next section, however, since we need relative operator boundedness of the kinetic energy with respect to the full Hamiltonian, which only holds for the transformed kinetic energy, as we shall see.
Before stating the main result of this section, we shall prove the following useful lemma. Its proof proceeds similarly to the one of Lemma 10 in [8]. For its statement, we introduce the Coulomb norm By the Hardy-Littlewood-Sobolev inequality (see, e.g., [18,Thm. 4.3]), this norm is dominated by the L 6{5 pR 3 q-norm. Let us introduce the notation p "´i∇ x " pp 1 , p 2 , p 3 q for the momentum operator. We shall also use p 2 for the Dirichlet Laplacian´∆ Ω on Ω.
Note that the bound holds trivially with the right side replaced by }h x } 2 2 N. The point of Lemma 5.1 is that functions that are more singular (in the x´y variable) can be handled, at the expense of the kinetic energy term p 2 .
Proof. For convenience of notation, let Ψ be a one-phonon vector; the general case works in the same way. We need to bound With Φpp, qq denoting the Fourier transform of |Ψpx, yq| (regarded as a function on R 3ˆR3 ), we have Φpp´q, qqkpqq dqˇˇˇˇ2 dp |Φpp, qq| 2 p 2 dq dp .
Note that the last factor is smaller than } ? N a p 2 Ψ} 2 (by the diamagnetic inequality). By writing the integral in x-space, one easily checks that hence our claim (5.2) is proven.
The main result of this section is the following.
Lemma 5.2. Assume that w x p¨q is such that and Then Applying this three times, we also get In particular, we conclude that apw x q`a : pw x q " ÿ j,k,l rp j , rp k , rp l , a : pp j p k p l |p|´6w x q´app j p k p l |p|´6w x qsss . (5.12) We introduce the notation B jkl " a : pp j p k p l |p|´6w x q´app j p k p l |p|´6w x q, and rewrite the triple commutator as using the invariance of B jkl under exchange of indices. The Cauchy-Schwarz inequality implies that for any λ ą 0. Moreover, We also have C jk " ÿ l rp l , B jkl s " a : pp j p k |p|´4w x q`app j p k |p|´4w x q (5.17) and for any λ ą 0. Furthermore, we can bound By Lemma 5.1, the first term on the right side is bounded by 4}u jk } 2 C p 2 N, and hence In combination with (5.12), (5.13) and (5.16), this concludes the proof of the lemma.
In the following, we shall apply this bound to the large momentum part of the interaction, in order to quantify the effect of an ultraviolet cutoff on the ground state energy. Because the Coulomb norm in (5.8) estimates the off-diagonal decay, we cannot use a sharp cutoff, however, and need to work with a smooth one instead. In fact, we shall apply Lemma 5.2 with w x pyq " zp´∆ Ω qpx, yq for zptq " t´1 {2´1´e´t{Λ 2¯2 (5.21) for some Λ ą 0. The function z is non-negative, and behaves like t 3{2 Λ´4 for t ! Λ 2 . Moreover, zptq´t´1 {2 falls off like t´1 {2 e´t {Λ 2 for t " Λ 2 . We shall show in Appendix C that the various norms appearing in (5.6)-(5.8) can be bounded, up to a multiplicative constant, by the equivalent expressions for Ω " R 3 , which can easily be estimated using Fourier transforms. In fact, we have where e n and ϕ n denote the eigenvalues and eigenfunctions of´∆ Ω . In particular, from (C.13) we deduce that In the same way, we obtain the bound sup xPΩ max j,k,l }p j p k p l |p|´6w x } 2 À Λ´5 {2 . We collect these results in the following corollary.

Gross Transformation
In this section we shall investigate the effect of a unitary Gross transformation [13,24] on the Hamiltonian (2.1). Let tf x u xPΩ Ă L 2 pΩq be a family of functions, parametrized by x P Ω, such that ∇ x f x P L 2 pΩq for all x P Ω. We consider a unitary transformation in L 2 pΩq b F of the form U " e apα 2 fxq´a : pα 2 fxq .
(6.1) (This operator acts by 'multiplication' with respect to the x variable.) For g P L 2 pΩq we have UapgqU : " apgq`xg|f x y and Ua : pgqU : " a : pgq`xf x |gy (6.2) and hence Moreover, for p "´i∇ x , We shall choose f x real-valued, hence the last term vanishes. Then Up 2 U : " p 2`α4`a: ppf x q`appf x q˘2 2α 2 p¨appf x q`2α 2 a : ppf x q¨p`α 2 app 2 f x q`α 2 a : pp 2 f x q . (6.5) For the Hamiltonian (2.1), we thus have UHU : " p 2`α4`a: ppf x q`appf x q˘2`2α 2 p¨appf x q`2α 2 a : ppf x q¨p We shall choose f x such that α 2 p 2 f x`fx´vx " g x , i.e., f¨pyq "`´α 2 ∆ Ω`1˘´1 pg¨pyq`v¨pyqq @y P Ω (6.7) for some g x P L 2 pΩq with sup xPΩ }g x } 2 ă 8. In fact, g x " 0 would be possible, but it will be more convenient to choose g x pyq " ξp´∆ Ω qpx, yq for ξptq "´t´1 {2 θpK 2´t q (6.8) for t ă 0 1/2 for t " 0 1 for t ą 0. (6.9) Then }g x } 2 2 " ξ 2 p´∆ Ω qpx, xq (6.10) and, since ξptq 2 ď t´1e 1´t{K 2 , the fact that the heat kernel of ∆ Ω is dominated by the one of ∆ R 3 implies as in (C.2) that For the corresponding f x , we have Using the fact that one obtains in a similar way as above α´4K´3 . (6.14) Moreover, and, using (6.13) and (C. 13), With the above choice of the function f x (depending on α and the parameter K) we denote U by U K,α from now on. With the aid of the previous estimates, we can now prove the following proposition. Its proof follows along similar lines as the corresponding argument for Ω " R 3 in [12]. Proposition 6.1. For any ε ą 0 there are K ą 0 and C ą 0 such that for all α Á 1 and any Ψ P L 2 pΩq b F in the domain of p 2`N p1`εq}pp 2`N qΨ}`C}Ψ} ě }U K,α HU : K,α Ψ} ě p1´εq}pp 2`N qΨ}´C}Ψ} . (6.17) We remark that due to the singular nature of v x in the interaction term, it is essential to apply the unitary transformation U K,α . In its absence, the bound (6.17) fails to hold. In other words, the domain of H does not coincide with the domain of p 2`N , but the one of U K,α HU : K,α does for a suitable choice of K. Proof. From (6.6) we see that the terms to estimate are the following: where we used (6.16), for any δ ą 0, using (6.11), and finally, the term The first term on the right side of (6.21) can be estimated as in (6.20) above. For the second term, we write The L 2 -norms of g x and f x have already been bounded above, in (6.11) and (6.14), respectively. To bound the third function in h p1q x , we use 0 ď t´1 {2´p K 2`t q´1 {2 ď Kt´1 {2 pK 2`t q´1 {2 , and find that the square of its L 2 -norm is bounded by By using the Schwarz inequality we conclude that for any δ ą 0. The last term to estimate is aph p2q x qΨ. Since |h p2q x pyq| ď pK 2´∆ R 3 q´1 {2 px, yq, Lemma 5.1 implies that }aph p2q x qΨ} ď p2πq´3 {2ˆż The prefactor on the right side is equal to a constant times K´1 {2 . Moreover, we can bound } ? N a p 2 Ψ} ď 1 2 }pp 2`N qΨ}. In combination with (6.14) and (6.15), we hence arrive at the desired result, with K " ε´2 and C " ε´1.
From Proposition 6.1 we draw two important conclusions. First, the ground state energy of H is uniformly bounded in α, for large α. Second, in any state of bounded energy, in the sense that }HΨ} ď const., both }U : K,α p 2 U K,α Ψ} and }U : K,α NU K,α Ψ} are uniformly bounded (for suitable K independent of α). In particular, we conclude that in order to compute the ground state energy, it suffices to consider wave functions Ψ having this property.
We have, by a similar computation as in (6.4), U : K,α p 2 U K,α " pp´A K,α q 2 with A K,α " α 2`a: ppf x q`appf x q˘(6.28) and U : K,α NU K,α " N´apf x q´a : pf x q`}f x } 2 2 . (6.29) Since }f x } 2 is uniformly bounded, as shown in (6.14) above, it easily follows that uniform boundedness of }U : K,α NU K,α Ψ} is equivalent to the one of }NΨ}.

Proof of Theorem 2.3: Lower Bound
7.1. Ultraviolet Cutoff. The first step in the lower bound is to introduce an ultraviolet cutoff in the interaction. Corollary 5.1 together with Proposition 6.1 will allow us to quantify its effect on the ground state energy. for α Á 1 and Λ Á 1.
Note that in order for the error introduced in (7.3) to be negligible compared to α´2, it is sufficient to choose Λ " α κ with κ ą 4{5. Proof.
(7.4) Corollary 5.1 states that apw x q`a : pw x q À`p 2`N`1˘2`Λ1´5{2`α´1 Λ 1´3{2˘( 7.5) for α Á 1. We now apply the unitary Gross transformation (6.1), with f x given in (6.12), and K chosen such that Proposition 6.1 holds for some fixed 0 ă ε ă 1, say ε " 1{2. We have U : K,α apw x qU K,α " apw x q`xw x |f x y (7.6) and sup xPΩ |xw x |f x y| À α´2Λ 1´1 (7.7) which can easily be seen by noting that xw x |f x y " pzηqp´∆ Ω qpx, xq (with z and η defined in (7.4) and (6.12), respectively) and using that |zptqηptq| À α´2pt`Λ 2 q´2, proceeding as in (C.2) to bound the expression in terms of the one for Ω " R 3 . Proposition 6.1 thus implies that apw x q`a : pw x q À pH`Cq 2`Λ1´5{2`α´1 Λ 1´3{2`α´2 Λ 1´1˘( 7.8) for a suitable constant C ą 0 (independent of α for α Á 1). For computing the ground state energy, it is clearly sufficient to consider wave functions in the spectral subspace of H corresponding to |H| ď C for a suitable constant C. We thus conclude that Step 2. We shall now further truncate u Λ 1 x . In fact, we shall replace it bỹ v Λ x pyq " With the aid of (C.5), one checks that and hence, using the fact that ? N is uniformly bounded for states with bounded energy, the error for introducing this additional cutoff is at most of order Λ 1{2 e´p Λ{Λ 1 q 2 {2 .
Step 3. Finally, we want to further simplifyṽ Λ x and replace it by v Λ x in (7.2). We claim that the ground state energy can only decrease under this replacement. This is the content of the following lemma. Then epλ 1 , . . . , λ N q " inf spec "´∆ Ω´a pu x q´a : pu x q`N ‰ (7.14) is decreasing in each λ j .
(7.16) By multiplying the functions a n i 1 ,...,in with an appropriate phase factor, we can make sure that ż Ω a n i 1 ,...,in pxqa n`1 i 1 ,...,in,j pxqϕ j pxq dx ě 0 (7.17) for all n ě 0, 1 ď j ď N and all i 1 , . . . , i n , and this can clearly only decrease the energy. When computing the ground state energy, it suffices to consider Ψs with such property, in which case the energy is clearly monotone decreasing in all the λ j .

7.2.
Final Lower Bound. The starting point of the proof of the lower bound is Proposition 7.1, which quantifies the error in replacing H by H Λ in (7.1) for computing the ground state energy. We are thus left with giving a lower bound on inf spec H Λ .
We choose, for simplicity, Λ in such a way that Λ 2 is not an eigenvalue of´∆ Ω . Let Π denote the projection Π " θpΛ 2`∆ Ω q and N " dim ran Π .
For later purposes we note that one has the Weyl asymptotics has finite rank. The Fock space F pL 2 pΩqq naturally factors into a tensor product F pΠL 2 pΩqq b F pp1´ΠqL 2 pΩqq, and H Λ is of the form A b 1`1 b N ą , where A acts on L 2 pΩq b F pΠL 2 pΩqq and N ą " ř nąN a : pϕ n qapϕ n q is the number operator on F pp1´ΠqL 2 pΩqq. In particular, inf spec H Λ " inf spec A.
As in Section 4 we shall use the Q-space representation (with a different basis, however), which identifies F pΠL 2 pΩqq with L 2 pR N q via the representation ϕ " Πϕ " N ÿ n"1 λ n ϕ n , (7.22) thus identifying a function ϕ P ran Π with a point pλ 1 , . . . , λ N q P R N . In this representation, we have 2α 2˙( 7.23) For a lower bound, we can replace´∆ Ω`Vϕ pxq by the infimum of its spectrum, for any fixed ϕ P ran Π. In particular, we have inf spec H Λ ě inf spec K (7.24) where K is the operator on L 2 pR N q with F P defined in (2.8). Here F P pϕq is a function of pλ 1 , . . . , λ N q via the identification (7.22). We now introduce an IMS type localization. Let χ : R`Ñ r0, 1s be a smooth function with χptq " 1 for t ď 1{2, χptq " 0 for t ě 1. Let ε ą 0, and let j 1 and j 2 denote the multiplication operators in L 2 pR N q 1´χpε´1}p´∆ Ω q´1 {2 pϕ´ϕ P q} 2 q 2 . (7.26) Then clearly j 2 1`j 2 2 " 1 and K " j 1 Kj 1`j2 Kj 2´E (7.27) where E is the IMS localization error It is easy to see that that E À α´4ε´2, independently of N. In particular, the localization error is negligible if ε " α´1.
On the support of j 1 , we can use the bound (3.11) on F P . This gives for C a positive constant. Now ϕ P will not necessarily be in the range of Π. However, since 1´K´εCL is positive for ε small enough, we can replace ϕ P by its closest point (in the norm defined via 1´K´εCL) in the range of Π for a lower bound. That is, xϕ´ϕ P |1´K´εCL|ϕ´ϕ P y ě xϕ´y|Πp1´K´εCLqΠ|ϕ´yy (7.30) where y " pΠp1´K´εCLqΠq´1Πp1´K´εCLqϕ P . The shift by y can be removed by a unitary transformation, without affecting the ground state energy. Hence This is of the correct form if N Ñ 8 and ε Ñ 0 as α Ñ 8.
On the support of j 2 , we use the bound (3.25) instead. We have, for any η ě 0, where we have used the fact that }p´∆ Ω q´1 {2 pϕ´ϕ P q} 2 ě ε{2 on the support of j 2 . We choose η independent of α (and hence also independent of Λ and ε) and small enough such that the operator in the last line is positive. Proceeding as in the case of j 1 above, we obtain ˙.

(7.33)
From the Weyl asymptotics (7.20) one checks that the trace diverges like N 2{3 " Λ 2 for large Λ. Hence if we choose Λα´1 ď const. ε with a sufficiently small constant, the term in parenthesis in (7.33) is actually larger than e P . Since we will choose Λ " α κ with κ ą 4{5, this is compatible with the condition ε ! 1 as long as κ ă 1. We thus conclude that if Λα´1 ď const. ε and ε is small enough, we have the bound inf spec K ě e P´1 2α 2 Tr´1´a1´ΠpK`εCLqΠ¯´const. α´4ε´2 .
For a lower bound, we can further drop the Π's in the second term on the right side, and replace them by 1. Note that }K`εCL} ď ν ă 1 for small enough ε, and the function f ptq " 1´?1´t is Lipschitz continuous and convex on r0, νs. We utilize the following simple lemma.
The convexity of f implies that f pxg j |A|g j yq ď xg j |f pAq|g j y, which yields the desired result. In combination with (7.3) and (7.24), this is our final lower bound.

Appendix A. Equivalent Formulation of Assumption 2
In this appendix we shall explain how Assumption 2 can be verified via a spectral analysis of the Hessian of E P at its minimizer ψ P ě 0, which is assumed to be unique. We partly follow ideas in [6,Sec. 2].
There is also another Hessian defined for purely imaginary perturbations of ψ P , but it is trivially given by the linear operator defined by the equation (A.1) and plays no role here.
Note that Z P ψ P " 0. We now show that if ψ P spans the kernel of Z P , then Assumption 2 holds.

Appendix B. Bounds on Solutions of Poisson's Equation
We consider solutions u of the equation´∆u " f in an open set Ω Ă R d with boundary conditions u " 0 on BΩ. We are interested in bounds on derivatives of u in terms of derivatives of f , uniformly on small balls, possibly intersecting the boundary of Ω. While we use these bounds only for d " 3, it requires no extra effort to prove them in arbitrary dimension d ě 2.
B.1. Statement of the Inequality. Let k P N and δ P p0, 1q. We say that an open set Ω Ă R d is a C k,δ set if there are constants r 0 ą 0 and M ă 8 such that for any x P BΩ there is a function Γ : ty 1 P R d´1 : |y 1 | ă r 0 u Ñ R satisfying Γp0q " 0, ∇Γp0q " 0 and such that, after a translation and a rotation (which maps x to 0 and the exterior unit normal at x to p0, . . . , 0,´1q, and is denoted by T x ), T x pΩ X B r 0 pxqq " tpy 1 , y d q P R d´1ˆR : |y 1 | ă r 0 , y d ą Γpy 1 qu X B r 0 p0q .

(B.2)
Here and below we use the notation |B k f pxq| " p ř |β|"k |B β f pxq| 2 q 1{2 and similarly |B k f pxq´B k f pyq| " p ř |β|"k |B β f pxq´B β f pyq| 2 q 1{2 , with B β " B β 1 1¨¨¨B β d d for β P N d 0 , and |β| " ř d j"1 β j . The above definition of a C k,δ set is standard (see, e.g., [11,Sec. 6.2]), except possibly for the choice of the r 0 dependence in (B.1). Our choice ensures scale invariance in the sense that if Ω is scaled by a factor λ, r 0 gets multiplied by λ while M stays the same.
Theorem B.1. Let k P N, 0 ă δ ă 1, R 0 ą 0 and Ω Ă R d be an open C k,δ set. Then we have, for all a P Ω and all R ď R 0 , if k " 1 The constants in these bounds depend only on d, k, δ, M and R 0 {r 0 .
Dropping the Hölder semi-norm on the left side and estimating it on the right side in terms of one higher derivative, we obtain Corollary B.1. Let k P N, 0 ă δ ă 1, R 0 ą 0 and Ω Ă R d be an open C k,δ set. Then we have for all a P Ω and all R ď R 0 , The constants in these bounds depend only on d, k, δ, M and R 0 {r 0 .

B.2. Local
Estimates. The more difficult assertion in Theorem B.1 is for balls such that B 2R paq X BΩ ‰ H. The strategy in this case will be to flatten the boundary, but this results in a second order elliptic equation with variable coefficients. In this subsection we state and prove bounds on solutions of such equations for domains with a flat boundary portion.
Let Ω Ă R d :" R d´1ˆp 0, 8q be an open set with an open boundary portion T on BR d . We emphasize explicitly that the case T " H is allowed. For x, y P Ω we write, following [11,Sect. 4.4], and |u| pσq k,δ,ΩYT :" One readily checks that these norms satisfy |f g| and λ ą 0 is a uniform lower bound on the lowest eigenvalue of the symmetric matrix defined by a r,s .
For us the bound with g " 0 suffices, but g appears naturally in the proof.
Proof. A similar, but less precise bound appears in [11,Corollary 8.36]. Since its proof is sketched only very briefly, we provide some more details. The starting point is [11, (4.46)], which proves the lemma in the case L "´∆ and Ω " B R px 0 q X R d with x 0 P R d . By the same argument as in the proof of [11,Theorem 4.12] (which is not given, but which is similar to the proof of [11,Theorem 4.8]), this bound leads to Lemma B.1 for L "´∆, but for general Ω. Using a simple change of variables as in the proof of [11, Lemma 6.1] we obtain the lemma for L "´∇¨A∇ with a constant matrix A again for a general Ω. Finally, using the perturbation argument as in the proof of [11,Lemma 6.4] (which again is not given, but which is similar to the proof of [11,Theorem 6.2]) we obtain the lemma. and λ ą 0 is a uniform lower bound on the lowest eigenvalue of the symmetric matrix defined by a r,s .
Proof. Lemma B.2 with k " 2 coincides with [11,Lemma 6.4]. Estimates similar to, but less precise than our statement for k ě 3 are stated as [11, Problem 6.2], but without any details. We shall show that for any integer k ě 2 and any σ ě 0, where the implicit constant depends only on d, k, δ, σ, λ and Λ. We will prove this by induction on k. First, let k " 2. For σ " 0 the claimed inequality is [11,Lemma 6.4] (whose proof is not given, but which is similar to the proof of [11,Theorem 6.2]). The proof for σ ą 0 follows by the same argument. Now let k ě 3 and σ ě 0. We assume the inequality has already been shown for all smaller values of k and for all values of σ. For 1 ď j ď d´1 the function v " B j u satisfies we immediately obtain the bound in this case. (Of course, in order to prove the bounds much simpler versions of Lemmas B.1 and B.2 would suffice.) Now assume that distpa, BΩq ă 2R. We set Without loss of generality we assume M ě 1 2 , hence r 1 ď r 0 . We will first assume that R ď r 1 {4, which implies that if p P BΩ is chosen with |p´a| " distpa, BΩq, then (B.28) (Indeed, if |y´a| ă 2R, then |y´p| ď |y´a|`|a´p| ă 2R`distpa, BΩq ď 4R ď r 1 .) Therefore, we can work in the boundary coordinates from the definition of a C k,δ domain centered at the point p. After a translation and a rotation we may assume that p " 0 and that there is a function Γ : ty 1 P R d´1 : |y 1 | ă r 0 u Ñ R with Γp0q " 0, ∇Γp0q " 0 and Ω X B r 0 p0q " tpy 1 , y d q P R d´1ˆR : |y 1 | ă r 0 , y d ą Γpy 1 qu X B r 0 p0q . (B.29) We introduce the change of variables Φ : Ω X B r 0 p0q Ñ R d , The following lemma shows that decreasing r 0 to r 1 ensures that Φ is bi-Lipschitz.
LetΩ " ΦpB 2R paq X Ωq. This is an open set in R d with a boundary portion T " ΦpB 2R paq X BΩq on BR d . For a function g on B 2R paq X Ω we define a functiong onΩ byg pxq " gpΦ´1pxqq .
We claim that Moreover, for j ď d´1, we have B jg " B j g`B d gB j Γ, and B dg " B d g. Since |B j Γ| ď M, we see that |Bg| À |Bg|. When computing a second derivative, also a term like B d gB j B k Γ appears. Bounding |B j B k Γ| ď Mr´1 0 and R À r 0 , we obtain |B 2g | À |B 2 g|`R´1|Bg|.
The arguments for higher derivatives and for the Hölder term are similar. After these preliminaries we now return to our differential equation. We havé ∆u " f in Ω X B 2R paq and u " 0 on BΩ X B 2R paq. Therefore the functions upxq " upΦ´1pxqq ,f pxq " f pΦ´1pxqq (B.38) satisfy with the operator (B.41) A straightforward computation shows that the smallest eigenvalue of the matrix defined by a r,s is given by 1`1 2 pp∇Γq 2´a p∇Γq 4`4 p∇Γq 2 q. The function t Þ Ñ 1`1 2 pt´?t 2`4 tq is positive for t ě 0 and strictly decreasing to 0 as t Ñ 8. Therefore, since |∇Γ| ď M by our definition of C k,δ smoothness, we see that the lowest eigenvalue is uniformly bounded below by some λ ą 0 depending only on M. Moreover, using the definition of a C k,δ -set and the fact that R À r 0 , we deduce from (B.36) that ÿ (B.46) We claim that the left side of (B.45) is bounded from below by a constant (depending only on d, k, δ and M) times The proof of the latter fact is similar to that of (B.36). Namely, for x P B R paq X Ω, one has distpΦpxq, BΩzT q ě 1 2 distpx, BpB 2R paq X ΩqzpB 2R paq X BΩqq ě The case of larger R 0 is readily reduced to the previous case by covering the ball B R paq with finitely many smaller balls of size r 1 {4. As long as R 0 {r 0 is bounded, this only modifies the constants in the bounds.

Appendix C. Bounds on the Kernel of Functions of the Dirichlet Laplacian
In this appendix we will use the bounds in Appendix B, specifically Corollary B.1, to obtain estimates on derivatives of the integral kernel of various functions of the Dirichlet Laplacian ∆ Ω for Ω Ă R d . We work in arbitrary dimension d ě 1.
(C.1) Therefore, by Bernstein's theorem we infer that for any completely monotone function f on r0, 8q, we have This bound is used in the main text multiple times, for instance with f ptq " t´1e´t {K 2 and f ptq " pt`K 2 q´3.
To motivate the following, we shall first derive a more general but slightly worse bound on the diagonal x " y, assuming only that f is non-increasing. Assuming that Ω is bounded (or more generally that the spectrum of´∆ Ω is discrete) we shall denote the eigenvalues of´∆ Ω (in increasing order and repeated according to their multiplicities) by e n , and the corresponding eigenfunctions by ϕ n . According to (C.1) we have for any K ą 0 C.2. Bounds on the Diagonal. We now use the same method to derive bounds on ř n f pe n q|B β ϕ n pxq| 2 . To do so we shall use Corollary B.1 to prove the following.
We apply (C.6) with g the characteristic function of te ď K 2 u for some K ą 0, R " K´1 and R 0 " e´1 where we have used (C.5) in the last step. More generally, we obtain for any nonincreasing function f with lim tÑ8 t d{2`|β| f ptq " 0 that ÿ n f pe n q|B β ϕ n pxq| 2 "´ż 8 0 ÿ enďE |B β ϕ n pxq| 2 f 1 pEq dE (C. 13) We note that the validity of (C.12) is shown in [15,Thm. 17.5.3] if Ω has C 8 boundary. Following the proof there (which is based on regularity theory in L 2based Sobolev spaces) one sees that a certain finite number of derivatives is actually sufficient, but the result is not as precise as ours, which only requires C |β|,δ regularity of the boundary.
C.3. Offdiagonal Bounds. In this section we shall derive a bound on the derivatives of the kernel of certain functions of the Dirichlet Laplacian, valid even away from the diagonal. These bounds are much less general than the ones in the previous two subsections, however. For simplicity we only consider the particular class of functions needed in the main text, but the method obviously extend to other functions as well.