The Strongly Coupled Polaron on the Torus: Quantum Corrections to the Pekar Asymptotics
aa r X i v : . [ m a t h - ph ] F e b THE STRONGLY COUPLED POLARON ON THE TORUS: QUANTUMCORRECTIONS TO THE PEKAR ASYMPTOTICS
DARIO FELICIANGELI AND ROBERT SEIRINGERA
BSTRACT . We investigate the Fröhlich polaron model on a three-dimensional torus, andgive a proof of the second-order quantum corrections to its ground-state energy in thestrong-coupling limit. Compared to previous work in the confined case, the translationalsymmetry (and its breaking in the Pekar approximation) makes the analysis substantiallymore challenging.
1. I
NTRODUCTION
The underlying physical system we are interested in studying is that of a charged particle(e.g., an electron) interacting with the quantized optical modes of a polar crystal (calledphonons). In this situation, the electron excites the phonons by inducing a polarizationfield, which, in turn, interacts with the electron. In the case of a ‘large polaron’ (i.e., whenthe De Broglie wave-length of the electron is much larger than the lattice spacing in themedium), this system is described by the Fröhlich Hamiltonian [10], which represents asimple and well-studied model of non-relativistic quantum field theory (see [1, 8, 11, 20,27, 28] for properties, results and further references).A key parameter that appears in the problem is the coupling constant, usually denotedby α . We study the strong coupling regime of the model, i.e., its asymptotic behavioras α → ∞ . In this limit, the ground state energy of the Fröhlich Hamiltonian agreesto leading order with the prediction of the Pekar approximation [24], which assumes aclassical behavior for the phonon field. This was first proved in [4], using a path integralapproach (see also [21] and [22], for recent work on the polaron process [28]). Later, theresult was improved in [18], by providing explicit bounds on the leading order correctionterm.The object of our study is, precisely, the main correction to the classical (Pekar) approxi-mation of the polaron model, i.e., the leading error term in the aforementioned asymptoticsfor the ground state energy. Such correction is expected to be of order O ( α − ) smallerthan the leading term, and arises from the quantum fluctuations about the classical limit[2]. This claim was first verified rigorously in [9], where both the electron and the phononfield are confined to a bounded domain (of linear size adjusted to the natural length scaleset by the Pekar ansatz) with Dirichlet boundary conditions. Such restriction breaks trans-lation invariance and simplifies the structure of the Pekar problem in comparison with theunconfined case, guaranteeing, at least in the case of the domain being a ball [6], unique-ness up to phase of the Pekar minimizers and non-degeneracy of the Hessian of the Pekarfunctional. We build upon the strategy developed in [9] to treat the ultraviolet singularity ofthe model, which in turn relies on multiple application of the Lieb–Yamazaki commutatormethod [19] and a subsequent use of Nelson’s Gross transformation [13, 23].The key novelty of the present work is to deal with a translation invariant setting. Weinvestigate the quantum correction to the Pekar approximation of the polaron model on a Date : February 2, 2021. torus, and prove the validity of the predictions in [2] also in this setting. As a first step, weanalyze the structure of the set of minimizers of the corresponding Pekar functional, prov-ing uniqueness of minimizers up to symmetries, which was so far known to hold only in theunconfined case [16, 15] and on balls with Dirichlet boundary conditions [6]. The trans-lation invariance leads to a degeneracy of the Hessian of the Pekar functional and corre-sponding zero modes, substantially complicating the analysis of the quantum fluctuations.In order to ‘flatten’ the surface of minimizers, we introduce a convenient diffeomorphisminspired by formal computations in [14], which effectively allows us to decouple the zeromodes. 2. S
ETTING AND M AIN R ESULTS
The Model.
We consider a -dimensional flat torus of side length L > . We denoteby ∆ L the Laplacian on T L and by ∆ − L ( x, y ) the integral kernel of its ‘inverse’, which wedefine by ( ∆ L (cid:2) ∆ − L ( · , y ) (cid:3) = δ y ´ T L ∆ − L ( x, y ) dx = 0 . (2.1.1)An explicit formula for ∆ − L ( x, y ) is given by − ∆ − L ( x, y ) = X = k ∈ πL Z | k | e ik · ( x − y ) L , (2.1.2)which, for any x ∈ T L , yields an L function of y , its Fourier coefficients being in ℓ .Analogously we define ∆ − sL for any s > . In the following, we identify T L with the box [ − L/ , L/ ⊂ R , and the Laplacian with the corresponding one on [ − L/ , L/ withperiodic boundary conditions.Let v L ( y ) := − ∆ − / L (0 , y ) = X = k ∈ πL Z | k | e − ik · y L , (2.1.3)and v xL ( y ) := v L ( y − x ) . The Fröhlich Hamiltonian [10] for the polaron is given by H L := − ∆ L ⊗
11 + 11 ⊗ N − a ( v xL ) − a † ( v xL )= − ∆ L ⊗
11 + 11 ⊗ X k ∈ πL Z a † k a k − L / X = k ∈ πL Z | k | (cid:16) a k e ik · x + a † k e − ik · x (cid:17) , (2.1.4)acting on L ( T L ) ⊗ F ( L ( T L )) , where F ( L ( T L )) denotes the bosonic Fock space over L ( T L ) . The number operator, denoted by N , accounts for the field energy, whereas − ∆ L accounts for the electron kinetic energy. The creation and annihilation operators for aplane wave of momentum k are denoted by a † k and a k , respectively, and they are assumedto satisfy the rescaled canonical commutation relations [ a k , a † j ] = α − δ k,j . (2.1.5)In light of (2.1.5), N has spectrum σ ( N ) = α − { , , , . . . } . We note that the definition(2.1.4) is somewhat formal, since v L L ( T L ) . It is nevertheless possible to define H L viathe associated quadratic form, and to find a suitable domain on which it is self-adjoint andbounded from below (see [12], or Remark 4.1 in Section 4 below).We shall investigate the ground state energy of H L , for fixed L and α → ∞ . TRONGLY COUPLED POLARON ON THE TORUS 3
Remark 2.1.
By rescaling all lengths by α , H L is unitarily equivalent to the operator α − e H L , where e H L can be written compactly as e H L = − ∆ α − L − √ α (cid:2) ˜ a ( v xα − L ) + ˜ a † ( v xα − L ) (cid:3) + e N , (2.1.6)with the creation and annihilation operators ˜ a † and ˜ a now satisfying the (un-scaled) canoni-cal commutation relations [˜ a ( f ) , ˜ a † ( g )] = h f | g i , and ˜ N the corresponding number operator.Large α hence corresponds to the strong-coupling limit of a polaron confined to a torus ofside length Lα − . We find it more convenient to work in the variables defined in (2.1.4),however. Remark 2.2.
The Fröhlich polaron model is typically considered without confinement,i.e., as a model on L ( R ) ⊗ F ( L ( R )) with electron-phonon coupling function given by ( − ∆ R ) − / ( x, y ) = (2 /π ) / | x − y | − . In the confined case studied in [9], R was replacedby a bounded domain Ω , and thus the electron-phonon coupling function was given by ( − ∆ Ω ) − / ( x, y ) , where − ∆ Ω denotes the Dirichlet Laplacian on Ω . The latter setting,similarly to ours, has the advantage of guaranteeing compactness for the correspondinginverse Laplacian, which is a key technical ingredient both for [9] and our main results. Inaddition, for generic domains Ω the Pekar functional has a unique minimizer up to phase(which is proved in [6] for Ω a ball, and enters the analysis in [9] for general Ω as anassumption). Compared with [9], setting the problem on the torus (or on R ) introducesthe extra difficulty of having to deal with translation invariance and a whole continuum ofPekar minimizers. Hence the present work can be seen as a first step in the direction ofgeneralizing the results of [9] to the case of R .2.2. Pekar Functional(s).
For ψ ∈ H ( T L ) , k ψ k = 1 , and ϕ ∈ L R ( T L ) , we introducethe classical energy functional corresponding to (2.1.4) as G L ( ψ, ϕ ) := h ψ | h ϕ | ψ i + k ϕ k , (2.2.1)where h ϕ is the Schrödinger operator h ϕ := − ∆ L + V ϕ , V ϕ := 2∆ − / L ϕ. (2.2.2)We define the Pekar energy as e L := min ψ,ϕ G L ( ψ, ϕ ) . (2.2.3)In the case of R , it was shown in [4] and [18] that the infimum of the spectrum of theFröhlich Hamiltonian converges to the minimum of the corresponding classical energyfunctional as α → ∞ . In [9], it was shown that the same holds for the model confined to abounded domain with Dirichlet boundary conditions and the subleading correction in thisasymptotics was computed. Our goal is to extend the results of [9] to the case of T L .We define the two functionals E L ( ψ ) := min ϕ G L ( ψ, ϕ ) , F L ( ϕ ) := min ψ G L ( ψ, ϕ ) , (2.2.4)and their respective sets of minimizers M E L := (cid:8) ψ ∈ H ( T L ) | k ψ k = 1 , E L ( ψ ) = e L (cid:9) , (2.2.5) M F L := { ϕ ∈ L R ( T L ) | F L ( ϕ ) = e L } . (2.2.6) DARIO FELICIANGELI AND ROBERT SEIRINGER
Clearly, E L is invariant under translations and changes of phase and F L is invariant undertranslations. It is thus useful to introduce the notation Θ L ( ψ ) := { e iθ ψ y ( · ) := e iθ ψ ( · − y ) | θ ∈ [0 , π ) , y ∈ T L } , (2.2.7) Ω L ( ϕ ) = { ϕ y | y ∈ T L } , (2.2.8)for ψ ∈ H ( T L ) and ϕ ∈ L R ( T L ) , respectively.Our first result, Theorem 2.1 (or, more precisely, Corollary 2.1) is a fundamental ingre-dient to prove our main result, Theorem 2.2. It concerns the uniqueness of minimizers of E L up to symmetries and shows the validity of a quadratic lower bound for E L in termsof the H -distance from the surface of minimizers. We shall prove these properties forsufficiently large L . Theorem 2.1 (Uniqueness of Minimizers and Coercivity for E L ) . There exist L > and apositive constant κ independent of L , such that for L > L there exists < ψ L ∈ C ∞ ( T L ) such that e L < , M E L = Θ L ( ψ L ) . (2.2.9) Moreover ψ yL = ψ L for any = y ∈ T L and, for any L -normalized f ∈ H ( T L ) , E L ( f ) − e L ≥ κ dist H (cid:0) M E L , f (cid:1) . (2.2.10)These properties of E L translate easily to analogous properties for the functional F L , asstated in the following corollary. Corollary 2.1 (Uniqueness of Minimizers and Coercivity for F L ) . For
L > L (where L is the same as in Theorem 2.1) there exists ϕ L ∈ C ∞ ( T L ) such that M F L = Ω L ( ϕ L ) . (2.2.11) Moreover, with ψ L as in Theorem 2.1, we have ϕ L = σ ψ L := ( − ∆ L ) − / | ψ L | , ψ L = unique positive g.s. of h ϕ L . (2.2.12) Finally, there exists κ ′ > independent of L such that, for all ϕ ∈ L ( T L ) , F L ( ϕ ) − e L ≥ min y ∈ T L h ϕ − ϕ yL | − (11 + κ ′ ( − ∆ L ) / ) − | ϕ − ϕ yL i + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − / ˆ T L ϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.2.13) and this implies F L ( ϕ ) − e L ≥ τ L dist L ( M F L , ϕ ) (2.2.14) with τ L := κ ′ (2 π/L ) κ ′ (2 π/L ) . In the case of R , similar results are known to hold. In particular, the analogue of (2.2.9)was shown in [16] and the analogue of (2.2.10) follows from the results in [15]. In thecase of a bounded domain with Dirichlet boundary conditions, an equivalent formulationof Theorem 2.1 was taken as working assumption in [9]. In the case of a ball in R withDirichlet boundary conditions, the analogue of Theorem 2.1 was proved in [6]. In both thecase of R and of balls, rotational symmetry plays a key role in the proof of these results.Rotational symmetry is not present in our setting, hence a different approach is required.Our method of proof of Theorem 2.1 relies on a comparison of the models on T L and R ,for large L . As a consequence, our analysis does not yield quantitative estimates on L . TRONGLY COUPLED POLARON ON THE TORUS 5
To state our main result, which also holds in the case
L > L , we need to introduce theHessian of the functional F L at its unique (up to translations) minimizer ϕ L , lim ε → ε ( F L ( ϕ L + εφ ) − e L ) =: h φ | H F L ϕ L | φ i ∀ φ ∈ L R ( T L ) . (2.2.15)An explicit computation gives (see Proposition 3.4) H F L ϕ L = 11 − − ∆ L ) − / ψ L Q ψ L h ϕ L − inf spec h ϕ L ψ L ( − ∆ L ) − / , (2.2.16)where h ϕ L is defined in (2.2.2), ψ L is interpreted as a multiplication operator and Q ψ L :=11 − | ψ L i h ψ L | . Clearly, by minimality of ϕ L , H F L ϕ L ≥ , and it is also easy to see that H F L ϕ L ≤ . We shall show that H F L ϕ L has a three-dimensional kernel, given by span {∇ ϕ L } ,corresponding to the invariance under translations of the functional. Note that we could de-fine the Hessian of F L at any other minimizer ϕ yL , obtaining a unitarily equivalent operator H F L ϕ yL .2.3. Main Result.
Recall the definition (2.2.3) for the Pekar energy e L as well as (2.2.16)for the Hessian of F L at its minimizers, for L > L . Our main result is as follows. Theorem 2.2.
For any
L > L , as α → ∞ inf spec H L = e L − α Tr (cid:18) − q H F L ϕ L (cid:19) + o ( α − ) . (2.3.1) More precisely, the bounds − C L α − / ≤ α inf spec H L − α e L + 12 Tr (cid:18) − q H F L ϕ L (cid:19) ≤ C L α − / (2.3.2) hold for some C L > and α sufficiently large. The trace appearing in (2.3.1) and (2.3.2) is over L ( T L ) . Note that, since H F L ϕ L ≤ , thecoefficient of α − in (2.3.1) is negative.In the case of bounded domains with Dirichlet boundary conditions, an analogue ofTheorem 2.2 was proven in [9] (where logarithmic corrections appear in the bounds thatcorrespond to (2.3.2) as a consequence of technical complications due to the boundary).Showing the validity of an analogous result on R still remains an open problem, however.Indeed, the constant C L appearing in the lower bound in (2.3.2) diverges as L → ∞ . Onthe other hand, our method of proof used in Section 4.1 to show the upper bound in (2.3.2)does apply, with little modifications, to the full space case. In any case, both the upper andlower bound are expected to hold in the case of R as well [2, 14, 9, 27].Compared to the results obtained in [9], Theorem 2.2 deals with the additional com-plication of the invariance under translations of the problem, which implies that the setof minimizers of F L is a three-dimensional manifold. This substantially complicates theproof of the lower bound in (2.3.2), as we shall see in Section 4.3. In particular, we need toperform a precise local study around the manifold of minimizers Ω L ( ϕ L ) , which we carryout by introducing a suitable diffeomorphism (inspired by [14]). Remark 2.3 (Small L Regime) . As we show in Lemma 3.2, there exists L > such thatthe analogue of Theorem 2.1 for L < L can be proven with a few-line-argument. In thiscase, E L is simply non-negative and is therefore minimized by the constant function. Inparticular, e L = 0 and ϕ L = 0 . DARIO FELICIANGELI AND ROBERT SEIRINGER
Also an analogue of Theorem 2.2 can be proven in the regime
L < L , i.e., it is possibleto show that for L < L there exists C L > such that − C L α − / ≤ α inf spec H L + 12 X = k ∈ πL Z − s − L | k | ! ≤ C L α − / (2.3.3)for large α . In this case (unlike the regime L > L where the set of minimizers M F L is athree-dimensional manifold) M F L only consists of the function, and this allows to followessentially the same arguments of [9] (with only small modifications, which are also neededin the regime L > L and hence are discussed in this paper). We shall therefore not carryout the details of this analysis here.Whether uniqueness of Pekar minimizers up to symmetries holds for all L > (i.e., alsoin the regime L ≤ L ≤ L ) remains an open problem.Throughout the paper, we use the word universal to describe any constant (which isgenerally denoted by C ) or property that is independent of all the parameters involved andin particular independent of L , for L ≥ L (for some fixed L > ). Also, we write a . b whenever a ≤ Cb for some universal and positive C . We write C L whenever a constantdepends on L but is otherwise universal with respect to all other parameters. Finally, wewrite a . L b whenever a ≤ C L b for some positive C L .2.4. Structure of the Paper.
In Section 3 we study the properties of the Pekar functionals E L and F L , showing the validity of Theorem 2.1, Corollary 2.1, as well as some otherimportant properties of F L . In Section 4 we prove Theorem 2.2.3. P ROPERTIES OF THE P EKAR F UNCTIONALS
In this section we derive important properties of the functionals E L and F L , introducedin (2.2.4). In Section 3.1, we show the validity of Theorem 2.1, relying on the comparisonof the models on T L and R for large L . In Section 3.2, we study the functional F L . Inparticular, we prove Corollary 2.1 and compute the Hessian of F L at its minimizers.Given a function f ∈ L ( T L ) and k ∈ πL Z , we denote by f k the k -th Fourier coefficientof f . We also denote ˆ f := f − L − ˆ T L f. (3.0.1)We shall use the following definition of fractional Sobolev semi-norms for functions f ∈ L ( T L ) , = s ∈ R : k f k H s ( T L ) = h f | ( − ∆ L ) s | f i = X = k ∈ πL Z | k | s | f k | . (3.0.2)3.1. Study of E L . An important role in this analysis is played by the full-space Pekarfunctional, of which we recall the definition: let ψ ∈ H ( R ) be an L ( R ) -normalizedfunction, then E ( ψ ) := ˆ R |∇ ψ ( x ) | dx − ˆ R ˆ R ρ ψ ( x )( − ∆ R ) − ( x, y ) ρ ψ ( y ) dxdy =: T ( ψ ) − W ( ψ ) , (3.1.1) TRONGLY COUPLED POLARON ON THE TORUS 7 where ρ ψ := | ψ | and ( − ∆ R ) − ( x, y ) = (4 π ) − | x − y | − . By a simple completion of thesquare, it is straightforward to see that E L , defined in (2.2.4), can also be written as E L ( ψ ) = ˆ T L |∇ ψ ( x ) | dx − ˆ T L ˆ T L ρ ψ ( x )( − ∆ L ) − ( x, y ) ρ ψ ( y ) dxdy =: T L ( ψ ) − W L ( ψ ) , (3.1.2)for any L -normalized ψ ∈ H ( T L ) . To compare the two, we need the following Lemma. Lemma 3.1.
There exists a universal constant C such that sup x,y ∈ T L (cid:12)(cid:12)(cid:12) ( − ∆ − L ( x, y ) − (4 π ) − (dist T L ( x, y )) − (cid:12)(cid:12)(cid:12) ≤ CL . (3.1.3)
Proof.
We define F L ( x ) := − ∆ − L ( x, and F ( x ) = (4 π ) − | x | − and observe that ourstatement is equivalent to showing that k F L − F k L ∞ ([ − L/ ,L/ ) ≤ CL . (3.1.4)By definition, we have F L ( x ) = L F ( xL ) . Hence, (3.1.4) is equivalent to k F − F k L ∞ ([ − / , / ) ≤ C. (3.1.5)Again by definition, F − F is harmonic (distributionally and hence also classically) on ( R \ { Z } ) ∪ { } (when F , and only F , is extended to the whole space by periodic-ity). Thus we conclude that F − F is in C ∞ (( − , ) and, in particular, bounded on [ − / , / . (cid:3) The previous discussion, combined with Lemma 3.1, suggests that E L formally con-verges to E as L → ∞ . As we shall see, this convergence can be made rigorous, andallows to infer properties of E L by comparing it to E , in the large L regime.We recall here the main known results about the full-space Pekar functional. As shownin [16], E admits a unique positive and radially decreasing minimizer Ψ which is alsosmooth, the set of minimizers of E coincides with Θ( Ψ ) := { e iθ Ψ y | θ ∈ [0 , π ) , y ∈ R } , (3.1.6)and Ψ satisfies the Euler–Lagrange equation ( − ∆ R + V σ Ψ − µ Ψ ) Ψ = 0 , (3.1.7)with V σ Ψ = 2∆ − R | Ψ | , µ Ψ = T ( Ψ ) − W ( Ψ ) . (3.1.8)We denote by e ∞ the infimum of E over L -normalized functions in H ( R ) , i.e., e ∞ := E ( Ψ ) . (3.1.9)Furthermore, as was shown in [15], the Hessian of E at its minimizers is strictly positiveabove the trivial zero modes resulting from the invariance under translations and changesof phase. This implies the validity of the following Theorem, which is not stated explicitlyin [15] but can be obtained by standard arguments (see, e.g., [5, Appendix A], [7]) as aconsequence of the results therein contained. Theorem A.
There exists a constant
C > , such that, for any L -normalized f ∈ H ( R ) E ( f ) − e ∞ ≥ C dist H (Θ( Ψ ) , f ) . (3.1.10) DARIO FELICIANGELI AND ROBERT SEIRINGER
We now dwell on the study of the properties of E L . In Section 3.1.1 we derive an impor-tant preliminary result, namely Proposition 3.1. It formalizes in a mathematical useful waythe concept of E L converging to E . In Section 3.1.2, we study the Hessian of E L , show-ing that it converges (in the sense of Proposition 3.2) to the Hessian of E and therefore isstrictly positive above its trivial zero modes for large L . Finally, in Section 3.1.3 we usethe results obtained in Sections 3.1.1 and 3.1.2 to show the validity of Theorem 2.1.We remark that our approach differs from the one used on R and on balls to show, for therelated E -functional, uniqueness of minimizers and strict positivity of the Hessian (see [16]and [15] for the case of R and [6] for the case of balls). In those cases, rotational symmetryallows to first show uniqueness of minimizers and then helps to derive the positivity of theHessian at the minimizers. We take somewhat the opposite road: comparing E L to E , wefirst show that minimizers (even if not unique) all localize around the full-space minimizers(see Proposition 3.1) and that the Hessian at each minimizer is universally strictly positive(see Proposition 3.2) for large L . We then use these two properties to derive, as a final step,uniqueness of minimizers.3.1.1. Preliminary Results.
The next Lemma proves the existence of minimizers for any
L > . Moreover, it shows that there exists L > such that, for L < L , E L is strictlypositive on any non-constant L -normalized function, as already mentioned in Remark 2.3. Lemma 3.2.
For any
L > , e L in (2.2.3) is attained, and there exists a universal constant C > such that e L > − C . Moreover, there exists L > such that, for L < L , E L ( ψ ) > for any non-constant L -normalized ψ .Proof. We consider any L -normalized ψ ∈ H ( T L ) and begin by observing that in termsof the Fourier coefficients we have W L ( ψ ) = X = k ∈ πL Z | ( ρ ψ ) k | | k | , (3.1.11) ( ρ ψ ) k = X j ∈ πL Z ¯ ψ j ψ j + k L / = ( ρ ˆ ψ ) k + ¯ ψ ψ k L / + ¯ ψ − k ψ L / . (3.1.12)By Parseval’s identity | ψ | ≤ and thus, using the Cauchy–Schwarz inequality, we candeduce that | ( ρ ψ ) k | ≤ ( L − , | ( ρ ˆ ψ ) k | + L ( | ψ k | + | ψ − k | ) . (3.1.13)Therefore W L ( ψ ) ≤ X = k ∈ πL Z | ( ρ ˆ ψ ) k | | k | + 6 L X = k ∈ πL Z | ψ k | | k | ≤ W L ( ˆ ψ ) + 6(2 π ) L k ˆ ψ k L ( T L ) . (3.1.14)We can bound both terms on the r.h.s. in two different ways, one which is good for small L and one which is good for all the other L . Indeed, by applying estimate (3.1.13) and using TRONGLY COUPLED POLARON ON THE TORUS 9 the Poincaré-Sobolev inequality (see [17], chapter 8) on the zero-mean function ˆ ψ , we get W L ( ˆ ψ ) ≤ X = k ∈ πL Z | ( ρ ˆ ψ ) k | | k | / X = k ∈ πL Z | ( ρ ˆ ψ ) k | / . L k ( ρ ˆ ψ ) k k l ∞ k ˆ ψ k L ( T L ) . L / k ˆ ψ k L ( T L ) . L k ˆ ψ k L ( T L ) . LT L ( ˆ ψ ) = LT L ( ψ ) . (3.1.15)Moreover, L − k ˆ ψ k L ( T L ) . LT L ( ˆ ψ ) = LT L ( ψ ) . (3.1.16)Therefore, we can conclude that W L ( ψ ) . LT L ( ψ ) ⇒ E L ( ψ ) ≥ (1 − CL ) T L ( ψ ) . (3.1.17)Thus, for L < L := C − , either ψ ≡ const . and E L ( ψ ) = 0 or E L ( ψ ) & T L ( ψ ) > .Moreover, this also implies E L ( ψ ) & T L ( ψ ) ≥ (2 π ) L k ˆ ψ k + 12 T L ( ψ ) & dist H (cid:18) Θ L (cid:18) L / (cid:19) , ψ (cid:19) , (3.1.18)which is the analogue of (2.2.10) from Theorem 2.1 in the case L < L .We now proceed to study the more interesting regime L ≥ L . By Lemma 3.1, splitting dist − T L ( x, · ) into an L / part and the remaining L ∞ part (whose norms can be chosen to beproportional to ε and ε − , respectively, for any ε > ), and by applying again the Poincaré-Sobolev inequality, we obtain W L ( ˆ ψ ) ≤ ˆ T L × T L ρ ˆ ψ ( x ) ρ ˆ ψ ( y )4 π dist T L ( x, y ) dxdy + CL . ε k ˆ ψ k L ( T L ) + ε − + 1 ≤ T L ( ψ )6 + C. (3.1.19)Moreover, since L ≥ L , trivially L − k ˆ ψ k L ( T L ) . and we can conclude that for any L -normalized ψ ∈ H ( T L ) W L ( ψ ) ≤ T L ( ψ )2 + C ⇒ E L ( ψ ) ≥ T L ( ψ )2 − C. (3.1.20)From this we can infer that e L ≥ − C for any L . To show existence of minimizers, weobserve that by (3.1.20) any minimizing sequence ψ n on T L must be bounded in H ( T L ) .Therefore, there exists a subsequence (which we still denote by ψ n for simplicity) thatconverges weakly in H ( T L ) and strongly in L p ( T L ) , for any ≤ p < to some ψ (by theBanach-Alaoglu Theorem and the Rellich-Kondrachov embedding Theorem). The limitfunction ψ is L -normalized and T L ( ψ ) ≤ lim inf n →∞ T L ( ψ n ) (3.1.21)by weak lower semicontinuity of the norm. Using the L -convergence of ψ n to ψ and thefact that k · k ˚ H − ( T L ) . L k · k L ( T L ) , we finally obtain | W L ( ψ n ) − W L ( ψ ) | = (cid:16) k ρ ψ k ˚ H − ( T L ) + k ρ ψ n k ˚ H − ( T L ) (cid:17) (cid:12)(cid:12)(cid:12) k ρ ψ k ˚ H − ( T L ) − k ρ ψ n k ˚ H − ( T L ) (cid:12)(cid:12)(cid:12) . L k ρ ψ n − ρ ψ k ˚ H − ( T L ) . L k ρ ψ n − ρ ψ k L ( T L ) ≤ L k ψ n − ψ k L ( T L ) (cid:16) k ψ n k L ( T L ) + k ψ k L ( T L ) (cid:17) → . (3.1.22) This implies that E L ( ψ ) ≤ lim inf n →∞ E L ( ψ n ) = e L , (3.1.23)and thus that ψ is a minimizer. Note that, since E L ( ψ n ) → e L = E L ( ψ ) by definition of ψ n and, as shown, W L ( ψ n ) → W L ( ψ ) , it also holds T L ( ψ n ) = E L ( ψ n ) + W L ( ψ n ) → E L ( ψ ) + W L ( ψ ) = T L ( ψ ) (3.1.24)which implies that ψ n actually converges to ψ strongly in H ( T L ) . (cid:3) Once we have shown existence of minimizers, we need to investigate more carefullytheir properties. Some of them are derived in the following Lemma. Recall that V ψ = 2∆ − / L ψ, σ ψ = − ∆ − / L | ψ | , (3.1.25)and that, as stated above, we call any property universal which does not depend on L ≥ L . Lemma 3.3.
Let ψ ∈ M E L (as defined in (2.2.5) ). Then ψ satisfies the following Euler-Lagrange equation ( − ∆ L + V σ ψ − µ Lψ ) ψ = 0 , with µ Lψ = T L ( ψ ) − W L ( ψ ) . (3.1.26) Moreover, ψ ∈ C ∞ ( T L ) , is universally bounded in H ( T L ) (and therefore in L ∞ ( T L ) ), hasconstant phase and never vanishes. Finally, any L -normalized sequence f n ∈ H ( T L n ) such that E L n ( f n ) is universally bounded, is universally bounded in H ( T L n ) .Proof. The fact that sequences f n ∈ H ( T L n ) of L -normalized functions for which E L n is universally bounded are universally bounded in H ( T L n ) follows trivially from estimate(3.1.20). This immediately yields a universal bound on the H -norm of minimizers.The Euler–Lagrange equation (3.1.26) for the problem is derived by standard compu-tations omitted here. By Lemma 3.1 and by splitting (dist T L (0 , · )) − in its L / and L ∞ parts, we have | V σ ψ ( x ) | ≤ ˆ T L T L ( x, y ) | ψ ( y ) | dy + CL . (cid:16) k ψ k L ( T L ) + 1 (cid:17) . ( T L ( ψ ) + 1) . (3.1.27)Therefore, by the universal H -boundedness of minimizers, V σ ψ is universally bounded in L ∞ ( T L ) , for any ψ ∈ M E L . This immediately allows to conclude universal ˚ H (and hence H ) bounds for functions in M E L , using the Euler–Lagrange equation (3.1.26), Lemma 3.2and the universal H -boundedness of minimizers, which guarantee that ≥ µ Lψ = 2 E L ( ψ ) − T L ( ψ ) ≥ − C. Since L ≥ L , universal H -boundedness also implies universal L ∞ -boundedness of min-imizers by the Sobolev inequality.For any L > , any ψ ∈ M E L satisfies (3.1.26), is in H ( T L ) and is such that V σ ψ ∈ L ∞ ( T L ) . Therefore ψ also satisfies, for any λ > ψ = ( − ∆ L + λ ) − ( − V σ ψ + µ Lψ + λ ) ψ. (3.1.28)In particular, by a bootstrap argument we can conclude that ψ ∈ C ∞ ( T L ) . Moreover,picking λ > − µ Lψ + k V σ ψ k L ∞ ( T L ) and using that ( − ∆ L + λ ) − is positivity improving, wecan also conclude that if ψ ≥ then ψ > . By the convexity properties of the kineticenergy (see [17], Theorem 7.8), we have that T L ( | ψ | ) ≤ T L ( ψ ) which implies that if ψ ∈M E L then T L ( ψ ) = T L ( | ψ | ) and also | ψ | ∈ M E L . Hence both ψ and | ψ | are eigenfunctionsof the least and simple (by positivity of one of the eigenfunctions) eigenvalue µ Lψ = µ L | ψ | of TRONGLY COUPLED POLARON ON THE TORUS 11 the Schrödinger operator − ∆ L + V σ ψ , which allows us to infer that ψ has constant phaseand never vanishes. (cid:3) We now proceed to develop the tools that will allow to show the validity of Theorem 2.1.We begin with a simple Lemma.
Lemma 3.4.
For ψ ∈ H ( T L ) , k ρ ψ k ˚ H / ( T L ) . k ψ k H ( T L ) . (3.1.29) Proof.
We have k ρ ψ k H / ( T L ) = |h∇ ρ ψ |∇ (∆ − / L ρ ψ ) i| = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T L | ψ ( x ) |∇ ( | ψ ( x ) | ) · ∇ x X = k ∈ πL Z ( ρ ψ ) k | k | / e ik · x L / dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 ˆ T L | ψ ( x ) | ∂ i ( | ψ ( x ) | ) X = k ∈ πL Z k i ( ρ ψ ) k | k | / e ik · x L / dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.1.30)We define g i ( x ) := X = k ∈ πL Z k i ( ρ ψ ) k | k | / e ik · x L / , (3.1.31)and observe that ( g i ) = 0 and | ( g i ) k | = | k i ( ρ ψ ) k || k | / ≤ | ( ρ ψ ) k || k | / for k = 0 . These estimates onthe Fourier coefficients of g i imply that, for i = 1 , , , k g i k H / ( T L ) = X = k ∈ πL Z | k | / | ( g i ) k | ≤ X = k ∈ πL Z | ( ρ ψ ) k | ≤ k ψ k L ( T L ) . (3.1.32)Moreover, using the fractional Sobolev embeddings (see, for example, [3]) and that g i haszero mean, we have k g i k L ( T L ) . k g i k ˚ H / ( T L ) ≤ k ψ k L ( T L ) . (3.1.33)Applying these results to (3.1.30) and using Hölder’s inequality two times, the Poincaré-Sobolev inequality and the convexity properties of the kinetic energy (see [17], Theorem7.8), we conclude k ρ ψ k H / ( T L ) . k ψ k L ( T L ) k g / i k L ( T L ) k∇ ( | ψ | ) k L ( T L ) ≤ k ψ k L ( T L ) k ψ k ˚ H ( T L ) ≤ k ψ k / L ( T L ) k ψ k / L ( T L ) k ψ k ˚ H ( T L ) . k ψ k H ( T L ) . (3.1.34) (cid:3) Our next goal is to show that e L → e ∞ as L → ∞ , and that in the large L regimethe states that are relevant for the minimization of E L are necessarily close to the full spaceminimizer (or any of its translates). This is a key ingredient for the discussion carried out inthe following sections, and is stated in a precise way in the next proposition. The coercivityresults obtained in [15] are of fundamental importance here as they guarantee that, at leastfor the full space model, low energy states are close to minimizers.We recall that the full-space Pekar functional, defined in (3.1.1), admits a unique positiveand radial minimizer Ψ which is also smooth (see (3.1.6)), and we introduce the notation Ψ L := Ψ χ [ − L/ ,L/ . (3.1.35) Note that Ψ L ∈ H ( T L ) , by radiality and regularity of Ψ . Proposition 3.1.
We have lim L →∞ e L = e ∞ . (3.1.36) Moreover, for any ε > there exist L ε and δ ε such that for any L > L ε and any L -normalized ψ ∈ H ( T L ) with E L ( ψ ) − e L < δ ε , dist H (Θ L ( ψ ) , Ψ L ) ≤ ε, | µ Lψ − µ Ψ | ≤ ε, (3.1.37) where Θ L ( ψ ) , Ψ L , µ Lψ and µ Ψ are defined in (2.2.7) , (3.1.35) , (3.1.26) and (3.1.8) , respec-tively.Proof. We first show that lim sup L →∞ e L ≤ e ∞ by using Ψ L as a trial state for E L . Observethat k Ψ L k L ( T L ) → and T L ( Ψ L ) → T ( Ψ ) as L → ∞ . To estimate the difference of theinteraction terms we note that Ψ L ( Ψ − Ψ L ) = 0 and therefore | W L ( Ψ L ) − W ( Ψ ) | ≤ | W L ( Ψ L ) − W ( Ψ L ) | + W ( Ψ − Ψ L ) + 2 (cid:10) ( Ψ − Ψ L ) (cid:12)(cid:12) ∆ − R Ψ L (cid:11) . (3.1.38)By dominated convergence, the last two terms converge to zero as L → ∞ . On the otherhand, by Lemma 3.1 and since Ψ is normalized | W L ( Ψ L ) − W ( Ψ L ) | ≤ CL + 14 π ˆ [ − L/ ,L/ Ψ L ( x ) Ψ L ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T L ( x, y ) − | x − y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdy. (3.1.39)Moreover, since dist T L ( x, y ) = | x − y | for x, y ∈ [ − L/ , L/ and using the symmetryand the positivity of the integral kernel and the fact that dist T L ( x, y ) ≤ | x − y | , we get ˆ [ − L/ ,L/ Ψ L ( x ) Ψ L ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T L ( x, y ) − | x − y | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdy ≤ ˆ [ − L/ ,L/ Ψ L ( x ) ˆ [ − L/ ,L/ ( Ψ L − Ψ L/ ) ( y )dist T L ( x, y ) dy ! dx. (3.1.40)Finally, by splitting dist − T L ( x, · ) in its L ∞ and L parts and using that Ψ is normalized, wecan bound the r.h.s. of (3.1.40) by (cid:0) C k Ψ L − Ψ L/ k + C k Ψ L − Ψ L/ k ∞ (cid:1) , which vanishesas L → ∞ , since Ψ ( x ) | x |→∞ −−−−→ . Putting the pieces together, we conclude | W L ( Ψ L ) − W ( Ψ L ) | = o L (1) . (3.1.41)This shows our first claim, since e L ≤ E L ( Ψ L / k Ψ L k ) = 1 k Ψ L k (cid:18) T L ( Ψ L ) − k Ψ L k W L ( Ψ L ) (cid:19) → e ∞ . (3.1.42)We now proceed to show that lim inf L →∞ e L ≥ e ∞ (3.1.43)and the validity of (3.1.37) using IMS localization. We shall show that for any L -normalizedsequence ψ n ∈ H ( T L n ) with L n → ∞ such that E L n ( ψ n ) − e L n → , (3.1.44) TRONGLY COUPLED POLARON ON THE TORUS 13 we have lim inf n →∞ E L n ( ψ n ) ≥ e ∞ , lim n →∞ dist H (Θ L n ( ψ n ) , Ψ L n ) = 0 , lim n →∞ | µ L n ψ n − µ Ψ | = 0 , (3.1.45)which implies the claim of the proposition.Pick η ∈ C ∞ ( R ) with supp ( η ) ⊂ B and k η k = 1 . We denote by η R the rescaled copyof η supported on B R with L -norm equal to . As long as R ≤ L/ , η R ∈ C ∞ ( T L ) andwe then consider the translates η yR for any y ∈ T L . Given ψ ∈ H ( T L ) , we also define ψ yR := ψη yR / k ψη yR k . (3.1.46)By standard properties of IMS localization, for any R ≤ L/ , we have ˆ T L T L ( ψ yR ) k ψη yR k dy = ˆ T L T L ( ψη yR ) dy = T L ( ψ ) + ´ |∇ η | R . (3.1.47)Moreover, by using that | ψ | = ´ T L | ψη yR | dy = ´ T L | ψ yR | k ψη yR k dy and completing thesquare W L ( ψ ) = ˆ T L h W L ( ψ yR ) − (cid:13)(cid:13) | ψ yR | − | ψ | (cid:13)(cid:13) H − ( T L ) i k ψη yR k dy. (3.1.48)Combining (3.1.47) and (3.1.48), we therefore obtain E L ( ψ ) + CR = ˆ T L h E L ( ψ yR ) + (cid:13)(cid:13) | ψ yR | − | ψ | (cid:13)(cid:13) H − ( T L ) i k ψη yR k dy. (3.1.49)Since the integrand on the r.h.s. is equal to the l.h.s. on average (indeed k ψη yR k dy is aprobability measure) there exists ¯ y ∈ T L such that E L ( ψ ¯ yR ) + (cid:13)(cid:13) | ψ ¯ yR | − | ψ | (cid:13)(cid:13) H − ( T L ) ≤ E L ( ψ ) + CR . (3.1.50)This fact has several consequences and it is particularly useful if we apply it to our sequence ψ n with a radius R = R n ≤ L n / (we take for simplicity R = L n / ). Indeed, by the abovediscussion and (3.1.44), we obtain that there exists ¯ y n ∈ T L n such that the L -normalizedfunctions ¯ ψ n := ψ n η ¯ y n L n / k ψ n η ¯ y n L n / k (3.1.51)are competitors both for the minimization of E L n and E (indeed, ¯ ψ n can then be thought ofas a function in C ∞ c ( R ) , supported on B L n / ) and satisfy E L n ( ¯ ψ n ) ≤ E L n ( ψ n ) + CL n ≤ e L n + o L n (1) , k ρ ¯ ψ n − ρ ψ n k H − ( T Ln ) ≤ CL n . (3.1.52)In other words, we can localize any element of our sequence ψ n to a ball of radius R = L n / with an energy expense of order L − n , and the localized function is close (in the senseof the second line of (3.1.52)) to ψ n itself, up to an error again of order L − n .Moreover T L n ( ¯ ψ n ) = T ( ¯ ψ n ) and, using Lemma 3.1 and the fact that dist T Ln ( x, y ) = | x − y | for all x, y ∈ B L n / , we have | W L n ( ¯ ψ n ) − W ( ¯ ψ n ) | . L n . (3.1.53) Therefore, using (3.1.52) e ∞ ≤ E ( ¯ ψ n ) ≤ E L n ( ¯ ψ ) + CL n ≤ e L n + o L n (1) , (3.1.54)which shows the first claim in (3.1.45). By Theorem A and (3.1.54), it also follows that dist H (cid:0) Θ( Ψ ) , ¯ ψ n (cid:1) n →∞ −−−→ . (3.1.55)Hence, up to an n -dependent translation and change of phase (which we can both assumeto be zero without loss of generality by suitably redefining ψ n ), ¯ ψ n H ( R ) −−−−→ Ψ , and theconvergence also holds in L p ( R ) for any ≤ p ≤ . From this and the second line of(3.1.52), we would like to deduce that also ψ n and Ψ L n are close. We first note that, bya simple application of Hölder’s inequality, it follows that for any f ∈ L ( T L ) with zeromean k f k L ( T L ) ≤ X = k ∈ πL Z | k | / | f k | / X = k ∈ πL Z | k | − | f k | / = k f k / H / ( T L ) k f k / H − ( T L ) . (3.1.56)We combine this with (3.1.52) and apply it to the zero mean function ( ρ ψ n − ρ ¯ ψ n ) , obtaining k ρ ¯ ψ n − ρ ψ n k L ( T Ln ) . k ρ ψ n k H / ( T Ln ) + k ρ ¯ ψ n k H / ( T Ln ) L / n / . (3.1.57)Applying Lemma 3.4 to ψ n and ¯ ψ n (which are uniformly bounded in H by Lemma 3.3)we conclude that ( ρ ψ n − ρ ¯ ψ n ) L −→ .As a consequence, since ψ n and ¯ ψ n have the same phase, ψ n and ¯ ψ n are arbitrarily closein L . Indeed, k ψ n − ¯ ψ n k L ( T Ln ) = ˆ T Ln (cid:12)(cid:12) | ψ n | − | ¯ ψ n | (cid:12)(cid:12) dx ≤ ˆ T Ln ( ρ ψ n − ρ ¯ ψ n ) dx n →∞ −−−→ . (3.1.58)By the identification of T L n with [ − L n / , L n / , we finally get k ψ n − Ψ k L ( R ) → , if ψ n is set to be outside [ − L n / , L n / . Moreover, ψ n converges to Ψ in L p ( R ) for any ≤ p < , since k ψ n k = 1 , ψ n L −→ Ψ , k Ψ k = 1 and k ψ n k p is uniformly bounded forany ≤ p ≤ .To show the second claim in (3.1.45), we need to show that the convergence actuallyholds in H ( T L n ) , i.e., that k ψ n − Ψ L n k H ( T Ln ) → . First, we show convergence in H ( B R ) for fixed R . Note that (cid:16) k ψ n k H ( T Ln ) − k Ψ k H ( R ) (cid:17) → , (3.1.59)since | T L n ( ψ n ) − T L n ( ¯ ψ n ) | = |E L n ( ψ n ) + W L n ( ψ n ) − E L n ( ¯ ψ n ) + W L n ( ¯ ψ n ) |≤ |E L n ( ψ n ) − E L n ( ¯ ψ n ) | + | W L n ( ψ n ) − W L n ( ¯ ψ n ) | → , (3.1.60)and T L n ( ¯ ψ n ) = T ( ¯ ψ n ) → T ( Ψ ) by H convergence. Moreover, given that ψ n is uniformlybounded in H ( B R ) and ψ n → Ψ in L ( B R ) , we have ψ n ⇀ Ψ in H ( B R ) for any R and this, together with (3.1.59) and weak lower semicontinuity of the norms, implies TRONGLY COUPLED POLARON ON THE TORUS 15 ψ n → Ψ in H ( B R ) for any R . Finally, for any ε > there exists R = R ( ε ) such that k Ψ k H ( B cR ) ≤ ε and, using strong H -convergence on balls and again (3.1.59), we obtain k ψ n − Ψ L n k H ( T Ln ) ≤ k ψ n − Ψ k H ( B R ) + k ψ n − Ψ k H ([ − L n / ,L n / \ B R ) ≤ k ψ n − Ψ k H ( B R ) + k ψ n k H ([ − L n / ,L n / \ B R ) + k Ψ k H ([ − L n / ,L n / \ B R ) ≤ k ψ n − Ψ k H ( B R ) + 2 ε + o n (1) → ε, (3.1.61)which concludes the proof of the second claim in (3.1.45).Finally, we show the third claim in (3.1.45). This simply follows from the previousbounds, which guarantee that E L n ( ψ n ) → e ∞ and T L n ( ψ n ) → T (Ψ) and hence µ Lψ n = T L n ( ψ n ) − W L n ( ψ n ) = 2 E L n ( ψ n ) − T L n ( ψ n ) → e ∞ − T (Ψ) = µ Ψ . (3.1.62) (cid:3) We conclude this section with a simple corollary of Proposition 3.1.
Corollary 3.1.
There exists L ∗ such that for L > L ∗ and any ψ ∈ M E L we have ψ = ψ y for = y ∈ T L .Proof. It is clearly sufficient to show the claim for ψ ∈ M E L such that dist H (Θ L ( ψ ) , Ψ L ) = k ψ − Ψ L k H ( T L ) (3.1.63)and for y ∈ T L such that | y | ≥ L/ (indeed, if the claim fails for some y ′ such that | y ′ | < L/ it also fails for some y such that | y | ≥ L/ ). For any such ψ and y , Proposition3.1 and the fact that Ψ = Ψ y for any y ∈ R guarantee the existence of L ∗ such that forany L > L ∗ we have k ψ − ψ y k H ( T L ) ≥ k Ψ yL − Ψ L k H ( T L ) − k ψ − Ψ L k H ( T L ) ≥ C > (3.1.64)and this completes the proof. (cid:3) Study of the Hessian of E L . In this section we study the Hessian of E L at its min-imizers, showing that it is strictly positive, universally, for L big enough. Positivity is ofcourse understood up to the trivial zero modes resulting from the symmetries of the prob-lem (translations and changes of phase). This is obtained by comparing E L with E andexploiting Theorem A.For any minimizer ψ ∈ M E L , the Hessian of E L at ψ is defined by lim ε → ε (cid:18) E L (cid:18) ψ + εf k ψ + εf k (cid:19) − e L (cid:19) = H E L ψ ( f ) ∀ f ∈ H ( T L ) . (3.1.65)An explicit computation gives H E L ψ ( f ) = h Im f | L Lψ | Im f i + h Re f | Q ψ ( L Lψ − X Lψ ) Q ψ | Re f i , (3.1.66)with Q ψ = 11 − | ψ ih ψ | and L Lψ := − ∆ L + V σ ψ − µ Lψ , X Lψ ( x, y ) := − ψ ( x )∆ − L ( x, y ) ψ ( y ) . (3.1.67)(We use the same notation for the operator X Lψ and its integral kernel for simplicity.) Werecall that µ Lψ = T L ( ψ ) − W L ( ψ ) and that V σ ψ = 2∆ − L ρ ψ and we note that L Lψ ψ = 0 isexactly the Euler–Lagrange equation derived in Lemma 3.3.By minimality of ψ , we already know that inf spec L Lψ = inf spec Q ψ ( L Lψ − X Lψ ) Q ψ ≥ , and it is actually equal to since ψ is in the kernel of both operators. Moreover, ker L Lψ =span { ψ } , since it is a Schrödinger operator of least (simple) eigenvalue . The situationis more complicated for Q ψ ( L Lψ − X Lψ ) Q ψ , whose kernel contains at least ψ and ∂ i ψ (by the translation invariance of the problem). Since both L Lψ and Q ψ ( L Lψ − X Lψ ) Q ψ have compact resolvents (they are given by bounded perturbations of − ∆ L ), they bothhave discrete spectrum. Our aim is two-fold: first we need to show that the kernel of Q ψ ( L Lψ − X Lψ ) Q ψ is exactly spanned by ψ and its partial derivatives, secondly we want toshow that the spectral gap (above the trivial zero modes) of both operators is bounded by auniversal positive constant.Before stating the main result of this section, we introduce the relevant full-space ob-jects: let again Ψ be the unique positive and radial full-space minimizer of the Pekar func-tional (3.1.1) and, analogously to (3.1.67), define L Ψ := − ∆ R + V σ Ψ − µ Ψ , X Ψ ( x, y ) := Ψ ( x )( − ∆ R ) − ( x, y ) Ψ ( y ) . (3.1.68)We introduce h ′∞ := inf f ∈ H R ( R , k f k f ∈ (span { Ψ } ) ⊥ h f | L Ψ | f i ,h ′′∞ := inf f ∈ H R ( R , k f k f ∈ (span { Ψ,∂ Ψ,∂ Ψ,∂ Ψ } ) ⊥ h f | L Ψ − X Ψ | f i . (3.1.69)We emphasize that the results contained in [15] imply that min { h ′∞ , h ′′∞ } > . Moreover,it is easy to see, using that V σ Ψ ( x ) . −| x | − for large x , that L Ψ has infinitely manyeigenvalues between , its least and simple eigenvalue with eigenfunction given by Ψ , and − µ Ψ , the bottom of its continuous spectrum. Since furthermore X Ψ is positive, this implies,in particular, that h ′′∞ , h ′∞ < − µ Ψ , (3.1.70)which we shall use later. Proposition 3.2.
For any
L > , we define h ′ L := inf ψ ∈M E L inf f ∈ H R ( T L ) , k f k f ∈ (span { ψ } ) ⊥ h f | L Lψ | f i , (3.1.71) h ′′ L := inf ψ ∈M E L inf f ∈ H R ( T L ) , k f k f ∈ (span { ψ,∂ ψ,∂ ψ,∂ ψ } ) ⊥ h f | L Lψ − X Lψ | f i . (3.1.72) Then lim inf L →∞ h ′ L ≥ h ′∞ , lim inf L →∞ h ′′ L ≥ h ′′∞ . (3.1.73)It is not difficult to show that lim sup L →∞ h ′ L ≤ h ′∞ , lim sup L →∞ h ′′ L ≤ h ′′∞ , (3.1.74)simply by considering localizations of the full-space optimizers and using Proposition 3.1.Hence there is actually equality in (3.1.73).To prove Proposition 3.2 we need the following two Lemmas. Lemma 3.5.
For ψ ∈ M E L , the operator Y Lψ with integral kernel Y Lψ ( x, y ) := ∆ − L ( x, y ) ψ ( y ) is universally bounded from L ( T L ) to L ∞ ( T L ) . This in particular implies that the opera-tors X Lψ , defined in (3.1.67) , are universally bounded from L ( T L ) to L ( T L ) . TRONGLY COUPLED POLARON ON THE TORUS 17
Proof.
Using Lemma 3.1 and the normalization of ψ , we have | Y Lψ ( f )( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ T L ∆ − L ( x, y ) ψ ( y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k f k + ˆ T L | ψ ( y ) f ( y ) | π dist T L ( x, y ) dy . k f k + ˆ B ( x ) | ψ ( y ) f ( y ) | dist T L ( x, y ) dy ≤ (1 + C k ψ k ∞ ) k f k . k f k . (3.1.75)To conclude, we also made use of the fact that the minimizers are universally bounded in L ∞ by Lemma 3.3. (cid:3) Recall the definition of Ψ L in (3.1.35). Lemma 3.6.
For any ε > , there exists R ′ ε and L ′ ε (with R ′ ε ≤ L ′ ε / ) such that for any L > L ′ ε , any normalized f in L ( T L ) supported on B cR ′ ε := [ − L/ , L/ \ B R ′ ε , and any ψ ∈ M E L such that k ψ − Ψ L k H ( T L ) = dist H (Θ L ( ψ ) , Ψ L ) (3.1.76) we have h f | L Lψ − X Lψ | f i ≥ − µ Ψ − ε. (3.1.77) Proof.
By definition of L Lψ and X Lψ , we have h f | L Lψ − X Lψ | f i = T L ( f ) − µ Lψ + h f | V σ ψ | f i − h f | X Lψ | f i≥ − µ Lψ + h f | V σ ψ | f i − h f | X Lψ | f i . (3.1.78)By Proposition 3.1, taking L ′ ε sufficiently large guarantees that | µ Lψ − µ Ψ | ≤ ε/ . (3.1.79)Thus we only need to show that h f | V σ ψ | f i and h f | X Lψ | f i can be made arbitrary small bytaking L ′ ε and R ′ ε sufficiently large. Since f is normalized and supported on B cR ′ ε , | h f | V σ ψ | f i | ≤ k V σ ψ k L ∞ (cid:16) B cR ′ ε (cid:17) . (3.1.80)Moreover, using Lemma 3.1, splitting the integral over B t ( x ) and B ct ( x ) (for some t > ),and assuming x ∈ B cR ′ ε , we find | V σ ψ ( x ) | ≤ CL + C ˆ T L | ψ ( y ) | dist T L ( x, y ) dy ≤ CL + Ct k ψ k L (cid:16) B cR ′ ε − t (cid:17) + 1 /t. (3.1.81)On the other hand, by Lemma 3.5, | h f | X Lψ | f i | ≤ C k f k ˆ T L ψ ( y ) | f ( y ) | dy ≤ C k χ B cR ′ ε ψ k . (3.1.82)Therefore, by applying Proposition 3.1, we can conclude that there exists L ′ ε and R ′ ε suchthat, for any L > L ′ ε and any L -normalized f supported on B cR ′ ε , we have h f | V σ ψ | f i − h f | X Lψ | f i ≥ − ε/ , (3.1.83)which concludes our proof. (cid:3) Proof of Proposition 3.2.
We only show the second inequality in (3.1.73), as its proof caneasily be modified to also show the first. Moreover, we observe that the second inequalityin (3.1.73) is equivalent to the statement that for any sequence ψ n ∈ M L n with L n → ∞ , lim inf n inf f ∈ H T Ln ) , k f k f ∈ span { ψn,∂ ψn,∂ ψn,∂ ψn }⊥ h f | L L n ψ n − X L n ψ n | f i ≥ h ′′∞ , (3.1.84)which we shall prove in the following.We consider ψ n ∈ M L n , L n → ∞ , and define h n := inf f ∈ H T Ln ) , k f k f ∈ span { ψn,∂ ψn,∂ ψn,∂ ψn }⊥ h f | L L n ψ n − X L n ψ n | f i . (3.1.85)By translation invariance of E L n and by Proposition 3.1, we can also restrict to sequences ψ n converging to Ψ in L ( R ) and such that k ψ n − Ψ L n k H ( T Ln ) → , where Ψ L n is definedin (3.1.35).Let now g n be a normalized function in L ( T L n ) , orthogonal to ψ n and its partial deriva-tives, realizing h n (which exists by compactness, and can be taken to be a real-valuedfunction). We define the following partition of unity ≤ η R , η R ≤ , with η iR ∈ C ∞ ( R ) , η iR ( x ) = η i ( x/R ) and η ( x ) = ( x ∈ B , x ∈ B c η = p − | η | . (3.1.86)We define η in := η iL n / and g in := η in g n / k η in g n k . (3.1.87)Standard properties of IMS localization imply that h n = h g n | L L n ψ n − X L n ψ n | g n i = X i =1 , k η in g n k (cid:10) g in (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g in (cid:11) − X i =1 , (cid:16) h g n ||∇ η in | | g n i + 2 h g n | [ η in , [ η in , X L n ψ n ]] | g n i (cid:17) . (3.1.88)Clearly, the first summand in the second sum is of order O ( L − n ) , by the scaling of η in . Forthe second summand, we observe that [ η in , [ η in , X L n ψ n ]]( x, y ) = ψ n ( x )( − ∆ L n ) − ( x, y ) ψ n ( y ) (cid:0) η in ( x ) − η in ( y ) (cid:1) , (3.1.89)and proceed to bound the Hilbert-Schmidt norm of both operators ( i = 1 , ), which willthen bound the last line of (3.1.88). We make use of Lemma 3.1 to obtain ˆ T Ln × T Ln | ∆ − L n ( x, y ) | ψ n ( x ) ψ n ( y ) (cid:0) η Ri ( x ) − η in ( y ) (cid:1) dxdy . L n + ˆ T Ln × T Ln ( η in ( x ) − η in ( y )) d T Ln ( x, y ) ψ n ( x ) ψ n ( y ) dxdy ≤ L n + k∇ η in k ∞ . (3.1.90)Therefore, also the second summand in the error terms is order L − n , which allows us toconclude that X i =1 , k η in g n k (cid:10) g in (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g in (cid:11) = h n + O ( L − n ) . (3.1.91) TRONGLY COUPLED POLARON ON THE TORUS 19
By Lemma 3.6 applied to g n (which is supported on B cL n / ) and (3.1.70), we find (cid:10) g n (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g n (cid:11) ≥ − µ Ψ + o n (1) > h ′′∞ + o n (1) . (3.1.92)Since the l.h.s. of (3.1.91) is a convex combination and ( L L n ψ n − X L n ψ n ) is uniformly boundedfrom below, (3.1.92) allows to restrict to sequences ψ n such that k η n g n k ≥ C (3.1.93)uniformly in n and (cid:10) g n (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g n (cid:11) ≤ h n + o n (1) , (3.1.94)since our claim holds on any sequence for which (3.1.93) and (3.1.94) are not simultane-ously satisfied. Using (3.1.93) it is easy to see that g n is almost orthogonal to ψ n , in thesense that | (cid:10) g n (cid:12)(cid:12) ψ n (cid:11) | = 1 k g n η n k | (cid:10) g n ( η n − (cid:12)(cid:12) ψ n (cid:11) | ≤ C k (1 − η n ) ψ n k ≤ C k χ B cLn/ ψ n k n →∞ −−−→ . (3.1.95)Here we used the L -convergence of ψ n to Ψ . Clearly, the same computation (togetherwith the H -convergence of ψ n to Ψ ) shows that g n is also almost orthogonal to the partialderivatives of ψ n .To conclude, we wish to modify g n in order to obtain a function ˜ g n which satisfiesthe constraints (i.e., is a competitor) of the full-space variational problem introduced in(3.1.69). We also wish to have h ˜ g n | L Ψ − X Ψ | ˜ g n i = (cid:10) g n (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g n (cid:11) + o n (1) . (3.1.96)Indeed, (3.1.96) together with (3.1.94) and the fact that ˜ g n is a competitor on R , wouldimply that h n ≥ (cid:10) g n (cid:12)(cid:12) L L n ψ n − X L n ψ n (cid:12)(cid:12) g n (cid:11) − o n (1) = h ˜ g n | L Ψ − X Ψ | ˜ g n i − o n (1) ≥ h ′′∞ − o n (1) , (3.1.97)which finally yields the proof of the Proposition also for sequences ψ n satisfying (3.1.93)and (3.1.94).We have a natural candidate for ˜ g n , which is simply ˜ g n := (11 − P ) g n k (11 − P ) g n k , (3.1.98)with P ( g n ) := Ψ h Ψ | g n i + P i =1 , , ∂ i Ψ k ∂ i Ψ k D ∂ i Ψ k ∂ i Ψ k (cid:12)(cid:12)(cid:12) g n E . Clearly ˜ g n is a competitor forthe full space minimization and we are only left with the task of proving that ˜ g n satisfies(3.1.96). We observe that, since g n is almost orthogonal to ψ n and its partial derivatives,and using Proposition 3.1, | (cid:10) Ψ (cid:12)(cid:12) g n (cid:11) | ≤ k Ψ − ψ n k L ( B Ln/ ) + | (cid:10) ψ n (cid:12)(cid:12) g n (cid:11) | = o n (1) , | (cid:10) ∂ i Ψ (cid:12)(cid:12) g n (cid:11) | ≤ k Ψ − ψ L k H ( B Ln/ ) + | (cid:10) ∂ i ψ n (cid:12)(cid:12) g n (cid:11) | = o n (1) . (3.1.99)Therefore kP ( g n ) k → and k (11 − P ) g n k → . (3.1.100) Hence, the normalization factor does not play any role in the proof of (3.1.96). Moreover (cid:10) (11 − P ) g n (cid:12)(cid:12) ( L Ψ − X Ψ ) (cid:12)(cid:12) (11 − P ) g n (cid:11) = (cid:10) g n (cid:12)(cid:12) ( L Ψ − X Ψ ) (cid:12)(cid:12) g n (cid:11) + (cid:10) P ( g n ) (cid:12)(cid:12) ( L Ψ − X Ψ ) (cid:12)(cid:12) P ( g n ) (cid:11) − (cid:10) g n (cid:12)(cid:12) ( L Ψ − X Ψ ) (cid:12)(cid:12) P ( g n ) (cid:11) , (3.1.101)and thus we can conclude that also P ( g n ) does not play any role in the proof of (3.1.96),since ( L Ψ − X Ψ ) P is a bounded operator ( P has finite dimensional range contained in thedomain of ( L Ψ − X Ψ ) ), P is a projection and kP ( g n ) k → . With this discussion, wereduced our problem to showing that (cid:10) g n (cid:12)(cid:12) ( L Ψ − X Ψ ) (cid:12)(cid:12) g n (cid:11) = (cid:10) g n (cid:12)(cid:12) ( L L n ψ n − X L n ψ n ) (cid:12)(cid:12) g n (cid:11) + o n (1) . (3.1.102)Clearly the kinetic energy terms coincide for every n and µ L n ψ n → µ , by Proposition 3.1.Therefore we only need to prove that | (cid:10) g n (cid:12)(cid:12) V σ ψn − V σ Ψ (cid:12)(cid:12) g n (cid:11) | , | (cid:10) g n (cid:12)(cid:12) X L n ψ n − X Ψ (cid:12)(cid:12) g n (cid:11) | → . (3.1.103)For the first term, using that g n is supported on B L n / , we have | (cid:10) g n (cid:12)(cid:12) V σ ψn − V σ Ψ (cid:12)(cid:12) g n (cid:11) | ≤ k V σ Ψ − V σ ψn k L ∞ ( B Ln/ ) . (3.1.104)If we define Ψ R := χ B R Ψ and ( ψ n ) R := χ B R ψ n we have V σ Ψ = V σ ΨR + V σ [ Ψ − ΨR ] and V σ ψn = V σ ( ψn ) R + V σ [ ψn − ( ψn ) R ] . We consider R = R ( n ) = L n / and observe that | V σ [ Ψ − ΨR ] ( x ) | = 2 ˆ R ( − ∆ R ) − ( x, y )( Ψ − Ψ R ) dy . k Ψ − Ψ R k + k Ψ − Ψ R k → . (3.1.105)Similar computations, together with Lemma 3.1, yield similar estimates for | V σ [ ψn − ( ψn ) R ] ( x ) | .Moreover, since dist T Ln ( x, y ) = | x − y | for x, y ∈ B L n / , we have, for any x ∈ B L n / | ( V σ ΨR − V σ ( ψn ) R )( x ) | . (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B Ln/ | x − y | ( Ψ ( y ) − ψ n ( y ))( Ψ ( y ) + ψ n ( y )) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 1 L n . k Ψ + ψ n k ∞ k Ψ − ψ n k + k Ψ − ψ n k k Ψ + ψ n k + 1 L n → . (3.1.106)Here we used again Lemma 3.1, the convergence of ψ n to Ψ and the universal L ∞ -bounded-ness of minimizers. Putting the pieces together we obtain k V σ Ψ − V σ ψn k L ∞ ( B Ln/ ) ≤ k V σ [ Ψ − ΨR ] k ∞ + k V σ [ ψn − ( ψn ) R ] k ∞ + k V σ ΨR − V σ ( ψn ) R k L ∞ ( B R ( n ) ) → , (3.1.107)as desired. The study is similar for h g n | X L n ψ n − X Ψ | g n i , hence we shall not write it downexplicitly.We conclude that (3.1.102) holds and, by the discussion above, the proof is complete. (cid:3) Proof of Theorem 2.1.
In this section we first prove universal local bounds for E L around minimizers. These are a direct consequence of the results on the Hessian in theprevious subsection, the proof follows along the lines of [7], [9, Appendix A] and [5, Ap-pendix A]. Such universal local bounds yield universal local uniqueness of minimizers, i.e.,the statement that minimizers that are not equivalent (i.e., not obtained one from the otherby translations and changes of phase) must be universally apart (in H ( T L ) ). Together with TRONGLY COUPLED POLARON ON THE TORUS 21
Proposition 3.1, this clearly implies uniqueness of minimizers for L big enough, which isthe first part of Theorem 2.1. A little extra effort will then complete the proof of Theorem2.1.In this section, for any ψ ∈ M E L and any f ∈ L ( T L ) , we write e iθ ψ y = P L Θ L ( ψ ) ( f ) ,respectively e iθ ψ y = P H Θ L ( ψ ) ( f ) , to mean that e iθ ψ y realizes the L -distance, respectivelythe H -distance, between f and Θ L ( ψ ) . Note that by compactness these always exist, butthey might not be unique. The possible lack of uniqueness is not a concern for our analysis,however. Proposition 3.3 (Universal Local Bounds) . There exist universal constants K > and K > and L ∗∗ > such that, for any L > L ∗∗ , any ψ ∈ M E L and any L -normalized f ∈ H ( T L ) with dist H (Θ L ( ψ ) , f ) ≤ K , (3.1.108) we have E L ( f ) − e L ≥ K k P L Θ L ( ψ ) ( f ) − f k H ( T L ) ≥ K dist H (Θ L ( ψ ) , f ) . (3.1.109) Proof.
We can restrict to positive ψ ∈ M E L and normalized f such that P L Θ L ( ψ ) ( f ) = ψ, (3.1.110)which clearly implies h ψ | f i ≥ , h Re f | ∂ i ψ i = 0 . (3.1.111)Under this assumption, we prove that if (3.1.108) holds then E L ( φ ) − e L ≥ K k ψ − f k H ( T L ) ≥ K dist H (Θ L ( ψ ) , f ) . (3.1.112)The general result follows immediately by invariance of E L under translations and changesof phase.We denote δ := f − ψ and proceed to expand E L around ψ : E L ( f ) = E L ( ψ + δ ) = e L + H E L ψ ( δ ) + Err ψ ( δ ) . (3.1.113)We recall that H E L ψ is simply the quadratic form associated to the Hessian of E L at ψ and itis defined in (3.1.66). We denote P ψ := | ψ i h ψ | . The last term, which we see as an errorcontribution, is explicitly given by Err ψ ( δ ) = − h Re δ | X Lψ | P ψ Re δ i + 4 h P ψ Re δ | X Lψ | P ψ Re δ i− (cid:10) | δ | (cid:12)(cid:12) − ∆ − L | ψ Re δ i + W L ( δ ) . (3.1.114)Our first goal is to estimate | Err ψ ( δ ) | . By (3.1.111) and the normalization of both ψ and f , we find k δ k = 2 − h ψ | f i . (3.1.115)Therefore, also using the positivity of ψ , we have P ψ Re δ = ψ ( h ψ | f i −
1) = − ψ k δ k . (3.1.116)We now apply Lemma 3.5 to obtain | h Re δ | X Lψ | P ψ Re δ i | . k Re δ k k P ψ Re δ k . k δ k , | h P ψ Re δ | X Lψ | P ψ Re δ i | . k P ψ Re δ k . k δ k , | (cid:10) | δ | (cid:12)(cid:12) − ∆ − L | ψ Re δ i | . k δ k k Re δ k ≤ k δ k . (3.1.117) Finally, by (3.1.20), W L ( δ ) = k δ k W L (cid:18) δ k δ k (cid:19) ≤ k δ k (cid:18) T L (cid:18) δ k δ k (cid:19) + C (cid:19) . k δ k k δ k H ( T L ) . (3.1.118)Recalling (3.1.108), we can estimate k δ k = dist L ( f, Θ L ( ψ )) ≤ dist H ( f, Θ L ( ψ )) ≤ K , (3.1.119)and this implies, combined with (3.1.117) and (3.1.118), that | Err ψ ( δ ) | . k δ k H ( T L ) . (3.1.120)We now want to bound H E L ψ ( δ ) . We fix < τ < min { h ′∞ , h ′′∞ } , where h ′∞ and h ′′∞ aredefined in (3.1.69). Proposition 3.2 implies that there exists L ∗∗ such that for L > L ∗∗ and ψ ∈ M E L , we have L Lψ ≥ τ Q ψ , Q ψ ( L Lψ − X Lψ ) Q ψ ≥ τ Q ′ ψ , (3.1.121)where we define Q ψ = 11 − P ψ and Q ′ ψ := 11 − P ψ − P i =1 , , P ∂ i ψ/ k ∂ i ψ k . We note that, by(3.1.111) and since ψ is orthogonal in L to its partial derivatives, we have Q ψ (Re f − ψ ) = Q ′ ψ (Re f − ψ ) . (3.1.122)Therefore, recalling the definition of H E L ψ given in (3.1.66), H E L ψ ( δ ) = h Im f | L Lψ | Im f i + h Re f − ψ | Q ψ ( L Lψ − X Lψ ) Q ψ | Re f − ψ i≥ τ ( k Q ψ Im f k + k Q ′ ψ (Re f − ψ ) k ) = τ k Q ψ δ k L ( T L ) . (3.1.123)Moreover, applying (3.1.115), k Q ψ δ k L ( T L ) = k δ k − h ψ | δ i = k δ k (cid:18) − k δ k (cid:19) ≥ k δ k , (3.1.124)and we can thus conclude that H E L ψ ( δ ) ≥ τ k δ k . (3.1.125)On the other hand, by the universal boundedness of V σ ψ in L ∞ ( T L ) and the universalboundedness of µ Lψ (see Proposition 3.1), we have, for some universal C > , L Lψ ≥ − ∆ L − C . (3.1.126)Similarly, also using Lemma 3.5, for some universal C > , Q ( L Lψ − X Lψ ) Q ≥ − ∆ L − C . (3.1.127)If we then define C := (max { C , C } + 1) , we can conclude the validity of the universalbound H E L ψ ( δ ) ≥ k δ k H ( T L ) − C k δ k L ( T L ) . (3.1.128)By interpolating between (3.1.125) and (3.1.128), we obtain H E L ψ ( δ ) ≥ ττ + 2 C k δ k H ( T L ) . (3.1.129)Using (3.1.120) and (3.1.129) in (3.1.113), we can conclude that there exists a universalconstant C such that for any L > L ∗∗ , any < ψ ∈ M E L and any normalized f satisfying(3.1.110), E L ( f ) − e L ≥ C k δ k H ( T L ) − C k δ k H ( T L ) . (3.1.130) TRONGLY COUPLED POLARON ON THE TORUS 23
In particular, for K sufficiently small, we can find a universal constant c such that (3.1.112)holds, as long as k δ k H ( T L ) = k P L Θ L ( ψ ) ( f ) − f k H ( T L ) ≤ c. (3.1.131)To conclude the proof, it only remains to show that there exists a universal K such that(3.1.131) holds as long as (3.1.108) holds. This can be achieved as follows. We have,using that both ψ and P H Θ L ( ψ ) ( f ) are in M E L and thus are universally bounded in H ( T L ) (by Lemma 3.3) and recalling that ψ = P L Θ L ( ψ ) ( f ) , k ψ − P H Θ L ( ψ ) ( f ) k ˚ H ( T L ) ≤ k ψ − P H Θ L ( ψ ) ( f ) k / L ( T L ) k − ∆ L ( ψ − P H Θ L ( ψ ) ( f )) k / L ( T L ) . k ψ − P H Θ L ( ψ ) ( f ) k / L ( T L ) ≤ (cid:16) dist L (Θ L ( ψ ) , f ) + k f − P H Θ L ( ψ ) ( f ) k L ( T L ) (cid:17) / . dist / H (Θ L ( ψ ) , f ) . (3.1.132)Therefore, for some universal C k f − ψ k H ( T L ) ≤ dist H (Θ L ( ψ ) , f ) + C dist / H (Θ L ( ψ ) , f ) , (3.1.133)and it suffices to take K ≤ (cid:2) ( − C + √ C + 4 c ) / (cid:3) to conclude our discussion. (cid:3) We are ready to prove Theorem 2.1.
Proof of Theorem 2.1.
Fix K as in Proposition 3.3. Using Proposition 3.1, we know thatthere exists L K / such that, for any L > L K / and any ψ ∈ M E L , we have dist H (Θ L ( ψ ) , Ψ L ) ≤ K / . (3.1.134)We claim that (2.2.9) holds with L := max { L K / , L ∗ , L ∗∗ } , where L ∗ is the same as inCorollary 3.1 and L ∗∗ is the same as in Proposition 3.3.Let L > L and ψ ∈ M E L . Since L > L ≥ L ∗ , we have ψ y = ψ for any = y ∈ T L .Moreover, since L > L ≥ L K / and using the triangle inequality, for any other ψ ∈ M E L we have dist H (Θ L ( ψ ) , ψ ) ≤ K . (3.1.135)Since L > L ≥ L ∗∗ , we can apply Proposition 3.3, finding K dist H (Θ L ( ψ ) , ψ ) ≤ E L ( ψ ) − e L = 0 , (3.1.136)i.e., ψ ∈ Θ L ( ψ ) , and (2.2.9) holds for L > L .For ψ ∈ M E L = Θ L ( ψ ) , and L > L , we now show the quadratic lower bound (2.2.10),independently of L . Lemma 3.3, which guarantees universal H -boundedness of mini-mizers, and estimate (3.1.20) ensure, by straightforward computations, that there exists < κ ∗ < / such that, if f ∈ L ( T L ) is normalized and satisfies E L ( f ) − e L < κ ∗ dist H (Θ L ( ψ ) , f ) , (3.1.137)then f is universally bounded in H ( T L ) and must satisfy E L ( f ) − e L < δ K , (3.1.138)where δ K is the δ ε from Proposition 3.1 with ε = K . On the other hand, Proposition3.1 and Proposition 3.3 combined with the fact that we have taken L ≥ L K / (and that trivially L K / ≥ L K ), guarantee that any L -normalized f satisfying (3.1.138) mustsatisfy E L ( f ) − e L ≥ K dist H (Θ L ( ψ ) , f ) . (3.1.139)Therefore the bound (2.2.10) from Theorem 2.1 holds with the universal constant κ :=min { κ ∗ , K } and our proof is complete. (cid:3) This concludes our study of E L . We now move on to the study of the functional F L .3.2. Study of F L . This section is structured as follows. In Section 3.2.1 we prove Corol-lary 2.1. In Section 3.2.2, we compute the Hessian of F L at its minimizers, showing thevalidity of (2.2.16). This allows to obtain a more precise lower bound for F L (comparedto the bounds (2.2.13) and (2.2.14) from Corollary 2.1), which holds locally around the -dimensional surface of minimizers M F L = Ω L ( ϕ L ) . Finally, in Section 3.2.3, we investi-gate closer the surface of minimizers Ω L ( ϕ L ) and the behavior of the functional F L close toit. In particular, we show that the Hessian of F L at its minimizers is strictly positive aboveits trivial zero modes and derive some key technical tools, which we exploit in Section 4.3.2.1. Proof of Corollary 2.1.
In this section, we show the validity of Corollary 2.1. Weneed the following Lemma. Recall that in our discussion constants are universal if they areindependent of L for L ≥ L > . Lemma 3.7.
For ψ, φ ∈ H ( T L ) , k ψ k = k φ k = 1 , h ρ ψ − ρ φ | ( − ∆ L ) − / | ρ ψ − ρ φ i . k| ψ | − | φ |k H ( T L ) . (3.2.1) Proof.
We define f ( x ) := | ψ ( x ) | + | φ ( x ) | and g ( x ) := | ψ ( x ) | − | φ ( x ) | . By the Hardy-Littlewood-Sobolev and the Sobolev inequality (see for example [3] for a comprehensiveoverview of such results on the torus), and using the normalization of φ and ψ we have h ρ ψ − ρ φ | ( − ∆ L ) − / | ρ ψ − ρ φ i = k ( − ∆ L ) − / ( f g ) k ≤ C k f g k / ≤ C k f k k g k ≤ C ′ k g k H ( T L ) = C ′ k| ψ | − | φ |k H ( T L ) , (3.2.2)which proves the Lemma. (cid:3) Proof of Corollary 2.1.
With ψ L as in Theorem 2.1, let ϕ L := σ ψ L ∈ C ∞ ( T L ) . Observingthat G L ( ψ, ϕ ) = E L ( ψ ) + k σ ψ − ϕ k , (3.2.3)and using Theorem 2.1 we can immediately conclude that in the regime L > L M F L = Ω L ( ϕ L ) . (3.2.4)It is also immediate, recalling the definition of G L in (2.2.1) and that ψ L > , to concludethat ψ L must be the unique positive ground state of h ϕ L .To prove (2.2.13), we first of all observe that if ϕ ∈ L ( T L ) , we have F L ( ϕ ) = | ( ϕ ) | + F L ( ˆ ϕ ) . (3.2.5)Therefore, it is sufficient to restrict to ϕ with zero-average and show that in this case F L ( ϕ ) − e L ≥ min y ∈ T L h ϕ − ϕ yL | − (11 + κ ′ ( − ∆ L ) / ) − | ϕ − ϕ yL i . (3.2.6) TRONGLY COUPLED POLARON ON THE TORUS 25
Using Theorem 2.1, we obtain G L ( ψ, ϕ ) − e L = E L ( ψ ) − e L + k ϕ − σ ψ k ≥ E L ( | ψ | ) − e L + k ϕ − σ ψ k ≥ κ dist H ( | ψ | , Θ( ψ L )) + k ϕ − σ ψ k = κ k| ψ | − ψ yL k H ( T L ) + k ϕ − σ ψ k , (3.2.7)for some y ∈ T L . We now apply Lemma 3.7 and recall that ϕ yL = σ ψ yL , obtaining with asimple completion of the square G L ( ψ, ϕ ) − e L ≥ κ ′ D ρ ψ − ρ ψ yL (cid:12)(cid:12)(cid:12) ( − ∆ L ) − / (cid:12)(cid:12)(cid:12) ρ ψ − ρ ψ yL E + k ϕ − σ ψ k = k F / ( σ ψ − ϕ yL ) + F − / ( ϕ yL − ϕ ) k + h ϕ − ϕ yL | − F − | ϕ − ϕ yL i , (3.2.8)where F = 11 + κ ′ ( − ∆ L ) / . Dropping the first term and minimizing over ψ yields ourclaim. Finally, (2.2.14) immediately follows from (2.2.13) and the spectral gap of theLaplacian, using the fact that ϕ L and all its translates have zero average since ϕ L = σ ψ L . (cid:3) The Hessian of F L . For any ϕ ∈ L R ( T L ) , we introduce the notation e ( ϕ ) := inf spec h ϕ , (3.2.9)and observe that F L , defined in (2.2.4), can equivalently be written as F L ( ϕ ) = k ϕ k + e ( ϕ ) , ϕ ∈ L R ( T L ) . (3.2.10)We compute the Hessian of F L at its minimizers using standard arguments in pertur-bation theory, showing the validity of expression (2.2.16). We need the following twoLemmas. Lemma 3.8.
For L ≥ L > , any ϕ ∈ L ( T L ) and any T > k ( − ∆ L + T ) − ϕ k = k ϕ ( − ∆ L + T ) − k ≤ C T k ϕ k L ( T L )+ L ∞ ( T L ) (3.2.11) for some constant C T > with lim T →∞ C T = 0 . Here ϕ is understood as a multiplicationoperator, k · k denotes the operator norm on L ( T L ) , and k ϕ k L ( T L )+ L ∞ ( T L ) := inf ϕ ϕ ϕϕ ∈ L T L ) , ϕ ∈ L ∞ ( T L ) (cid:16) k ϕ k L ( T L ) + k ϕ k L ∞ ( T L ) (cid:17) . (3.2.12)Note that k ϕ k L ( T L ) ≤ L / k ϕ k L ( T L )+ L ∞ ( T L ) ≤ L / k ϕ k L ( T L ) , (3.2.13)which clearly makes the two norms equivalent. Nevertheless, we find it more natural towork with a bound of the form (3.2.11), where C T is independent of L .Lemma 3.8 implies that, for any ϕ ∈ L ( T L ) + L ∞ ( T L ) , the multiplication operatorassociated with ϕ is infinitesimally relatively bounded with respect to − ∆ L . More pre-cisely, for any δ > , there exists C (cid:16) δ, k ϕ k L ( T L )+ L ∞ ( T L ) (cid:17) depending on ϕ only through k ϕ k L ( T L )+ L ∞ ( T L ) , such that for any f ∈ Dom ( − ∆ L ) k ϕf k ≤ δ k ∆ L f k + C (cid:16) δ, k ϕ k L ( T L )+ L ∞ ( T L ) (cid:17) k f k . (3.2.14)Whenever infinitesimal relative boundedness holds with a constant C ( δ ) uniform over aclass of operators, we will say that the class is uniformly infinitesimally relatively bounded. In this case, Lemma 3.8 ensures that multiplication operators associated to functions in ( L + L ∞ ) -balls are uniformly infinitesimally relatively bounded with respect to − ∆ L . Proof.
We first observe that, by self-adjointness of ( − ∆ L + T ) − , it is sufficient to showthat the claimed bound holds for k ϕ ( − ∆ L + T ) − k . For any f, ϕ ∈ L ( T L ) and anydecomposition of the form ϕ = ϕ + ϕ with ϕ ∈ L ( T L ) and ϕ ∈ L ∞ ( T L ) we have k ϕ ( − ∆ L + T ) − f k ≤ k ϕ k k ( − ∆ L + T ) − f k ∞ + k ϕ k ∞ k ( − ∆ L + T ) − f k ≤ k ϕ k k ( − ∆ L + T ) − f k ∞ + T − k ϕ k ∞ k f k . (3.2.15)Moreover, k ( − ∆ L + T ) − f k ∞ ≤ X k ∈ πL Z L / ( | k | + T ) | f k | ≤ L X k ∈ πL Z | k | + T ) / k f k ≤ C (cid:18) ˆ R | x | + T ) (cid:19) / k f k = CT − / k f k . (3.2.16)Therefore, picking C T := max (cid:8) T − , CT − / (cid:9) yields k ϕ ( − ∆ L + T ) − f k ≤ C T ( k ϕ k + k ϕ k ∞ ) k f k , (3.2.17)optimizing over ϕ and ϕ completes the proof. (cid:3) Lemma 3.9.
For ϕ ∈ L ( T L ) k ( − ∆ L ) − / ϕ k L ∞ ( T L )+ L ( T L ) . k ( − ∆ L + 1) − / ϕ k L ( T L ) . (3.2.18) Proof.
We write f = χ [0 , and f = χ [1 , + ∞ ) and ϕ = f (cid:2) ( − ∆ L ) − / (cid:3) ϕ, ϕ = f (cid:2) ( − ∆ L ) − / (cid:3) ϕ. (3.2.19)Clearly ( − ∆ L ) − / ϕ = ϕ + ϕ . Moreover k ( − ∆ L ) − / ϕ k L ∞ + L ≤ k ϕ k ∞ + k ϕ k ≤ X = k ∈ πL Z | k | < L | k | / X = k ∈ πL Z | k | < | ϕ k | / + X k ∈ πL Z | k |≥ | ϕ k | | k | / . X = k ∈ πL Z | k | < | ϕ k | / + X k ∈ πL Z | k |≥ | ϕ k | | k | / . X k ∈ πL Z | k | + 1 | ϕ k | / = C k ( − ∆ L + 1) − / ϕ k L ( T L ) . (3.2.20)This concludes the proof. (cid:3) Lemmas 3.8 and 3.9 together yield the following Corollary, whose proof is omitted as itis now straightforward.
TRONGLY COUPLED POLARON ON THE TORUS 27
Corollary 3.2.
For any ϕ such that k ( − ∆ L + 1) − / ϕ k is finite, the multiplication op-erator V ϕ (defined in (2.2.2) ) is infinitesimally relatively bounded with respect to ( − ∆ L ) .Moreover, for T > there exists C T such that k ( − ∆ L + T ) − V ϕ k ≤ C T k ( − ∆ L + 1) − / ϕ k , and C T ց as T → ∞ . (3.2.21)In particular, Corollary 3.2 implies that the family of multiplication operators associatedto { V ϕ | k ( − ∆ L + 1) − / ϕ k ≤ M } is uniformly infinitesimally relatively bounded withrespect to − ∆ L for any M .With these tools at hand we now investigate F L close to its minimum and, in particular,compute the Hessian of F L at its minimizers. We follow very closely the analogous anal-ysis carried out in [9]. By translation invariance of the problem, it is clearly sufficient toperform the computation with respect to ϕ L , where ϕ L is the same as in Corollary 2.1. Proposition 3.4.
For
L > L let ϕ ∈ L R ( T L ) be such that k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k L ( T L ) ≤ ε L (3.2.22) for some ε L > small enough. Then |F L ( ϕ ) − e L − h ϕ − ϕ L | − K L | ϕ − ϕ L i| . L k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k h ϕ − ϕ L | J L | ϕ − ϕ L i , (3.2.23) where K L := 4( − ∆ L ) − / ψ L Q ψ L h ϕ L − e ( ϕ L ) ψ L ( − ∆ L ) − / ,J L = 4( − ∆ L ) − / ψ L ( − ∆ L + 1) − ψ L ( − ∆ L ) − / , (3.2.24) and ψ L , which we recall is the (positive) ground state of h ϕ L , is understood, in the expres-sions for K L and J L , as a multiplication operator. Note that this implies that H F L ϕ L = 11 − K L , as claimed in (2.2.16). In particular, K L ≤ by minimality of ϕ L . It is also clear, by definition, that K L ≥ . We emphasize that J L is trace class, being the square of ( − ∆ L + 1) − / ψ L ( − ∆ L ) − / , which is Hilbert-Schmidtsince ψ L is in L , as a function of x , and f ( k ) := ( | k | + 1) − / | k | − is in L , as a functionof k . From the trace class property of J L , together with the boundedness of ( − ∆ L +1) / Q ψL h ϕL − e ( ϕ L ) ( − ∆ L + 1) / (which follows from Corollary 3.2), we immediately infer thetrace class property of K L . We even show in Lemma 3.10 that J L , K L . L ( − ∆ L + 1) − .We shall in the following denote by K yL , respectively J yL , the unitary equivalent operatorsobtained from K L and J L by a translation by y . Note that K yL and J yL appear if one expands F L with respect to ϕ yL instead of ϕ L . Moreover, the invariance under translations of F L implies that span {∇ ϕ L } ⊂ ker(11 − K L ) . (3.2.25)We show in Section 3.2.3 that these two sets coincide. Finally, even though both ε L andthe estimate (3.2.23) in Proposition 3.4 depend on L , with a little extra work one can showthat the bound is actually uniform in L (for large L ). For simplicity we opt for the currentversion of Proposition 3.4, as it is sufficient for the purpose of our investigation, which isset on a torus of fixed linear size L > L . Proof.
We shall denote h := h ϕ L . By assumption (3.2.22) and since ϕ L ∈ L ( T L ) , wecan apply Corollary 3.2 to ϕ L and to ( ϕ − ϕ L ) . This way we see that V ϕ − ϕ L is uniformlyinfinitesimally relatively bounded with respect to h for any ϕ satisfying (3.2.22). It is clear that h admits a simple and isolated least eigenvalue e ( ϕ L ) . Standard resultsin perturbation theory then imply that there exist ε L > and a contour γ around e ( ϕ L ) such that for any ϕ satisfying (3.2.22) e ( ϕ ) is the only eigenvalue of h ϕ = h + V ϕ − ϕ L inside γ . (For fixed ϕ , the statement above is a standard result in perturbation theory, see[26, Theorem XII.8]; moreover it is also possible to get a ϕ -independent γ encircling e ( ϕ ) ,see [26, Theorem XII.11] and recall that V ϕ − ϕ L are uniformly infinitesimally relativelybounded with respect to h .) We can then write e ( ϕ ) = Tr ˆ γ zz − ( h + V ϕ − ϕ L ) dz πi . (3.2.26)Moreover, by the uniform infinitesimal relative boundedness of V ϕ − ϕ L with respect to h ,we have sup z ∈ γ k V ϕ − ϕ L ( z − h ) − k < , (3.2.27)for ε L sufficiently small. For any z ∈ γ , we can thus use the resolvent identity in the form z − h − V ϕ − ϕ L = (cid:18) − Q ψ L z − h V ϕ − ϕ L (cid:19) − Q ψ L z − h (cid:18) − Q ψ L z − h V ϕ − ϕ L (cid:19) − P ψ L z − h (cid:18) − V ϕ − ϕ L z − h (cid:19) − . (3.2.28)The first term is analytic inside the contour γ and hence it gives zero after integration wheninserted in (3.2.26). Inserting the second term of (3.2.28), which is rank one, in (3.2.26)and using Fubini’s Theorem to interchange the trace and the integral, we obtain e ( ϕ ) = ˆ γ zz − e ( ϕ L ) * ψ L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − V ϕ − ϕ L z − h (cid:19) − (cid:18) − Q ψ L z − h V ϕ − ϕ L (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ L + dz πi . (3.2.29)For simplicity, we introduce the notation A = V ϕ − ϕ L z − h , B = Q ψ L z − h V ϕ − ϕ L . (3.2.30)Because of (3.2.27), both A and B are smaller than in norm, uniformly in z ∈ γ . Weshall use the identity − A − B =11 + A + A ( A + B ) + B − B + A − A + A − A B + A − A B − B . (3.2.31)We insert the various terms in (3.2.29) and do the contour integration. The term gives e ( ϕ L ) . The term A , recalling that ( − ∆ L ) − / ρ ψ L = ϕ L , yields h ψ L | V ϕ − ϕ L | ψ L i = 2 h ϕ − ϕ L | ϕ L i . (3.2.32)A standard calculation shows that the term A ( A + B ) gives (cid:28) ψ L (cid:12)(cid:12)(cid:12)(cid:12) V ϕ − ϕ L Q ψ L e ( ϕ L ) − h V ϕ − ϕ L (cid:12)(cid:12)(cid:12)(cid:12) ψ L (cid:29) = − h ϕ − ϕ L | K L | ϕ − ϕ L i . (3.2.33) TRONGLY COUPLED POLARON ON THE TORUS 29
Furthermore, since Q ψ L ψ L = 0 , the term B (11 − B ) − yields zero. Recalling that F L ( ϕ ) = k ϕ k + e ( ϕ ) we obtain from (3.2.29) F L ( ϕ ) − F L ( ϕ L ) − h ϕ − ϕ L | − K L | ϕ − ϕ L i = ˆ γ zz − e ( ϕ L ) (cid:28) ψ L (cid:12)(cid:12)(cid:12)(cid:12) A − A + A (cid:18) A − A + 111 − A B − B (cid:19) B (cid:12)(cid:12)(cid:12)(cid:12) ψ L (cid:29) dz πi . (3.2.34)We observe that, since γ is uniformly bounded and uniformly bounded away from e ( ϕ L ) ,we can get rid of the integration, i.e., it suffices to bound ( I ) := sup z ∈ γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ψ L (cid:12)(cid:12)(cid:12)(cid:12) A − A (cid:12)(cid:12)(cid:12)(cid:12) ψ L (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) , ( II ) := sup z ∈ γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) ψ L (cid:12)(cid:12)(cid:12)(cid:12) A (cid:18) A − A + 111 − A B − B (cid:19) B (cid:12)(cid:12)(cid:12)(cid:12) ψ L (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) , (3.2.35)with the r.h.s. of (3.2.23) to conclude the proof. We note that h ϕ − ϕ L | J L | ϕ − ϕ L i = (cid:13)(cid:13) ( − ∆ L + 1) / V ϕ − ϕ L ψ L (cid:13)(cid:13) , (3.2.36)and that, by infinitesimal relative boundedness of V ϕ L with respect to ( − ∆ L ) and since γ is uniformly bounded away from e ( ϕ L ) , there exists some constant C L > such that sup z ∈ γ (cid:13)(cid:13) ( − ∆ L + 1) / ( z − h ) − k ( − ∆ L + 1) / (cid:13)(cid:13) ≤ C L for k = 1 , . (3.2.37)Therefore, ( I ) = sup z ∈ γ (cid:12)(cid:12) ( z − e ( ϕ L )) − h V ϕ − ϕ L ψ L | ( z − h ) − A (11 − A ) − | V ϕ − ϕ L ψ L i (cid:12)(cid:12) . L sup z ∈ γ (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ L + 1) / ( z − h ) − A − A ( − ∆ L + 1) / (cid:13)(cid:13)(cid:13)(cid:13) h ϕ − ϕ L | J L | ϕ − ϕ L i , . L sup z ∈ γ (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ L + 1) − / A − A ( − ∆ L + 1) / (cid:13)(cid:13)(cid:13)(cid:13) h ϕ − ϕ L | J L | ϕ − ϕ L i , (3.2.38) ( II ) ≤ sup z ∈ γ (cid:13)(cid:13)(cid:13)(cid:13) A − A + 111 − A B − B (cid:13)(cid:13)(cid:13)(cid:13) h ψ L | AA † | ψ L i / h ψ L | BB † | ψ L i / . L sup z ∈ γ (cid:13)(cid:13)(cid:13)(cid:13) A − A + 111 − A B − B (cid:13)(cid:13)(cid:13)(cid:13) h ϕ − ϕ L | J L | ϕ − ϕ L i . (3.2.39)Since A (11 − A ) − = V ϕ − ϕ L ( z − h ϕ ) − , (3.2.40)it follows that (cid:13)(cid:13)(cid:13)(cid:13) ( − ∆ L + 1) − / A − A ( − ∆ L + 1) / (cid:13)(cid:13)(cid:13)(cid:13) ≤ k ( − ∆ L + 1) − / V ϕ − ϕ L ( − ∆ L ) − / kk ( − ∆ L ) / ( z − h ϕ ) − ( − ∆ L ) / k . L k ( − ∆ L + 1) − ( ϕ − ϕ L ) k , (3.2.41)where we used the relative boundedness of h ϕ w.r.t to − ∆ L and Corollary 3.2. This yieldsthe right bound for ( I ) . Similar estimates yield the right bounds for k A (11 − A ) − k and k (11 − A ) − B (11 − B ) − k . L k B k , concluding the proof. (cid:3) As a final result of this subsection, we prove the following Lemma about the operators K L and J L . Lemma 3.10.
Let K L and J L be the operators defined in (3.2.24) . We have K L , J L . L ( − ∆ L + 1) − . (3.2.42) Proof.
We prove the result for J L . By the relative boundedness of h ϕ L with respect to − ∆ L the same proof applies to K L . We shall show that ( − ∆ L + 1)( − ∆ L ) − / ψ L ( − ∆ L + 1) − / is bounded as an operator on L ( T L ) . In fact, for f ∈ L ( T L ) , k ( − ∆ L + 1)( − ∆ L ) − / ψ L ( − ∆ L + 1) − / f k = X = k ∈ πL Z (cid:18) | k | + 1 | k | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ξ ∈ πL Z ( ψ L ) k − ξ f ξ ( | ξ | + 1) / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k ( − ∆ L + 1) / ψ L k X = k ∈ πL Z (cid:18) | k | + 1 | k | (cid:19) X ξ ∈ πL Z | f ξ | ( | k − ξ | + 1) ( | ξ | + 1) . L X ξ ∈ πL Z | f ξ | | ξ | + 1 X = k ∈ πL Z ( | k | + 1) | k | ( | k − ξ | + 1) . L k f k , (3.2.43)where we used that ψ L ∈ C ∞ ( T L ) and that P = k ∈ πL Z ( | k | +1) | k | ( | k − ξ | +1) . | ξ | + 1 . Therefore J L ≤ k ( − ∆ L +1)( − ∆ L ) − / ψ L ( − ∆ L +1) − / k ( − ∆ L +1) − . L ( − ∆ L +1) − , (3.2.44)as claimed. (cid:3) Local Properties of M F L and F L . For
L > L we introduce the notation Π L ∇ := L -projection onto span {∇ ϕ L } , (3.2.45)which is going to be used throughout this section and Section 4. According to Theorem 2.1,the condition L > L guarantees that ψ yL = ψ L for any ψ L ∈ M E L and any y = 0 , whichimplies that ran Π L ∇ is three dimensional (i.e that the partial derivatives of ϕ L are linearlyindependent); if not, there would exist ν ∈ S such that ∂ ν ψ L = 0 and this would imply ψ L = ψ yL for any y parallel to ν .For technical reasons, we also introduce a family of weighted norms which will beneeded in Section 4. For T ≥ , we define k ϕ k W T := h ϕ | W T | ϕ i / , (3.2.46)where W T acts in k -space as multiplication by W T ( k ) = ( | k | ≤ T ( | k | + 1) − | k | > T. (3.2.47)Note that k ϕ k W = h ϕ | ( − ∆ L + 1) − | ϕ i and k ϕ k W ∞ = k ϕ k .For the purpose of this section we could formulate the following Lemma only withrespect to k · k = k · k W ∞ , but we opt for this more general version since we shall need itin Section 4. Lemma 3.11.
For any
L > L , there exists ε ′ L (independent of T ) such that for any ϕ ∈ L R ( T L ) with dist W T ( ϕ, Ω L ( ϕ L )) ≤ ε ′ L there exist a unique couple ( y ϕ , v ϕ ) , depending on T , with y ϕ ∈ T L and v ϕ ∈ (span i =1 , , { W T ∂ i ϕ L } ) ⊥ , such that ϕ = ϕ y ϕ L + ( v ϕ ) y ϕ and k v ϕ k W T ≤ ε ′ L . (3.2.48) TRONGLY COUPLED POLARON ON THE TORUS 31
As Proposition 3.4 above, we opt for an L -dependent version of Lemma 3.11 for sim-plicity, as it is sufficient for our purposes. We nevertheless believe it is possible to provea corresponding statement that is uniform in L . Note that Lemma 3.11 is equivalent to thestatement that there exists a T -independent ε ′ L such that the W T -projection onto Ω L ( ϕ L ) isuniquely defined in an ε ′ L -neighborhood of Ω L ( ϕ L ) with respect to the W T -norm, and that,for any ϕ therein, ϕ y ϕ L characterizes the W T -projection of ϕ onto Ω L ( ϕ L ) , so that dist W T ( ϕ, Ω L ( ϕ L )) = k ϕ − ϕ y ϕ L k W T = k v ϕ k W T . (3.2.49) Proof.
We begin by observing that the Lemma is equivalent to showing that for any k·k W T -normalized v ∈ (span i =1 , , { W T ∂ i ϕ L } ) ⊥ , any ε ≤ ε ′ L and any = y ∈ T L we have ε < k ϕ L + εv − ϕ yL k W T . (3.2.50)Indeed, if the Lemma holds then ϕ = ϕ L + εv does not admit other decompositions of theform (3.2.48), which implies that, for any y = 0 , (3.2.50) holds (otherwise there wouldexist y = 0 minimizing the W T -distance of ϕ from Ω L ( ϕ L ) and such y would necessar-ily yield a second decomposition of the form (3.2.48)). On the other hand, if the statement(3.2.50) holds and the Lemma does not, then there exists ϕ such that dist W T ( ϕ, Ω L ( ϕ L )) ≤ ε ′ L and also such that ( y , v ) and ( y , v ) yield two different decompositions of the form(3.2.48) for ϕ (note that at least one decomposition of the form (3.2.48) always exist, asthere exist at least one element of Ω L ( ϕ L ) realizing the W T -distance of ϕ from Ω L ( ϕ L ) ).By considering ϕ − y (respectively ϕ − y ) we find k v k W T > k v k W T (respectively k v k W T > k v k W T ), which is clearly a contradiction. We shall hence proceed to prove the statement(3.2.50).Taylor’s formula and the regularity of ϕ L imply the existence of T -independent constant C L such that ϕ yL = ϕ L + y · ( ∇ ϕ L ) + g y , with k g y k W T ≤ k g y k ≤ C L | y | . (3.2.51)As remarked after (3.2.45), the kernel of Π L ∇ is three-dimensional, hence there exists aconstant C L independent of T such that min ν ∈ S k ν · ∇ ϕ L k W T ≥ min ν ∈ S k ν · ∇ ϕ L k W ≥ C L . (3.2.52)Therefore, using that v ⊥ W T ∇ ϕ L in combination with (3.2.51) and (3.2.52), we find, for | y | < ( C L − εC L ) / ( C L ) − , (3.2.53)that k ϕ L + εv − ϕ yL k W T = k εv − y · ( ∇ ϕ L ) − g y k W T ≥ (cid:0) ε + | y | C L (cid:1) / − C L | y | > ε, (3.2.54)i.e., that (3.2.50) holds for y satisfying (3.2.53). Furthermore, we have k ϕ L + εv − ϕ yL k W T ≥ ε + k ϕ L − ϕ yL k W T ( k ϕ L − ϕ yL k W T − ε ) , (3.2.55)and this implies that (3.2.50) holds for any y such that k ϕ L − ϕ yL k W T > ε. (3.2.56)Using again (3.2.52) and (3.2.51), there exist C L , c L , c L > independent of T such that k ϕ L − ϕ yL k W T = k y · ( ∇ ϕ L ) + g y k W T ≥ C L | y | − C L | y | ≥ C L | y | , for | y | ≤ c L , k ϕ L − ϕ yL k W T > c L for | y | > c L , (3.2.57) where the second line simply follows from k · k W T ≥ k · k W , the fact that ϕ L = ϕ yL forany = y ∈ [ − L/ , L/ and the continuity of ϕ L . Combining (3.2.56) and (3.2.57), weconclude that (3.2.50) holds if either | y | > c L or | y | > ε ( C L ) − . (3.2.58)Picking ε ′ L sufficiently small, the fact that (3.2.50) holds both under the conditions (3.2.53)and (3.2.58) shows that it holds for any y ∈ T L , and this completes the proof. (cid:3) We conclude our study of the Pekar functional F L by showing that ker(11 − K L ) =span {∇ ϕ L } = ran Π L ∇ . Since clearly ran Π L ∇ ⊂ ker(11 − K L ) , this is a consequence of thefollowing Proposition. Proposition 3.5.
Recalling the definition of τ L from Corollary 2.1, we have − K L ≥ τ L (11 − Π L ∇ ) . (3.2.59) Proof.
We need to show that for all normalized v ∈ ran(11 − Π L ∇ ) the bound h v | − K L | v i ≥ τ L (3.2.60)holds. Using Lemma 3.11 in the case T = ∞ , for any such v and ε small enough, denoting ϕ = ϕ L + εv , we obtain dist L ( ϕ, Ω L ( ϕ L )) = ε . (3.2.61)Moreover, since k ( − ∆ L + 1) − ( ϕ − ϕ L ) k ≤ ε k v k = ε , for ε small enough we can expand F L ( ϕ ) with respect to ϕ L using Proposition 3.4. Combining this with (2.2.14), we arriveat τ L ε ≤ F L ( ϕ L + εv ) − e L ≤ ε h v | − K L | v i + ε h v | J L | v i . (3.2.62)Since ε can be taken arbitrarily small, the proof is complete. (cid:3)
4. P
ROOF OF M AIN R ESULTS
In this Section we give the proof of Theorem 2.2. In Section 4.1 we prove the upperbound in (2.3.2). In Section 4.2 we estimate the cost of substituting the full Hamiltonian H L with a cut-off Hamiltonian depending only on finitely many phonon modes, a key stepin providing a lower bound for inf spec H L . Finally, in Section 4.3, we show the validityof the lower bound in (2.3.2).The approach used in Sections 4.1 and 4.2 follows closely the one used in [9], even if,in our setting, minor complications arise in the proof of the upper bound due the presenceof the zero modes of the Hessian. For the lower bound in Section 4.3, however, a sub-stantial amount of additional work is needed to deal with the translation invariance, whichcomplicates the proof significantly.4.1. Upper Bound.
In this section we construct a trial state, which will be used to obtainan upper bound on the ground state energy of H L for fixed L > L . This is carried outusing the Q -space representation of the bosonic Fock space F ( L ( T L )) (see [25]). Eventhough the estimates contained in this section are L -dependent, we believe it is possible,with little modifications to the proof, to obtain the same upper bound with the same errorestimates uniformly in L . TRONGLY COUPLED POLARON ON THE TORUS 33
Our trial state depends non-trivially only on finitely many phonon variables, and weproceed to describe it more in detail. If one picks Π to be a real finite rank projection on L ( T L ) , then F ( L ( T L )) ∼ = F (Π L ( T L )) ⊗ F ((11 − Π) L ( T L )) . (4.1.1)The first factor F (Π L ( T L )) can isomorphically be identified with L ( R N ) , where N is thecomplex dimension of ran Π . In particular, there is a one-to-one correspondence betweenany real ϕ ∈ ran Π and λ = ( λ , . . . , λ N ) ∈ R N , explicitly given by ϕ = N X i =1 λ i ϕ i ∼ = ( λ , . . . , λ N ) = λ, (4.1.2)where { ϕ i } Ni =1 denotes an orthonormal basis of ran Π consisting of real-valued functions.The trial state we use corresponds to the vacuum in the second factor F ((11 − Π) L ( T L )) and shall hence be written only as a function of x (the electron variable) and λ (the finitelymany phonon variables selected by Π ). We begin by specifying some properties we wish Π to satisfy. Consider ϕ L from Corollary 2.1 and define Π to be a projection of the form Π = Π ′ + Π L ∇ , where Π L ∇ is defined in (3.2.45) and Π ′ is an ( N − -dimensional projectionon span {∇ ϕ L } ⊥ = ran(11 − Π L ∇ ) that will be further specified later but will always be takenso that ϕ L ∈ ran Π . Our trial state is of the form Ψ( x, ϕ ) = G ( ϕ ) η ( ϕ ) ψ ϕ ( x ) , (4.1.3)where • x ∈ T L and ϕ is a real element of ran Π (identified with λ ∈ R N as in (4.1.2)), • G ( ϕ ) is a Gaussian factor explicitly given by G ( ϕ ) = exp (cid:16) − α h ϕ − ϕ L | [Π(11 − K L )Π] / | ϕ − ϕ L i (cid:17) , (4.1.4) • η is a ‘localization factor’ given by η ( ϕ ) = χ (cid:16) ε − k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k L ( T L ) (cid:17) , (4.1.5)for some < ε < ε L (with ε L as in Proposition 3.4), where ≤ χ ≤ is a smoothcut-off function such that χ ( t ) = 1 for t ≤ / and χ ( t ) = 0 for t ≥ , • ψ ϕ is the unique positive ground state of h ϕ = − ∆ L + V ϕ .We note that our state actually depends on two parameters ( N and ε ) and, of course, on thespecific choice of Π ′ . We choose { ϕ i } i =1 ,...,N to be a real orthonormal basis of eigenfunc-tions of [Π(11 − K L )Π] corresponding to eigenvalues µ i = 0 for i = 1 , , and µ i ≥ τ L > for i = 4 , . . . , N . Recalling Proposition 3.5, this amounts to choosing { ϕ i } i =1 , , to be areal orthonormal basis of ran Π L ∇ and { ϕ i } i =4 ,...,N to be a real orthonormal basis of eigen-functions of [Π ′ (11 − K L )Π ′ ] . With this choice, we have (with a slight abuse of notation) G ( ϕ ) = G ( λ , . . . , λ N ) = exp − α N X i =4 µ / i ( λ i − λ Li ) ! , (4.1.6)where ϕ L ∼ = λ L = (0 , , , λ L , . . . , λ LN ) , since ϕ L ∈ ran Π by construction, and the firstthree coordinates are since ϕ L ∈ (cid:0) ran Π L ∇ (cid:1) ⊥ . We first show that even if G does not have finite L ( R N ) -norm, Ψ does due to thepresence of η . We define T ε := {k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k ≤ ε } ⊂ R N (4.1.7)and γ L := inf ϕ ∈ ran Π L ∇k ϕ k h ϕ | ( − ∆ L + 1) − | ϕ i > . (4.1.8)Then, on T ε , recalling that Π L ∇ ϕ L = 0 , we have γ / L q λ + λ + λ = γ / L k Π L ∇ ϕ k ≤ k ( − ∆ L + 1) − / Π L ∇ ( ϕ − ϕ L ) k ≤ k ( − ∆ L + 1) − / Π ′ ( ϕ − ϕ L ) k + ε ≤ k Π ′ ( ϕ − ϕ L ) k + ε = N X i =4 ( λ i − λ Li ) ! / + ε (4.1.9)and this implies, using the normalization of ψ ϕ , that k Ψ k = ˆ R N G ( λ , . . . , λ N ) η ( λ ) dλ . . . dλ N ≤ ˆ R N G ( λ , . . . , λ N ) T ε ( λ ) dλ · · · dλ N ≤ π ˆ R N − G ( λ , . . . , λ N ) γ − / L N X i =4 ( λ i − λ Li ) ! / + ε dλ · · · dλ N < ∞ . (4.1.10)We spend a few words to motivate our choice of Ψ . The absolute value squared of Ψ has to be interpreted as a probability density over the couples ( ϕ, x ) , with ϕ beinga classical state for the phonon field and x the position of the electron. In the electroncoordinate, our Ψ corresponds to the ground state of h ϕ for any value of ϕ . This implies,by straightforward computations, that the expectation value of the Fröhlich Hamiltonian in Ψ equals the one of e ( ϕ ) + N , e ( ϕ ) being the ground state energy of h ϕ and N the numberoperator. Moreover, because of the factor η , we are localizing our state to the regime wherethe Hessian expansion of e ( ϕ ) from Proposition 3.4 holds. To leading order, this effectivelymakes our system formally correspond to a system of infinitely many harmonic oscillatorswith frequencies given by the eigenvalues of (11 − K L ) / , with a Gaussian ground state.To carry out this analysis out rigorously, we need to choose a suitable finite rank projection Π , as detailed in the remained of this section.We are now ready to delve into the details of the proof. It is easy to see that the interac-tion term appearing in the Fröhlich Hamiltonian acts in the Q -space representation as themultiplication by V ϕ ( x ) . Therefore, since Ψ corresponds to the vacuum on (11 − Π) L ( T L ) and only depends on x through the factor ψ ϕ ( x ) , the g.s. of h ϕ , it follows that h Ψ | H L | Ψ i = h Ψ | e ( ϕ ) + N | Ψ i (4.1.11)where ϕ = Π ϕ ∼ = λ ∈ R N and the inner product on the r.h.s. is naturally interpreted as theone on L ( T L ) ⊗ L ( R N ) . In the Q -space representation, the number operator takes theform N = N X n =1 (cid:18) − α ∂ λ n + λ n − α (cid:19) = 14 α ( − ∆ λ ) + | λ | − N α . (4.1.12) TRONGLY COUPLED POLARON ON THE TORUS 35
Using the fact that η is supported on the set T ε defined in (4.1.7), we can use the Hessianexpansion from Proposition 3.4 to obtain bounds on e ( λ ) . Consequently, for a suitablepositive constant C L , h Ψ | H L | Ψ i ≤ h Ψ | e L + h ϕ − ϕ L | − K L + εC L J L | ϕ − ϕ L i| Ψ i + (cid:28) Ψ (cid:12)(cid:12)(cid:12)(cid:12) α ( − ∆ λ ) − N α (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:29) = (cid:18) e L − α Tr(Π) (cid:19) k Ψ k + A + B, (4.1.13)with A = * Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ( − ∆ λ ) + N X i =4 µ i ( λ i − λ Li ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ + , (4.1.14) B = εC L h Ψ | h ϕ − ϕ L | J L | ϕ − ϕ L i| Ψ i . (4.1.15)We shall now proceed to bound A and B . First, we recall that by Lemma 3.10 J L . L ( − ∆ L + 1) − . (4.1.16)Therefore, since η is supported on T ε , we have B . L ε k Ψ k . (4.1.17)To bound A a bit more work is required. A direct calculation shows that " α ( − ∆ λ ) + N X i =4 µ i ( λ i − λ Li ) G = 12 α Tr (cid:16) [Π(11 − K L )Π] / (cid:17) G. (4.1.18)The previous identity, together with straightforward manipulations involving integrationby parts, shows that A = 14 α h ψ ϕ Gη | ψ ϕ ( − ∆ λ G ) η i + ˆ T L × R N G |∇ λ ( ηψ ϕ ) | ! + * Ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =4 µ i ( λ − λ Li ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ + ≤ α Tr (cid:16) [Π(11 − K L )Π] / (cid:17) k Ψ k + 12 α " ˆ T L × R N G η |∇ λ ψ ϕ | + ˆ T L × R N G |∇ λ η | | ψ ϕ | =: 12 α Tr (cid:16) [Π(11 − K L )Π] / (cid:17) k Ψ k + A + A . (4.1.19)We proceed to bound A . By standard first order perturbation theory (using that thephase of ψ ϕ is chosen so that it is the unique positive minimizer of h ϕ ) we have ∂ λ n ψ ϕ = − Q ψ ϕ h ϕ − e ( ϕ ) V ϕ n ψ ϕ , (4.1.20) where we recall that Q ψ ϕ = 11 − | ψ ϕ i h ψ ϕ | . This implies that, for fixed ϕ , ˆ T L |∇ λ ψ ϕ ( x ) | dx = N X n =1 (cid:13)(cid:13)(cid:13)(cid:13) Q ψ ϕ h ϕ − e ( ϕ ) V ϕ n ψ ϕ (cid:13)(cid:13)(cid:13)(cid:13) L ( T L ) = N X n =1 h ϕ n | ( − ∆ L ) − / ψ ϕ (cid:18) Q ψ ϕ h ϕ − e ( ϕ ) (cid:19) ψ ϕ ( − ∆ L ) − / | ϕ n i = Tr Π( − ∆ L ) − / ψ ϕ (cid:18) Q ψ ϕ h ϕ − e ( ϕ ) (cid:19) ψ ϕ ( − ∆ L ) − / Π ! , (4.1.21)where ψ ϕ is interpreted as a multiplication operator in the last two expressions. Since ( − ∆ L + 1) / (cid:16) Q ψϕ h ϕ − e ( ϕ ) (cid:17) ( − ∆ L + 1) / is uniformly bounded over the support of η (thepotential V ϕ being uniformly infinitesimally relatively bounded with respect to − ∆ L byCorollary 3.2) and recalling the normalization of ψ ϕ , we get Tr Π( − ∆ L ) − / ψ ϕ (cid:18) Q ψ ϕ h ϕ − e ( ϕ ) (cid:19) ψ ϕ ( − ∆ L ) − / Π ! . L Tr (cid:0) Π( − ∆ L ) − / ψ ϕ ( − ∆ L + 1) − ψ ϕ ( − ∆ L ) − / Π (cid:1) . L . (4.1.22)In summary, we conclude that A . L α k Ψ k . (4.1.23)Finally, we proceed to bound A . Recalling the definition of η and T ε , we see that |∇ λ η | = (cid:12)(cid:12)(cid:12) ∇ λ h χ (cid:16) ε − k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k L ( T L ) (cid:17)i(cid:12)(cid:12)(cid:12) . ε − T ε ( ϕ ) (cid:12)(cid:12)(cid:12) ∇ λ k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k L ( T L ) (cid:12)(cid:12)(cid:12) . ε − T ε ( ϕ ) k ( − ∆ L + 1) − ( ϕ − ϕ L ) k k ( − ∆ L + 1) − / ( ϕ − ϕ L ) k ≤ T ε ( ϕ ) ε − , (4.1.24)where we used that η is supported on T ε and that χ is smooth and compactly supported.Therefore, using also the normalization of ψ ϕ , we obtain A . α ε k T ε G k L ( R N ) . (4.1.25)We now need to bound k T ε G k L ( R N ) in terms of k Ψ k = k ηG k L ( R N ) . We define S ν := { ϕ ∈ ran Π | k Π ′ ( ϕ − ϕ L ) k ≤ ν } (4.1.26)and observe that on S ν ∩ T ε we have, by the triangle inequality, k ( − ∆ L + 1) − / Π L ∇ ϕ k ≤ ε + ν, (4.1.27)and that on S cν G ( λ ) ≤ exp (cid:16) − α τ / L ν (cid:17) , (4.1.28) TRONGLY COUPLED POLARON ON THE TORUS 37 where we used that [Π(11 − K L )Π] / ≥ τ / L Π ′ (with τ L being the constant appearing inProposition 3.5). We then have, using (4.1.27), that k T ε G k = k T ε ∩ S ν G k + k T ε ∩ S cν G k ≤ ˆ {k ( − ∆ L +1) − / Π L ∇ ϕ k ≤ ε + ν }∩ S ν G dλ . . . dλ N + ˆ T ε ∩ S cν G dλ . . . dλ N . (4.1.29)We now perform the change of variables ( λ , λ , λ ) = 3( λ ′ , λ ′ , λ ′ ) in the first integraland the change of variables λ − λ L = 2( λ ′ − λ L ) in the second integral and fix ν = ε/ ,obtaining k T ε G k ≤ ˆ {k ( − ∆ L +1) − / Π L ∇ ϕ k ≤ ( ε + ν ) / }∩ S ν G dλ + 2 N ˆ T ε/ ∩ S cν/ G ( λ ′ ) dλ ′ ≤ (cid:16)
27 + 2 N exp (cid:16) − α τ / L ν / (cid:17)(cid:17) ˆ T ε/ G dλ ≤ (cid:16)
27 + 2 N exp (cid:16) − α τ / L ν / (cid:17)(cid:17) k Ψ k , (4.1.30)where in the second step we used that {k ( − ∆ L + 1) − / Π L ∇ ϕ k ≤ ( ε + ν ) / } ∩ S ν ⊂ T ε/ by the triangle inequality if ν = ε/ , and (4.1.28) to estimate the Gaussian factor on S cν/ .Therefore, as long as √ N ≤ C L αε for a sufficiently small C L , we conclude that A . α ε k Ψ k . (4.1.31)Plugging estimates (4.1.17), (4.1.19), (4.1.23), and (4.1.31) into (4.1.13), we infer, for √ N ≤ C L αε , that for a sufficiently large C L h Ψ | H L | Ψ ih Ψ | Ψ i ≤ e L − α Tr (cid:16) Π − [Π(11 − K L )Π] / (cid:17) + C L ( ε + α − ε − ) . (4.1.32)We now proceed to choose a real orthonormal basis for ran Π which is convenient tobound the r.h.s. of (4.1.32). Let { g j } j ∈ N be an orthonormal basis of eigenfunctions of K L with corresponding eigenvalue k j , ordered such that k j +1 ≥ k j . By Proposition 3.5 wehave k j = 1 for j = 1 , , and k j < for j > . Moreover, Π L ∇ coincides with theprojection onto span { g , g , g } . We pick Π ′ to be the projection onto span { g , . . . , g N } if ϕ L is spanned by { g , . . . , g N } and onto span { g , . . . , g N − , ϕ L } otherwise. With thischoice the eigenvalues µ i of Π(11 − K L )Π appearing in the Gaussian factor G are equal to µ j = 1 − k j , j = 1 , . . . , N − , µ N = ( − k N if ϕ L ∈ span { g , . . . , g N } , h ˜ ϕ L | − K L | ˜ ϕ L i otherwise , (4.1.33)with ˜ ϕ L := ϕ L − P N − j =4 g j h g j | ϕ L ik ϕ L − P N − j =4 g j h g j | ϕ L ik . In any case Tr (cid:16) Π − [Π(11 − K L )Π] / (cid:17) ≥ N − X j =1 (1 − (1 − k j ) / ) = Tr (cid:0) − (11 − K L ) / (cid:1) − ∞ X j = N (1 − (1 − k j ) / ) . (4.1.34) In order to estimate P ∞ j = N (1 − (1 − k j ) / ) , we note that Lemma 3.10 implies that k j . L ( l j + 1) − , where l j denotes the ordered eigenvalues of − ∆ L . Since l j ∼ j / for j ≫ ,we have ∞ X j = N (1 − (1 − k j ) / ) . L N − / . (4.1.35)This allows us to conclude that h Ψ | H L | Ψ ih Ψ | Ψ i ≤ e L − α Tr (cid:0) − (11 − K L ) / (cid:1) + C L ( ε + α − ε − + α − N − / ) , (4.1.36)as long as √ N ≤ C L αε . The error term is minimized, under this constraint, for ε ∼ α − / and N ∼ α ε ∼ α / , which yields h Ψ | H L | Ψ ih Ψ | Ψ i ≤ e L − α Tr (cid:0) − (11 − K L ) / (cid:1) + C L α − / , (4.1.37)as claimed in (2.3.2).4.2. The Cutoff Hamiltonian.
As a first step to derive the lower bound in (2.3.2), weshow that it is possible to apply an ultraviolet cutoff of size Λ to H L at an expense of order Λ − / (this is proven in Proposition 4.3 in Section 4.2.3). Our approach follows closelythe one in [9]. It relies on an application of a triple Lieb–Yamazaki bound (extending themethod of [19]) which we carry out in Section 4.2.1), and on a consequent use (in Section4.2.2) of a Gross transformation [13, 23].We shall in the following, for any real-valued f ∈ L ( T L ) , denote Φ( f ) := a † ( f ) + a ( f ) , (4.2.1) Π( f ) := Φ( if ) = i ( a † ( f ) − a ( f )) . (4.2.2)We recall that the interaction term in the Fröhlich Hamiltonian is given by − a † ( v xL ) − a ( v xL ) = − Φ( v xL ) , (4.2.3)where v L was defined in (2.1.3) and a and a † satisfy the rescaled commutation relations(2.1.5). We shall apply an ultraviolet cutoff of size Λ in k -space, which amounts to substi-tuting the interaction term with − a † ( v xL, Λ ) − a ( v xL, Λ ) = − Φ( v xL, Λ ) , (4.2.4)where v L, Λ ( y ) := X = k ∈ πL Z | k | < Λ | k | e − ik · y L . (4.2.5)To quantify the expense of such a cutoff we clearly need to bound − a † ( w xL, Λ ) − a ( w xL, Λ ) = − Φ( w xL, Λ ) , (4.2.6)where w L, Λ ( y ) = v L ( y ) − v L, Λ ( y ) = X k ∈ πL Z | k |≥ Λ | k | e − ik · y L . (4.2.7) TRONGLY COUPLED POLARON ON THE TORUS 39
Triple Lieb–Yamazaki Bounds.
Let us introduce the notation p = ( p , p , p ) = − i ∇ x for the electron momentum operator. Note that on any function of the form f ( x, y ) = f ( y − x ) , such as w xL, Λ for example, the operator p simply acts as multiplication by k in k -space and agrees, up to a sign, with − i ∇ y .The purpose of this section is to prove the following Proposition. Proposition 4.1.
Let w L, Λ be defined as in (4.2.7) and Λ > . Then a † ( w xL, Λ ) + a ( w xL, Λ ) = Φ( w xL, Λ ) . ( | p | + N + 1) (Λ − / + α − Λ − / ) , (4.2.8) as quadratic forms on L ( T L ) ⊗ F ( L ( T L )) . We first need the following Lemma.
Lemma 4.1.
Let w L, Λ be defined as in (4.2.7) and Λ > . Then for any j, l, m ∈ { , , } a † (cid:2) ( ∂ j ∂ l ∂ m ( − ∆ L ) − w L, Λ ) x (cid:3) a (cid:2) ( ∂ j ∂ l ∂ m ( − ∆ L ) − w L, Λ ) x (cid:3) . Λ − N , (4.2.9) k ∂ j ∂ l ( − ∆ L ) − w L, Λ k L ( T L ) . Λ − , (4.2.10) a † (cid:2) ( ∂ j ∂ l ( − ∆ L ) − w L, Λ ) x (cid:3) a (cid:2) ( ∂ j ∂ l ( − ∆ L ) − w L, Λ ) x (cid:3) . Λ − ( | p | + L − Λ − ) N , (4.2.11) as quadratic forms on L ( T L ) ⊗ F ( L ( T L )) .Proof. For any j, l, m ∈ { , , } , (4.2.9) follows from a † ( g ) a ( g ) ≤ k g k N for g ∈ L ( T L ) , and then proceeding along the same lines of the proof of (4.2.10). To prove(4.2.10) we estimate k ∂ j ∂ l ( − ∆ L ) − w L, Λ k L ( T L ) = 1 L X | k |≥ Λ k ∈ πL Z k j k l | k | . ˆ B c Λ | t | dt = 4 π − . (4.2.12)If we denote f xj,l := ( − ∂ j ∂ l ( − ∆ L ) − w L, Λ ) x , in order to show (4.2.11) it suffices to provethat (cid:12)(cid:12) f xj,l (cid:11) (cid:10) f xj,l (cid:12)(cid:12) . Λ − (cid:0) | p | + Λ − (cid:1) on L ( T L ) ⊗ L ( T L ) , (4.2.13)where the bracket notation refers to the second factor in the tensor product, i.e., the leftside is a rank-one projection on the second factor parametrized by x , which acts via multi-plication on the first factor. For any Ψ ∈ L ( T L ) ⊗ L ( T L ) with Fourier coefficients Ψ q,k ,we have D Ψ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) f xj,l (cid:11) (cid:10) f xj,l (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Ψ E = ˆ dx (cid:12)(cid:12)(cid:12)(cid:12) ˆ dyf xj,l ( y )Ψ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) = X q ∈ πL Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ πL Z | k |≥ Λ k j k l L / | k | Ψ q − k,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X q ∈ πL Z X k ∈ πL Z | k |≥ Λ , k = q L | k | | q − k | X k ∈ πL Z | q − k | | Ψ q − k,k | + X q ∈ πL Z | q |≥ Λ | Ψ ,q | L | q | ≤ sup q ∈ πL Z X k ∈ πL Z | k |≥ Λ , k = q L − | k | | q − k | h Ψ || p | | Ψ i + L − Λ − k Ψ k . h Ψ | Λ − ( | p | + L − Λ − ) | Ψ i , (4.2.14) which shows our claim. We only need to justify the last step, i.e., that the supremumappearing in (4.2.14) is bounded by C Λ − . We have X = k ∈ πL Z | k |≥ Λ , k = q L − | k | | q − k | . ˆ B c Λ | x | | q − x | dx = Λ − ˆ B c | x | | Λ − q − x | ≤ Λ − ˆ B (Λ − q ) | Λ − q − x | + ˆ B c | x | − ! ≤ π − . (4.2.15)This concludes the proof. (cid:3) We are now able to prove Proposition 4.1.
Proof of Proposition 4.1.
Following the approach by Lieb and Yamazaki in [19], we have X j =1 [ p j , a ( p j | p | − w xL, Λ )] = − a ( w xL, Λ ) . (4.2.16)Applying this three times, we obtain X j,k,l =1 [ p j , [ p k , [ p l , a ( p j p k p l | p | − w xL, Λ )]]] = − a ( w xL, Λ ) . (4.2.17)Similarly, X j,k,l =1 [ p j , [ p k , [ p l , a † ( p j p k p l | p | − w xL, Λ )]]] = a † ( w xL, Λ ) . (4.2.18)Therefore, if we define B jkl := a † ( p j p k p l | p | − w xL, Λ ) − a ( p j p k p l | p | − w xL, Λ )= a † (cid:2) ( ∂ j ∂ l ∂ m ( − ∆ L ) − w L, Λ ) x (cid:3) − a (cid:2) ( ∂ j ∂ l ∂ m ( − ∆ L ) − w L, Λ ) x (cid:3) , (4.2.19)we have a † ( w xL, Λ ) + a ( w xL, Λ ) = Φ( w xL, Λ ) = X j,k,l =1 [ p j , [ p k , [ p l , B jkl ]]] . (4.2.20)Using that B † jkl = − B jkl and that B jkl is invariant under exchange of indices, we arrive at Φ( w xL, Λ ) = X j,k,l =1 (cid:16) p j p k [ p l , B jkl ] + [ B † jkl , p l ] p j p k (cid:17) − X j,k,l =1 (cid:16) p j p k B jkl p l + p l B † jkl p j p k (cid:17) . (4.2.21)By the Cauchy–Schwarz inequality, we have for any λ > − p j p k B jkl p l − p l B † jkl p j p k ≤ λp j p k + λ − p l B † jkl B jkl p l . (4.2.22)Moreover, using (4.2.9) and the rescaled commutation relations (2.1.5) satisfied by a and a † , we have B † jkl B jkl ≤ C (cid:0) N + 2 α − (cid:1) Λ − . (4.2.23) TRONGLY COUPLED POLARON ON THE TORUS 41
Using (4.2.22) and (4.2.23) and picking λ = C / Λ − / we conclude that − X j,k,l =1 (cid:16) p j p k B jkl p l + p l B † jkl p j p k (cid:17) . Λ − / (cid:0) | p | + 3 | p | (4 N + 2 α − ) (cid:1) . (4.2.24)We now define C jk := X l =1 [ p l , B jkl ] = a † ( p j p k | p | − w xL, Λ ) + a ( p j p k | p | − w xL, Λ )= a † (cid:2) ( ∂ j ∂ k ( − ∆ L ) − w L, Λ ) x (cid:3) + a (cid:2) ( ∂ j ∂ k ( − ∆ L ) − w L, Λ ) x (cid:3) = C † jk . (4.2.25)Using (4.2.10), (4.2.11) and the Cauchy-Schwarz inequality, we have for any λ > p j p k C jk + C jk p j p k ≤ λp j p k + λ − C jk . (4.2.26)Moreover, C jk ≤ a † ( p j p k | p | − w xL, Λ ) a ( p j p k | p | − w xL, Λ ) + 2 α − k p j p k | p | − w xL, Λ k . Λ − ( | p | + Λ − ) N + α − Λ − . (4.2.27)Picking λ = Λ − / + α − Λ − / , we therefore conclude that X j,k,l =1 (cid:16) p j p k [ p l , B jkl ] + [ B † jkl , p l ] p j p k (cid:17) . (Λ − / + α − Λ − / )[ | p | + N ( | p | + L − Λ − ) + 1] . (4.2.28)Applying (4.2.24) and (4.2.28) in (4.2.21), we finally obtain Φ( w xL, Λ ) . (Λ − / + α − Λ − / ) (cid:2) | p | + N ( | p | + L − Λ − ) + 1 (cid:3) + Λ − / (cid:0) | p | + 3 | p | (4 N + 2 α − ) (cid:1) . ( | p | + N + 1) (Λ − / + α − Λ − / ) , (4.2.29)as claimed. (cid:3) Gross Transformation.
The bound (4.2.8), derived in Proposition 4.1, is not imme-diately useful as it stands. In order to relate the r.h.s. of (4.2.8) to the square of the FröhlichHamiltonian H L in (2.1.4), we shall apply a Gross transformation [13], [23].For a real-valued f ∈ H ( T L ) , recalling that f x ( · ) = f ( · − x ) , we consider thefollowing unitary transformation on L ( T L ) ⊗ F U = e a ( α f x ) − a † ( α f x ) = e i Π( α f x ) , (4.2.30)where U is understood to act as a ‘multiplication’ with respect to the x variable. For any g ∈ L ( T L ) , we have U a ( g ) U † = a ( g ) + h g | f x i and U a † ( g ) U † = a † ( g ) + h f x | g i , (4.2.31)and therefore U N U † = N + Φ( f x ) + k f k . (4.2.32)Moreover, U pU † = p + α Φ( pf x ) = p + α Φ[( i ∇ f ) x ] . (4.2.33) This implies that
U p U † = p + α (Φ[( i ∇ f ) x ]) + 2 α p · a [( i ∇ f ) x ] + 2 α a † [( i ∇ f ) x ] · p + α Φ[( − ∆ L f ) x ] . (4.2.34)Therefore, we also have U H L U † = | p | + α (Φ[( i ∇ f ) x ]) + 2 α p · a [( i ∇ f ) x ] + 2 α a † [( i ∇ f ) x ] · p + Φ[( − α ∆ L f + f − v L ) x ] + N + k f k − h v L | f i . (4.2.35)We denote g = − α ∆ L f + f − v L , (4.2.36)and we shall pick f ( y ) = h ( − α ∆ L + 1) − ( − ∆ L ) − / χ B cK ( − ∆ L ) i (0 , y )= X | k |≥ Kk ∈ πL Z α | k | + 1) | k | e − ik · y L (4.2.37)for some K > . Recalling (4.2.5), this implies that g ( y ) = − v L,K ( y ) = − X = k ∈ πL Z | k | For any ε > there exist K ε > and C ε > such that, for all α & and any Ψ ∈ L ( T L ) ⊗ F in the domain of | p | + N (1 − ε ) k ( | p | + N )Ψ k − C ε k Ψ k ≤ k U K ε α H L ( U K ε α ) † Ψ k ≤ (1 + ε ) k ( | p | + N )Ψ k + C ε k Ψ k . (4.2.43) TRONGLY COUPLED POLARON ON THE TORUS 43 Proof. We shall use the following standard (given the rescaled commutation relations satis-fied by a and a † ) properties, which hold for any Ψ ∈ F , any f ∈ L ( T L ) and any function h : [0 , ∞ ) → R k a ( f )Ψ k ≤ k f k k√ N Ψ k , k a † ( f )Ψ k ≤ k f k k√ N + α − Ψ k , (4.2.44) h ( N + α − ) a = ah ( N ) , h ( N ) a † = a † h ( N + α − ) . (4.2.45)It is then straightforward, with the aid of the estimates (4.2.39), (4.2.40), (4.2.41) and(4.2.42), to show, for any Ψ ∈ L ( T L ) ⊗ F , any δ > and any K > , that α k (Φ[( i ∇ f ) x ]) Ψ k . α k∇ f k k ( N + α − )Ψ k . K − k ( N + α − )Ψ k , (4.2.46) k Φ( g x )Ψ k . K / k√ N + α − Ψ k . δ k ( N + α − )Ψ k + δ − K k Ψ k , (4.2.47) α k a † [( i ∇ f ) x ] · p Ψ k . K − / k√ N + α − p | p | Ψ k . K − / k ( | p | + N + α )Ψ k . (4.2.48)It remains to bound the term k α p · a [( i ∇ f ) x ]Ψ k ≤ k α a [( i ∇ f ) x ] · p Ψ k + k a [( − α ∆ L f ) x ]Ψ k =: (I) + (II) . (4.2.49)As in (4.2.48), we can easily bound (I) . K − / k ( | p | + N + α − )Ψ k . (4.2.50)By (4.2.36) and (4.2.38) and recalling (4.2.5) and (4.2.7), we have a [( − α ∆ L f ) x ] = a [( g − f + v L ) x ] = − a ( f x ) + a ( w xL,K ) . (4.2.51)With the same arguments used in the proof of Lemma 4.1 we obtain k a ( w xL,K )Ψ k . K − / k p N ( | p | + K − )Ψ k , (4.2.52)and therefore, using (4.2.40) to bound k a ( f x )Ψ k , we arrive at (I I ) . α − K − / k√ N Ψ k + K − / k p N ( | p | + K − )Ψ k . α − K − / ( k ( N + α − )Ψ k + k Ψ k ) + K − / k ( | p | + N + K − )Ψ k . (4.2.53)Combining (4.2.46), (4.2.47), (4.2.48), (4.2.50), (4.2.53), (4.2.40) and (4.2.41) with (4.2.35),we obtain, for any K ≥ k U Kα H L ( U kα ) † Ψ k ≤ [1 + C ( K − / + δ )] k ( | p | + N )Ψ k + C ( δ − K + 3 α − K − ) k Ψ k , (4.2.54) k U Kα H L ( U Kα ) † Ψ k ≥ [1 − C ( K − / + δ )] k ( | p | + N )Ψ k − C ( δ − K + 3 α − K − ) k Ψ k , (4.2.55)which allows to conclude the proof by picking K ε ∼ ε − , δ ∼ ε and C ε ∼ ε − . (cid:3) Remark 4.1. Proposition 4.2 has as an important consequence the fact that the groundstate energy of H L is uniformly bounded for α & . Final Estimates for Cut-off Hamiltonian. With Propositions 4.1 and 4.2 at hand,we are finally ready to prove the main result of this section. Note that all the estimatesperformed in this section are actually independent of L . Proposition 4.3. Let H Λ L = − ∆ L − Φ( v xL, Λ ) + N , (4.2.56) where v L, Λ is defined in (4.2.5) . Then, for any Λ & and α & , inf spec H L − inf spec H Λ L & − (Λ − / + α − Λ − / + α − Λ − ) . (4.2.57)Note that for the error term introduced in (4.2.57) to be negligible compared to α − itsuffices to pick Λ ≫ α / . Proof. We begin by recalling that Proposition 4.1 implies that a ( w xL, Λ ) + a † ( w xL, Λ ) = Φ( w xL, Λ ) . (Λ − / + α − Λ − / )( | p | + N + 1) . (4.2.58)Applying the unitary Gross transformation U Kα introduced in the previous subsection (with f defined in (4.2.37) and K large enough for Proposition 4.2 to hold for some < ε < )to both sides of the previous inequality and recalling (4.2.31), we obtain ( U Kα ) † Φ( w xL, Λ ) U Kα = Φ( w xL, Λ ) + 2 h f | w L, Λ i . (Λ − / + α − Λ − / )( U Kα ) † ( | p | + N + 1) U Kα . (4.2.59)Proposition 4.2 implies that ( U Kα ) † ( | p | + N + 1) U Kα . ( H L + C ) , (4.2.60)where C is a positive constant (independent of α for α & ). Recalling the definitions of f and w L, Λ we also have | h f | w L, Λ i | ≤ X = k ∈ πL Z | k | > Λ L ( α | k | + 1) | k | . α − Λ − , (4.2.61)and this allows us to conclude, in combination with (4.2.59) and (4.2.60), that Φ( w xL, Λ ) . (Λ − / + α − Λ − / + α − Λ − )( H L + C ) . (4.2.62)Hence h Ψ | H L | Ψ i ≥ h Ψ | H Λ L | Ψ i − (Λ − / + α − Λ − / + α − Λ − ) h Ψ | ( H L + C ) | Ψ i . (4.2.63)By Remark 4.1, to compute the ground state energy of H L it is clearly sufficient to restrictto the spectral subspace relative to | H L | ≤ C for some suitable C , which then yields(4.2.57). This concludes the proof and the section. (cid:3) Final Lower Bound. In this section we show the validity of the lower bound in(2.3.2), thus completing the proof of Theorem 2.2. With Proposition 4.3 at hand, we havegood estimates on the cost of substituting H L with H Λ L and, in particular, we know that thedifference between the ground state energies of the two is negligible for Λ ≫ α / . We arethus left with the task of giving a lower bound on inf spec H Λ L .While the previous steps in the lower bound follow closely the analogous strategy in [9],the translation invariance of our model leads to substantial complications in the subsequentsteps, and the analysis given in this subsection is the main novel part of our proof. Incontrast to the case considered in [9], the set of minimizers M F L = Ω L ( ϕ L ) is a three-dimensional manifold, and in order to decouple the resulting zero-modes of the Hessian of TRONGLY COUPLED POLARON ON THE TORUS 45 the Pekar functional we find it necessary introduce a suitable diffeomorphism that ’flattens’the manifold of minimizers and the region close to it. Special attention also has to be paidon the metric in which this closeness is measured, necessitating the introduction of thefamily of norms in (3.2.47).We emphasize that the non-uniformity in L also results from the subsequent analysis,where the compactness of resolvent of − ∆ L enters in an essential way.Let Π denote the projection ran Π = span (cid:26) L − / e ik · x , k ∈ πL Z , | k | ≤ Λ (cid:27) , N = dim C ran Π . (4.3.1)For later use we note that N ∼ (cid:18) L π (cid:19) Λ as Λ → ∞ . (4.3.2)The Fock space F ( L ( T L )) naturally factorizes into the tensor product F (Π L ( T L )) ⊗F ((11 − Π) L ( T L )) and H Λ L is of the form A ⊗ 11 + 11 ⊗ N > , with A acting on L ( T L ) ⊗F (Π L ( T L )) and N > being the number operator on F ((11 − Π) L ( T L )) . In particular, inf spec H Λ L = inf spec A .As in Section 4.1, we can, for any L -orthonormal basis of real-valued functions { f n } of ran Π , identify F (Π L ( T L )) with L ( R N ) through the Q -space representation (see [25]).In particular, any real-valued ϕ ∈ ran Π corresponds to a point λ ∈ R N via ϕ = Π ϕ = N X n =1 λ n f n ∼ = ( λ , . . . , λ N ) = λ. (4.3.3)Note that, compared to Section 4.1, we are using a different choice of Π here for thedecomposition L ( T L ) = ran Π ⊕ (ran Π) ⊥ .In the representation given by (4.3.3), the operator A is given by A = − ∆ L + V ϕ ( x ) + N X n =1 (cid:18) − α ∂ λ n + λ n − α (cid:19) (4.3.4)on L ( T L ) ⊗ L ( R N ) . For a lower bound, we can replace h ϕ = − ∆ L + V ϕ with the infimumof its spectrum e ( ϕ ) , obtaining inf spec H Λ L ≥ inf spec K , (4.3.5)where K is the operator on L ( R N ) defined as K = − α N X n =1 ∂ λ n − N α + F L ( ϕ ) = 14 α ( − ∆ λ ) − N α + F L ( λ ) , (4.3.6)where F L , which is understood as a multiplication operator in (4.3.6), can be seen as afunction of ϕ ∈ span R { f j } Nj =1 or λ ∈ R N through the identification (4.3.3).Using IMS localization we shall split R N into two regions, one localized around thesurface of minimizers of F L , i.e., M F L = Ω L ( ϕ L ) , and the other localized away from it.On each of these regions we can bound F L from below with the estimates contained inProposition 3.4 and in Corollary 2.1, respectively. Because of the prefactor α − in front of − ∆ λ the outer region turns out to be negligible compared to the inner one (at least if wedefine the inner and outer region with respect to an appropriate norm). At the same time,employing an appropriate diffeomorphism, the inner region can be treated as if Ω L ( ϕ L ) was a a flat torus, leading to a system of harmonic oscillators whose ground state energycan be calculated explicitly.The start be specifying the norm with respect to which we measure closeness to Ω L ( ϕ L ) .Recall the definition of the W T -norms given in (3.2.47). Note that for T ≥ Λ the L -normcoincides with the W T -norm on ran Π , which makes < T < Λ the relevant regime forour discussion. In fact, we shall pick ≪ T ≪ Λ / , α / ≪ Λ , (4.3.7)where T ≫ is needed for the inner region to yield the right contribution, and T ≪ Λ / ensures that the outer region contribution is negligible.We proceed by introducing an IMS type localization with respect to k · k W T . Let χ : R + → [0 , be a smooth function such that χ ( t ) = 1 for t ≤ / and χ ( t ) = 0 for t ≥ .Let ε > and let j and j denote the multiplication operators on L ( R N ) j = χ (cid:0) ε − dist W T ( ϕ, Ω L ( ϕ L )) (cid:1) , j = q − j . (4.3.8)Then K = j K j + j K j − E , (4.3.9)where E is the IMS localization error given by E = 14 α N X n =1 (cid:0) | ∂ λ n j | + | ∂ λ n j | (cid:1) . (4.3.10)To bound E we apply Lemma 3.11, which states that for ε sufficiently small, for any ϕ ∈ supp j , there exists a unique y ϕ ∈ T L such that dist W T ( ϕ, Ω L ( ϕ L )) = (cid:10) ϕ − ϕ y ϕ L (cid:12)(cid:12) W T (cid:12)(cid:12) ϕ − ϕ y ϕ L (cid:11) . (4.3.11)Likewise, for any n ∈ { , . . . , N } and any h sufficiently small there exists a unique y n,h ∈ T L such that dist W T ( ϕ + hf n , Ω L ( ϕ L )) = (cid:10) ϕ + hf n − ϕ y n,h L (cid:12)(cid:12) W T (cid:12)(cid:12) ϕ + hf n − ϕ y n,h L (cid:11) . (4.3.12)It is easy to see, using again Lemma 3.11, that lim h → y h,n = y ϕ for any n . Therefore, usingthat dist W T ( ϕ + hf n , Ω L ( ϕ L )) ≤ k ϕ − ϕ y ϕ L k W T and dist W T ( ϕ, Ω L ( ϕ L )) ≤ k ϕ − ϕ y h,n L k W T ,we arrive at h f n | W T (cid:12)(cid:12) ϕ − ϕ y ϕ L (cid:11) = lim h → h f n | W T (cid:12)(cid:12) ϕ − ϕ y h,n L (cid:11) ≤ lim h → h − (cid:0) dist W T ( ϕ + hf n , Ω L ( ϕ L )) − dist W T ( ϕ, Ω L ( ϕ L )) (cid:1) ≤ h f n | W T (cid:12)(cid:12) ϕ − ϕ y ϕ L (cid:11) , (4.3.13)which shows that ∂ λ n dist W T ( ϕ, Ω L ( ϕ L )) = 2 h f n | W T ( ϕ − ϕ y ϕ L ) i . (4.3.14)Using that | χ ′ | , (cid:12)(cid:12)(cid:12)(cid:2) (1 − χ ) / (cid:3) ′ (cid:12)(cid:12)(cid:12) . [1 / , , for k = 1 , we obtain | [ ∂ λ n j k ] ( ϕ ) | . ε − (cid:12)(cid:12) ∂ λ n dist W T ( ϕ, Ω L ( ϕ L )) (cid:12)(cid:12) { dist WT ( ϕ, Ω L ( ϕ L )) ≤ ε } . ε − | h f n | W T ( ϕ − ϕ y ϕ L ) i| { dist WT ( ϕ, Ω L ( ϕ L )) ≤ ε } . (4.3.15) TRONGLY COUPLED POLARON ON THE TORUS 47 Summing over n , using that k W T k ≤ and that { f n } is an orthonormal system, we arriveat E . α − ε − , (4.3.16)and thus the localization error is negligible as long as ε ≫ α − . Hence, we are left withthe task of providing lower bounds for j K j and j K j under the constraint ε ≫ α − . Wecarry out these estimates in the next two subsections, 4.3.1 and 4.3.2. Finally, in Section4.3.3, we combine these bounds to prove the lower bound in (2.3.2).4.3.1. Bounds on j K j . Let us look closer at the intersection of the ε -neighborhood of Ω L ( ϕ L ) with respect to the W T -norm with ran Π , i.e., the set [ΠΩ L ( ϕ L )] ε,T := { ϕ ∈ ran Π | ¯ ϕ = ϕ, dist W T ( ϕ, Ω L ( ϕ L )) ≤ ε } = supp j ∩ ran Π . (4.3.17)In the following we shall show that this set is, for ε small enough, a tubular neighborhoodof ΠΩ L ( ϕ L ) , which can be mapped via a suitable diffeomorphism (given in Definition 4.1)to a tubular neighborhood of a flat torus.Since ϕ ∈ ran Π and Π commutes both with W T and with the transformation g g y for any y ∈ T L , we have dist W T ( ϕ, Ω L ( ϕ L )) = k (11 − Π) ϕ L k W T + dist W T ( ϕ, Ω L (Π ϕ L )) . (4.3.18)This implies that [ΠΩ L ( ϕ L )] ε,T is non-empty if and only if r T,ε := q ε − k (11 − Π) ϕ L k W T > . (4.3.19)Since ϕ L ∈ C ∞ ( T L ) , r T,ε > as long as ε & L Λ − h (4.3.20)for some h > and Λ sufficiently large. In particular, (4.3.20) is satisfied with h = 5 / for α large enough since, as discussed above, we need to pick ε ≫ α − and Λ ≫ α / for theIMS and the cutoff errors to be negligible.Lemma 3.11 implies that any ϕ ∈ [ΠΩ L ( ϕ L )] ε,T , for ε ≤ ε ′ L (independently of T and N ), admits a unique W T -projection ϕ y ϕ L onto Ω L ( ϕ L ) and ϕ = ϕ y ϕ L + ( v ϕ ) y ϕ , with v ϕ ∈ (span { W T ∇ ϕ L } ) ⊥ L . (4.3.21)Since W T and Π commute, Ω L ( ϕ L ) is ‘parallel’ to ran Π with respect to k · k W T , i.e., dist W T (ran Π , ϕ yL ) is independent of y and the W T -projection of ϕ yL onto Π is simply Π( ϕ yL ) = (Π ϕ L ) y . Therefore, for ε ≤ ε ′ L , any ϕ ∈ [ΠΩ L ( ϕ L )] ε,T admits a unique W T -projection (Π ϕ L ) y ϕ onto Ω L (Π ϕ L ) and (4.3.21) induces a unique decomposition of theform ϕ = (Π ϕ L ) y ϕ + ( η ϕ ) y ϕ , with η ϕ ∈ (span { Π W T ∇ ϕ L } ) ⊥ L , k η ϕ k W T ≤ r T,ε , (4.3.22)where η ϕ = Π v ϕ (note that (11 − Π) v ϕ = − (11 − Π) ϕ L ). This allows to introduce the fol-lowing diffeomorphism, which is a central object in our discussion. It maps [ΠΩ L ( ϕ L )] ε,T onto a tubular neighborhood of a flat torus. We shall call this diffeomorphism Gross coor-dinates , as it is inspired by an approach introduced in [14]. Definition 4.1 (Gross coordinates) . For B T, Λ ε := (cid:8) η ∈ span R { Π W T ∇ ϕ L } ⊥ L ∩ ran Π | k η k W T ≤ r T,ε (cid:9) ⊂ ran Π , (4.3.23)we define the Gross coordinates map u as u : [ΠΩ L ( ϕ L )] ε,T → T L × B T, Λ ε ,ϕ ( y ϕ , η ϕ ) , (4.3.24)where y ϕ and η ϕ are defined through the decomposition (4.3.22).By the discussion above it is clear that u is well-defined and invertible, for ε ≤ ε ′ L (defined in Lemma 3.11), with inverse u − given by u − : T L × B T, Λ ε → [ΠΩ L ( ϕ L )] ε,T ( y, η ) (Π ϕ L ) y + η y . (4.3.25)We emphasize that the whole aim of the discussion above is to show that u is well-defined,since once that has been shown the invertibility of u and the form of u − are obvious. Inother words, the map u − as defined in (4.3.25) is trivially-well defined, but it is injectiveand surjective with inverse u only thanks to the existence and uniqueness of the decompo-sition (4.3.22).To show that u is a smooth diffeomorphism, we prefer to work with its inverse u − ,which we proceed to write down more explicitly. For this purpose, we pick a real L -orthonormal basis { f k } Nk =1 of ran Π , such that f , f and f are an orthonormal basis of span { Π W T ∇ ϕ L } and f = Π ϕ L k Π ϕ L k . Note that span { Π W T ∇ ϕ L } is three dimensional, asremarked after (3.2.45), at least for N and T large enough, and that f is indeed orthog-onal to f , f and f since in k -space W T and Π are even multiplication operators whilethe partial derivatives are odd multiplication operators. We denote the projection onto span k =1 , , { Π W T ∂ k ϕ L } by Π L ∇ ,T := X k =1 | f k i h f k | . (4.3.26)Having fixed a real orthonormal L -basis, we can identify any real-valued function in ran Π (and hence also any function in [ΠΩ L ( ϕ L )] ε,T ) with a point ( λ , . . . , λ N ) via (4.3.3). Inthese coordinates, the orthogonal transformation that acts on functions in ran Π as thetranslation by y , i.e., ϕ ϕ y , reads R ( y ) := N X k =1 | f yk i h f k | , (4.3.27)and we can write B T, Λ ε in (4.3.23) as B T, Λ ε := η = ( η , . . . , η N ) ∈ span R { f , . . . , f N } (cid:12)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X k =4 η k f k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) W T ≤ r T,ε . (4.3.28)In this basis, we can write u − explicitly as u − ( y, η ) = (Π ϕ L ) y + η y = R ( y )(0 , , , k Π ϕ L k + η , η , . . . , η N ) . (4.3.29)The following Lemma uses this explicit expression for u − and shows that it is a smoothdiffeomorphism (therefore showing that the Gross coordinates map u is as well). TRONGLY COUPLED POLARON ON THE TORUS 49 Lemma 4.2. Let u − be the map defined in (4.3.29) . There exists ε L ≤ ε ′ L (independent of T and N ) and N L > such that for any ε ≤ ε L , any T > and any N > N L the map u − is a C ∞ -diffeomorphism from T L × B T, Λ ε onto [ΠΩ L ( ϕ L )] ε,T . Moreover, for ε ≤ ε L , | det Du − | and all its derivatives are uniformly bounded independently of T and N .Proof. We introduce the notation J ( y, η ) = Du − ( y, η ) and d ( y, η ) := | det J ( y, η ) | . Notethat R ( y ) in (4.3.27) satisfies R ( − y ) = R ( y ) − = R ( y ) t since { f yj } Nj =1 is an orthonormalbasis of ran Π for any y . Hence, for j = 1 , . . . , N we have ( u − ) j ( y, η ) = (cid:10) f j (cid:12)(cid:12) u − ( y, η ) (cid:11) = * R ( − y ) f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Π ϕ L + N X l =4 η l f l + . (4.3.30)This yields the smoothness of u − in η and in y (noting that { f j } Nj =1 ⊂ ran Π is a set ofsmooth functions for any N ). We proceed to compute J . We have, for ≤ k ≤ N , ∂ η k ( u − ) j ( y, η ) = h R ( − y ) f j | f k i = h f j | R ( y ) f k i , (4.3.31)and ∂ y k ( u − ) j ( y, η ) = * f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ y k R ( y ) Π ϕ L + N X l =4 η l f l !+ = − * f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ( y ) ∂ k Π ϕ L + N X l =4 η l f l !+ (4.3.32)for ≤ k ≤ . Therefore J ( y, η ) = R ( y ) " X k =1 | v k i h f k | + X k ≥ | f k i h f k | = R ( y ) − Π L ∇ ,T + X k =1 | v k i h f k | ! =: R ( y ) J ( η ) , (4.3.33)where v k ( η ) := − ∂ k u − (0 , η ) = − ∂ k (cid:16) Π ϕ L + P Nl =4 η l f l (cid:17) . Since R ( y ) is orthogonal, wesee that d = | det J | (implying, in particular, that d is independent of y ).Observe that J = (cid:18) A A (cid:19) , (4.3.34)where A is the × matrix given by ( A ) jk = h f j | v k i = * f j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∂ k Π ϕ L + N X l =4 η l f l !+ , j, k ∈ { , , } , (4.3.35)and A is the ( N − × matrix defined by ( A ) jk = * f j +3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∂ k Π ϕ L + N X l =4 η l f l !+ j ∈ { , . . . , N − } , k ∈ { , , } . (4.3.36)Since J is the identity in the bottom-right ( N − × ( N − corner and in the top-right × ( N − corner, d = | det A | . On ran Π L ∇ ,T the operators ∂ k with k = 1 , , and W − T are uniformly bounded in N and T . Recall also that k η k W T ≤ ε L . Hence, for someconstant C L independent of N and T , and for any j, k ∈ { , , } , we have | ( A ) jk | ≤ k ∂ k f j k k Π ϕ L k + k W − T ∂ k f j k W T k η k W T ≤ C L . (4.3.37)Moreover, for any j, k ∈ { , , } and any l, l , l ∈ { , . . . , N } , we also have ∂ η l ( A ) jk = h ∂ k f j | f l i , ∂ η l ∂ η l ( A ) jk = 0 . (4.3.38)Clearly, (4.3.37) and (4.3.38) together with the fact that d = | det A | show that d and allits derivatives are uniformly bounded in N and T . To show that there exists ε L and N L such that d ≥ C L > for all ε ≤ ε L , T > and N > N L , we show that the image ofthe -dimensional unit sphere under A is uniformly bounded away from , which clearlyyields our claim. For this purpose, we observe that the k -th column of A is given by Π L ∇ ,T h − ∂ k (cid:16) Π ϕ L + P Nl =4 η l f l (cid:17)i and therefore, for any unit vector a = ( a , a , a ) ∈ R , A a = X k =1 a k Π L ∇ ,T " − ∂ k Π ϕ L + N X l =4 η l f l ! = − Π L ∇ ,T ∂ a u − (0 , η ) , (4.3.39)where we denote P k =1 a k ∂ k = ∂ a . To bound the norm of A a from below, it is thensufficient to test ∂ a u − (0 , η ) against one normalized element of ran Π L ∇ ,T , say Π W T ∂ a ϕ L k Π W T ∂ a ϕ L k .We obtain k A a k = k Π L ∇ ,T ∂ a u − (0 , η ) k ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* Π W T ∂ a ϕ L k Π W T ∂ a ϕ L k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ a Π ϕ L + N X l =4 η l f l !+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k Π W T ∂ a ϕ L k − (cid:12)(cid:12)(cid:12) k Π W / T ∂ a ϕ L k − (cid:10) Π ∂ a ϕ L (cid:12)(cid:12) η (cid:11) W T (cid:12)(cid:12)(cid:12) ≥ k ∂ a ϕ L k − (cid:16) k Π W / ∂ a ϕ L k − k Π ∂ a ϕ L k W T k η k W T (cid:17) ≥ k ∂ a ϕ L k − (cid:16) k Π W / ∂ a ϕ L k − ε k ∂ a ϕ L k (cid:17) , (4.3.40)where we used that k η k W T ≤ ε , ≤ W T ≤ and Π ≤ , and ( · ) + denotes the positivepart. As remarked after (3.2.45), ∂ a ϕ L = ( − ∆ L ) − / ∂ a | ψ L | = 0 and since ϕ L ∈ C ∞ , ∂ a ϕ L and ∂ a ϕ L are uniformly bounded in a . We can thus find N L > and ε L such that ther.h.s. of (4.3.40) is bounded from below by some constant C L > uniformly for T > , N > N L and ε ≤ ε L . This shows that A (and hence J ) is invertible at every point and that d ≥ C L > uniformly in T > , N > N L and ε ≤ ε L , as claimed. This concludes theproof. (cid:3) Since u is a diffeomorphism, we can introduce a unitary operator that lifts u − to L ,defined by U : L ( T L × B T, Λ ε ) −→ L ([ΠΩ L ( ϕ L )] ε,T ) U ( ψ ) := | det ( Du ) | / ψ ◦ u. (4.3.41)Recall that j is supported in [ΠΩ L ( ϕ L )] ε,T , hence we can apply U to j K j , obtaining anoperator that acts on functions on T L × R N − that are supported in T L × B T, Λ ε . In particular, j K j ≥ j inf spec H ( T L × B T, Λ ε )[ U ∗ K U ] , (4.3.42)where the subscript indicates that the operator has to be understood as the correspondingquadratic form with form domain H ( T L × B T, Λ ε ) (i.e., with Dirichlet boundary conditions TRONGLY COUPLED POLARON ON THE TORUS 51 on the boundary of B T, Λ ε ). We are hence left with the task of giving a lower bound on inf spec H ( T L × B T, Λ ε )[ U ∗ K U ] , which will be done in the remainder of this subsection.Recalling the definition of K given in (4.3.6), we proceed to find a convenient lowerbound for U ∗ F L U . Any (Π ϕ L ) y ϕ + ( w ϕ ) y ϕ = ϕ ∈ [ΠΩ L ( ϕ L )] ε,T satisfies (3.2.22) with ϕ y ϕ L in place of ϕ L , and we can therefore expand F L ( ϕ ) using Proposition 3.4, obtaining F L ( ϕ ) − e L ≥ h ( w ϕ ) y ϕ − ((11 − Π) ϕ L ) y ϕ | − K y ϕ L − εC L J y ϕ L | ( w ϕ ) y ϕ − ((11 − Π) ϕ L ) y ϕ i = h ϕ L | (11 − Π)(11 − K L − εC L J L )(11 − Π) | ϕ L i− h (11 − Π) ϕ L | − K L − εC L J L | w ϕ i + h w ϕ | − K L − εC L J L | w ϕ i . (4.3.43)Since K L and J L are trace class operators, (11 − Π)(11 − K L − εC L J L )(11 − Π) > (4.3.44)holds for Λ sufficiently large and ε sufficiently small. Moreover, since ϕ L ∈ C ∞ ( T L ) | h (11 − Π) ϕ L | − K L − εC L J L | w ϕ i |≤ k W − / T (11 − K L − εC L J L )(11 − Π) ϕ L k k w ϕ k W T = O ( ε Λ − h ) (4.3.45)for arbitrary h > and uniformly in T . This implies that, for any ϕ = (Π ϕ L ) y ϕ + ( w ϕ ) y ϕ ∈ [ΠΩ L ( ϕ L )] ε,T , any Λ sufficiently large, any ε sufficiently small and an arbitrary h F L ( ϕ ) = F L ((Π ϕ L ) y ϕ + ( w ϕ ) y ϕ ) ≥ e L − O ( ε Λ − h ) + h w ϕ | − K L − εC L J L | w ϕ i . (4.3.46)Therefore, if we define the [( N − × ( N − -matrix M with coefficients M k,j := h f k +3 | − K L − εC L J L | f j +3 i , (4.3.47)then, by (4.3.46), the multiplication operator U ∗ F L U satisfies ( U ∗ F L U )( y, η ) ≥ e L + h η | M | η i − O ( ε Λ − h ) . (4.3.48)It is easy to see that M is a positive matrix, at least for ε sufficiently small and T and Λ sufficiently large. Indeed, the positivity of M is equivalent to the positivity of (11 − K L − εC L J L ) on ran(Π − Π L ∇ ,T ) and, by Proposition 3.5, (11 − K L − εC L J L ) is positive on anyvector space with trivial intersection with ran Π L ∇ . Clearly, since Π L ∇ ,T → Π L ∇ as T → ∞ ,the bound M ≥ c L > (4.3.49)holds, uniformly in T , Λ and for ε sufficiently small.We now proceed to bound − U ∗ ∆ λ U from below. Lemma 4.3. Let U be the unitary transformation defined in (4.3.41) . There exists C L > ,independent of N , T and ε , such that, for ε ≤ ε L , T > and N > N L U ∗ ( − ∆ λ ) U ≥ − ∆ η − C L . (4.3.50) Proof. Since (4.3.33) shows that J ( y, η ) = R ( y ) J ( η ) with R ( y ) orthogonal, we have U ∗ ( − ∆ λ ) U = − d − / ∇ · d / (cid:2) J − ( J − ) t (cid:3) d / ∇ d − / = − d − / ∇ · d / (cid:2) J − ( J − ) t (cid:3) d / ∇ d − / , (4.3.51) with d ( y, η ) = | det J ( y, η ) | and ∇ denoting the gradient with respect to ( y, η ) ∈ R N .Recalling the expression (4.3.34) for J , we find J − = (cid:18) A − − A A − (cid:19) = (cid:18) (cid:19) + (cid:18) A − − A A − (cid:19) =: (1 − Π L ∇ ,T ) + D. (4.3.52)Since D (11 − Π L ∇ ,T ) = (11 − Π L ∇ ,T ) D t = 0 , we have J − ( J − ) t = (11 − Π L ∇ ,T ) + DD t ≥ − Π L ∇ ,T . (4.3.53)With (4.3.51) and (4.3.53), we thus obtain U ∗ ( − ∆ λ ) U ≥ − d − / ∇ · d / (cid:0) − Π L ∇ ,T (cid:1) d / ∇ d − / = − ∆ η − (2 d ) − |∇ d | + (2 d ) − ∆ d. (4.3.54)Lemma 4.2 guarantees that d and all its derivatives are bounded, and d is bounded awayfrom uniformly in N > N L , T > and ε ≤ ε L , leading to (4.3.50). (cid:3) In combination, (4.3.48), (4.3.50) and the positivity of M imply that j K j ≥ j inf spec H ( T L × B T, Λ ε )( U ∗ K U ) (4.3.55) ≥ j (cid:18) e L − N α − O ( ε Λ − h ) − O ( α − ) + inf spec L ( R N ) (cid:20) − α ∆ η + h η | M | η i (cid:21)(cid:19) = j (cid:18) e L − α ( N − Tr (cid:0) M / (cid:1) ) − O ( ε Λ − h ) − O ( α − ) (cid:19) . (4.3.56)Note that since we are taking Λ ≫ α / , ε ≪ and h > was arbitrary, picking h = 5 allows to absorb the error term O ( ε Λ − h ) in the error term O ( α − ) . Recalling the definitionof M given in (4.3.47), we have Tr (cid:0) M / (cid:1) = Tr hq (Π − Π L ∇ ,T )(11 − K L − εC L J L )(Π − Π L ∇ ,T ) i . (4.3.57)With { t j } N − j =1 an orthonormal basis of ran(Π − Π L ∇ ,T ) of eigenfunctions of (Π − Π L ∇ ,T )(11 − K L − εC L J L )(Π − Π L ∇ ,T ) , we can write Tr (cid:0) M / (cid:1) = N − X j =1 h t j | − K L − εC L J L | t j i / = N − X j =1 " h t j | − K L | t j i / − εC L ξ / j h t j | J L | t j i (4.3.58)for some { ξ j } N − j =1 satisfying c L ≤ h t j | − K L − εC L J L | t j i ≤ ξ j ≤ h t j | − K L | t j i ≤ (4.3.59)for T and Λ large enough and ε small enough, where we used (4.3.49) for the lower bound.Using the concavity of the square root and the trace class property of J L , we conclude that Tr (cid:0) M / (cid:1) ≥ N − X j =1 h t j | p − K L | t j i − εC L Tr( J L ) = Tr h (Π − Π L ∇ ,T ) p − K L i − εC L . (4.3.60)Since ϕ L ∈ C ∞ and recalling (4.3.7), for an arbitrary h > we can bound k Π L ∇ − Π L ∇ ,T k . L min { Λ , T } − h = T − h , (4.3.61) TRONGLY COUPLED POLARON ON THE TORUS 53 which also implies the same estimate for the trace-norm of the difference of Π L ∇ and Π L ∇ ,T ,both operators being of rank . Recalling that Π L ∇ projects onto ker(11 − K L ) , we finallyobtain Tr (cid:0) M / (cid:1) ≥ Tr h Π p − K L i − O ( ε ) − O ( T − h ) . (4.3.62)The error term O ( T − h ) forces T → ∞ as α → ∞ , but allows T to grow with an arbitrarilysmall power of α . By picking h to be sufficiently large we can absorb it in the error term O ( ε ) .We obtain the final lower bound j K j ≥ j (cid:20) e L − α Tr (cid:2) Π(11 − (11 − K L ) / ) (cid:3) − O ( εα − ) − O ( α − ) (cid:21) ≥ j (cid:20) e L − α Tr (cid:2) (11 − (11 − K L ) / ) (cid:3) − O ( εα − ) − O ( α − ) (cid:21) . (4.3.63)4.3.2. Bounds on j K j . We recall Corollary 2.1, which implies that, for any ϕ ∈ L R ( T L ) , F L ( ϕ ) ≥ e L + inf y ∈ T L h ϕ − ϕ yL | B | ϕ − ϕ yL i , (4.3.64)where B acts in k -space as the multiplication by B ( k ) = ( for k = 0 , − (1 + κ ′ | k | ) − for k = 0 . (4.3.65)Note that B − ηW T > for η > small enough (independently of T ). Moreover, for any ϕ in the support of j and any y ∈ T L , h ϕ − ϕ yL | W T | ϕ − ϕ yL i ≥ ε / . (4.3.66)Therefore, on the support of j , we have F L ( ϕ ) ≥ e L + inf y ∈ T L h ϕ − ϕ yL | B − ηW T | ϕ − ϕ yL i + ηε / . (4.3.67)By the Cauchy–Schwarz inequality, using that all the operators involved commute, we have h ϕ − ϕ yL | B − ηW T | ϕ − ϕ yL i ≥ h ϕ | (11 − W / γ )( B − ηW T ) | ϕ i + h ϕ L | (11 − W − / γ )( B − ηW T ) | ϕ L i (4.3.68)for any γ > . Note that the right hand side is independent of y . Since ϕ L ∈ C ∞ ( T L ) , theFourier coefficients of ϕ L satisfy (1 + | k | ) / | ( ϕ L ) k | ≤ C L,t γ − t for | k | ≥ γ (4.3.69)for any t > . Using the positivity of B − ηW T we can bound h ϕ L | (11 − W − / γ )( B − ηW T ) | ϕ L i ≥ − X k ∈ πL Z | k | >γ ( B ( k ) − ηW T ( k ))(1 + | k | ) / | ( ϕ L ) k | = − X k ∈ πL Z | k | >γ ( B ( k ) − ηW T ( k ))(1 + | k | ) (1 + | k | ) / | ( ϕ L ) k | ≥ − C L,t γ − t X k ∈ πL Z | k | >γ | k | ) & L γ − t − . (4.3.70) Therefore we conclude, using the positivity of − W / γ β and of B − ηW T , that j K j ≥ j inf spec (cid:20) e L − N α + ηε − O ( γ − t − ) − α ∆ λ + h ϕ | (11 − W / γ )( B − ηW T ) | ϕ i (cid:21) = j (cid:18) e L + ηε − O ( γ − t − ) − α Tr (cid:20) Π (cid:18) − q (11 − W / γ )( B − ηW T ) (cid:19)(cid:21)(cid:19) . (4.3.71)We need to estimate the behavior in N = rank Π , T and γ of the trace appearing in thelast equation, which equals Tr (cid:20) Π (cid:18) − q (11 − W / γ )( B − ηW T ) (cid:19)(cid:21) = X k ∈ πL Z | k |≤ Λ (cid:18) − q (1 − W γ ( k ) / )( B ( k ) − ηW T ( k )) (cid:19) . (4.3.72)The contribution to the sum from | k | ≤ max { γ, T } can be bounded by C ( L max { γ, T } ) .For | k | > max { γ, T } , W γ ( k ) = W T ( k ) = (1+ | k | ) − , and the coefficient under the squareroot in the last line of (4.3.72) behaves asymptotically for large momenta as − | k | − .Hence, recalling (4.3.2), we conclude that Tr (cid:20) Π (cid:18) − q (11 − W / γ )( B − ηW T ) (cid:19)(cid:21) ≤ O (cid:0) max { γ, T } (cid:1) + O (Λ ) . (4.3.73)Because of (4.3.7), the first term on the right hand side is negligible compared to the secondif we choose γ to equal α to some small enough power. Because t was arbitrary, we thusarrive at j K j ≥ j (cid:18) e L + ηε − O ( α − Λ ) (cid:19) . (4.3.74)Therefore, if ε ≥ C L α − Λ (4.3.75)for a sufficiently large constant C L , we conclude that for sufficiently large α and Λ j K j ≥ j e L . (4.3.76)4.3.3. Proof of Theorem 2.2, lower bound. By combining the results (4.3.63) and (4.3.76)of the previous two subsections with (4.3.9) and (4.3.16), we obtain K ≥ j K j + j K j + O ( α − ε − ) ≥ j (cid:20) e L − α Tr (cid:2) (11 − (11 − K L ) / ) (cid:3) + O ( εα − ) + O ( α − ) (cid:21) + j e L + O ( α − ε − ) ≥ e L − α Tr (cid:2) (11 − (11 − K L ) / ) (cid:3) + O ( εα − ) + O ( α − ) + O ( α − ε − ) (4.3.77) TRONGLY COUPLED POLARON ON THE TORUS 55 under the constraint (4.3.75). With Proposition 4.3 we can thus conclude that inf spec H L ≥ inf spec H Λ L + O (Λ − / ) + O ( α − Λ − / ) + O ( α − Λ − ) ≥ inf spec K + O (Λ − / ) + O ( α − Λ − / ) + O ( α − Λ − ) ≥ e L − α Tr (cid:2) (11 − (11 − K ) / ) (cid:3) + O ( εα − ) + O ( α − ) + O ( α − ε − )+ O (Λ − / ) + O ( α − Λ − / ) + O ( α − Λ − ) . (4.3.78)To minimize the error terms under the constraint (4.3.75), we pick ε ∼ α − / and Λ ∼ α / ,which yields the claimed estimate inf spec H L ≥ e L − α Tr (cid:2) (11 − (11 − K L ) / ) (cid:3) + O ( α − / ) . 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