
==== Front
Calc Var Partial Differ Equ
Calc Var Partial Differ Equ
Calculus of Variations and Partial Differential Equations
0944-2669
1432-0835
Springer Berlin Heidelberg Berlin/Heidelberg

2735
10.1007/s00526-024-02735-3
Article
Traveling waves and effective mass for the regularized Landau-Pekar equations
http://orcid.org/0000-0001-5059-4466
Rademacher Simone simone.rademacher@math.lmu.de

grid.5252.0 0000 0004 1936 973X Department of Mathematics, LMU Munich, Theresienstrasse 39, 80333 Munich, Germany
Communicated by Enno Lenzmann.

26 4 2024
26 4 2024
2024
63 5 1214 10 2023
4 4 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
We consider the regularized Landau-Pekar equations with positive speed of sound and prove the existence of subsonic traveling waves. We provide a definition of the effective mass for the regularized Landau-Pekar equations based on the energy-velocity expansion of subsonic traveling waves. Moreover we show that this definition of the effective mass agrees with the definition based on an energy-momentum expansion of low energy states.

Mathematics Subject Classification

35Q40
35Q55
35C07
35A15
http://dx.doi.org/10.13039/100019180 HORIZON EUROPE European Research Council 694227 Rademacher Simone issue-copyright-statement© Springer-Verlag GmbH Germany, part of Springer Nature 2024
==== Body
pmcIntroduction and main results

The polaron is quasi-particle that models an electron moving through an ionic crystal while interacting with its self-induced polarization field. The polarization field can be either described as a quantum field by the Fröhlich model [1] (called quantum polaron) or as a classical field by the Landau-Pekar equations [2–4] (called classical polaron). The Landau-Pekar equations describe the polaron as a pair (ψ,φ)∈H1(R3)×Lϵ2(R3) where ψ denotes the L2-normalized wave function of the electron and φ the classical field which is for a positive function ε>0 an element of1.1 Lε2(R3):=φ|‖εφ‖2<∞.

The Landau-Pekar equations are given by the coupled system of differential equations1.2 i∂tψt=hαφtψt,iε-1∂tφt=φt+ασψt

where α>0 denotes the coupling constant, m>0 the electron’s mass,1.3 σψ:=(2π)3/2vεϱ^ψ,hφ=-Δ2m+Vφ,withVφ(x)=2Re∫eik·xv(k)φ(k)dk

and where ϱψ:=|ψ|2,1.4 ε=1,andv(k)=1|k|.

The strong coupling limit is linked with a classical field approximation: For α→∞, the classical Landau-Pekar equations can be derived from the quantum dynamics generated by the Fröhlich model [5–10].

Effective mass problem for the Landau-Pekar equations

The dynamics of the polaron is closely related to the outstanding problem of its effective mass: Due to the interaction with the self-induced polarization field, the electron slows down. In physics this phenomenon is described by the emergence of a quasi-particle, the polaron, with an increased effective mass meff>0.

Based on the classical polaron, Landau and Pekar [2–4] formulated a famous quantitative prediction for the effective mass in the strong coupling limit. Their heuristic ideas (described in more detail in [11]) rely on the existence of traveling waves of the Landau-Pekar equations, i.e. solutions of (1.2) with initial data (ψv,φv)∈H1(R3)×Lε2(R3), ‖ψv‖2=1 satisfying1.5 ψt(x),φt(k)=eievtψv(x-vt),eiv·ktφv(k)

with phase ev∈R and velocity v∈R3. Traveling waves were, however, conjectured to not exists for v≠0 for the Landau-Pekar equations [11] due to a vanishing speed of sound.

Related to that, the corresponding energy functional to (1.2) does not dominate the total momentum. Thus a computation of the energy as function of conserved total momentum yields a constant function and therefore an infinite mass. Contrarily for the quantum Fröhlich model such an energy-momentum expansion allows to approach the quantum polaron’s effective mass (see [12–14] resp. [15–17] for recent progress based on different techniques).

However for the classical polaron, i.e. the Landau-Pekar equations, neither traveling wave solutions nor an energy-momentum expansion serve for a mathematical rigorous definition of the effective mass. To overcome these problems [11] provides a definition of the effective mass that is based on a novel energy-velocity expansion and verifies the quantitative prediction by Landau and Pekar for the classical polaron.

The goal of this paper is to verify Landau and Pekar’s heuristic approach for the effective mass, originally formulated for the non-regularized classical polaron, mathematical rigorously for a regularized classical polaron model, namely the regularized Landau-Pekar equations.

More precisely, we choose instead of (1.4) the functions ε,v to be sufficiently regular (see Assumptions 1.1, 1.2 below) and show that subsonic traveling wave solutions with non-vanishing velocity v≠0 do exist (Theorem 1) and serve for a definition of the effective mass (Theorem 2). Moreover, the resulting formula agrees with a definition of the effective mass through an energy-momentum expansion (Theorem 3) and, furthermore, with results obtained for the quantum regularized Fröhlich model [18].

Regularized Landau-Pekar equations

The regularized Landau-Pekar equations describe more generally a particle moving through an excitable medium. We impose the following assumptions on the functions ε,v and1.6 g=(2π)3/2v/ε-1/2^.

Assumption 1.1

(Regularity) Let ε,v be radial with ε>0 and such that g∈H2(R3) and |g^(k)|≥(1+|k|)-9/2 for all k∈R3.

Furthermore we consider underlying media of positive critical velocity vcrit>0 formulated in the assumption below. The critical velocity is often referred to as speed of sound of the medium.

Assumption 1.2

Let ε>0 satisfy infk∈R3ε(k)|k|:=vcrit for a constant vcrit>0.

We remark that Assumptions 1.1 and 1.2 exclude the (non-regularized) Landau-Pekar equations with ε,v given by (1.4) that, in particular, have vanishing speed of sound with the above definition.

The dynamical equations (1.2) for ε,v satisfying Assumption 1.1, 1.2 are well defined (see Lemma 2.1 below). Moreover the energy functional1.7 Gα(ψ,φ)=⟨ψ|hαφ|ψ⟩+‖ε1/2φ‖22with‖ψ‖2=1

where hφ is given by (1.3) is preserved along the dynamics. For the regularized Landau-Pekar equations we show that there exists subsonic traveling wave solutions with 0<|v|<vcrit.

Theorem 1

Let ε,v satisfy Assumptions 1.1 and 1.2 and |v|<vcrit. Then there exists a traveling wave solution of the form (1.5) with v≠0.

Theorem 1 follows from Proposition 2.3 and is proven in Sect. 4.

We can not treat the case of supersonic traveling waves |v|>vcrit. However we conjecture that supersonic traveling waves do not exist. This conjecture is based on the observation that for |v|>vcrit, the energy functional does not dominate the total momentum, similarly as for the non-regularized model discussed before.

Effective mass problem for the regularized Landau-Pekar equations

We provide two definitions for the effective mass. The first definition (Theorem 2) is based on an energy-velocity expansion of traveling wave solutions and inspired by ideas of Landau and Pekar. The second (Theorem 3) is based on an energy-momentum expansion for low energy states. Both definitions lead to the same formula for the effective mass and, in particular, verify the physicists’ predictions.

Traveling waves approach

We derive an energy-velocity expansion of low-energy traveling wave solutions (1.5) with small velocities. To be more precise we consider states (ψ,φ)∈H1(R3)×Lε2(R3)(i) with small energy, i.e. satisfying 1.8 Gα(ψ,φ)≤eα+κfor sufficiently smallκ>0(independent ofα)

(ii)v and which are traveling wave solutions of velocity v, i.e. let v<vcrit (uniformly in α), then (ψv,φv) solves (1.5) with velocity v and with phase ev≥-eα+v2/4.

The definition of the effective mass through traveling waves is based on their energy-velocity expansion, i.e. for states of the set1.9 Iv:={(ψ,φ)∈H1(R3)×Lε2(R3)|(i), (ii)vare satisfied}

we study the energy expansion1.10 EvTW:=inf{Gα(ψ,φ)|(ψ,φ)∈Iv}

around the ground state energy eα.

Theorem 2

Let ε,v satisfy Assumptions 1.1 and 1.2. Assume that for the pair of ground states (ψα,φα) of Gα given by (1.7) where φα=-ασψα, the minimizer ψα is unique up to translations and changes of phase. There exists α0>0 such that for all α≥α0 and αv≤1, we have1.11 EvTW=eα+m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22v22+O(αv3).

The energy expansion of Theorem 2 (i.e. (1.11)) is proven in Sect. 5.

The coefficients of the energy expansion are well defined as1.12 ‖kvε-3/2ϱ^ψα‖22=‖kvε-3/2ϱ^ψ~α‖22

for any1.13 ψα,ψ~α∈Θ(ψα):={eiωψαy=eiωψα(·-y)|y∈R3,ω∈[0,2π)}.

We define the effective mass as the second order coefficient of the expansion of EvTW around the ground state energy. It follows from the ground state’s approximation (see Proposition 2.2 below) that ϱ^ψα(k)→1 point-wise in the limit α→∞. Therefore in the strong coupling limit the leading order in α of the effective mass is given by1.14 limα→∞α-1meffTW:=limα→∞α-1limv→0EvTW-eαv2/2=2(2π)33‖kvε-3/2‖22

and agrees with the findings of for the quantum Fröhlich model [18].

Approach through energy-momentum-expansion

For the second approach we are interested in the infimum of Gα w.r.t. to the set of states (ψ,φ)∈H1(R3)×Lε2(R3) with small energy (i.e. satisfy (i)) and (ii)p with mean momentum p∈R3, i.e. 1.15 ⟨ψ^|p|ψ^⟩+⟨φ|p|φ⟩=p.

Thus we consider states of the set1.16 Ip:={(ψ,φ)∈H1(R3)×Lε2(R3)|such that(i),(ii)phold}.

The definition of the effective mass then relies on an expansion ofEp:=inf{Gα(ψ,φ)|ψ,φ∈Ip}

stated in the following theorem.

Theorem 3

Let ε,v satisfy Assumptions 1.1 and 1.2. Assume that for the pair of ground states (ψα,φα) of Gα given by (1.7) where φα=-ασψα, the minimizer ψα is unique up to translations and changes of phase. There exists α0>0 such that for all α≥α0 and α-1/4p≤11.17 Ep=eα+m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22-1p22+O(α-5/4p3).

There exists α0>0 such that for all α≥α0 and α-1/2p≤1, a pair of minimizers (ψp,φp) of Ep is a traveling wave solution (ψv′,φv′) to (4.1) with velocity v′=meff-1p+O(α-3/2p2) and 1.18 Ep=Gα(ψv′,φv′)+O(α-2p3).

Theorem 3(a), (b) are proven in Sect. 5.

We define the effective mass as the coefficient of the second order contribution of the energy-momentum expansion and thus in leading order in α given in the strong coupling limit by1.19 limα→∞α-1meff:=limα→∞α-1limp→0Ep-eαp2/2-1=2(2π)3α3‖kvε-3/2‖22

which agrees with the effective mass meffTW defined in (1.14) and findings from the quantum Fröhlich model [18].

In particular Theorem 3(b) shows that any minimizer of Ep is given by a traveling wave solution of velocity v′=meff-1p, thus, by approximate elements of the set Iv considered in Theorem 2.

We remark that traveling wave solutions for non-linear Schrödinger equations with non-vanishing speed of sound were studied in various other settings (see for example [19] for the Gross-Pitaevksi and [20] for pseudo-relativistic Hartree equation). We note that [19] considers a variational approach to traveling waves in the spirit of Theorem 3(b).

Structure of the paper

In Sect. 2 we collect properties and approximations of the ground state, ground state energy (Sect. 2.1, Proposition 2.1 resp. Proposition 2.2) and traveling wave solutions (Sect. 2.2, Proposition 2.3) that will be important to prove our main theorems. In Sect. 3 we prove Propositions 2.1, 2.2 on the ground state’s properties. For this we first show the existence of ground states for all α>0 in Sect. 3.1, then the approximation of the ground (state) by the harmonic oscillator in Sect. 3.2 and finally the positivity of the Hessian for large α>α0 yielding coercivity estimates. We combine those results in Sect. 3.4 to finally prove Propositions 2.1, 2.2. In Sect. 4 we prove afterwards Proposition 2.3 (yielding in Theorem 1) on the properties of traveling waves. In Sect. 5 we finally prove Theorems 2 and 3 on the two definitions of the effective mass bases on the results before.

Properties of the ground state and traveling waves

Properties of the ground state

For the regularized polaron’s ground state2.1 eα:=infψ,φGα(ψ,φ)

the infimum can be taken first w.r.t. to the phonon field yielding by a completion of the square to the choice2.2 φα:=-ασψ,whereσψ:=(2π)3/2vεϱ^ψ

with ϱψ=|ψ|2. The resulting energy functional for ψ∈H1(R3) is2.3 Eα(ψ):=infφGα(ψ,φ)=⟨ψ|-Δ2m-αh∗|ψ|2|ψ⟩withh=(2π)3/2v2/ε-1^.

Thus if ψα is an element of the manifold of minimizers MEα of the energy functional Eα defined by2.4 MEα:={ψ∈H1(R3)|‖ψ‖2=1,Eα(ψα)=eα},

then the pair (ψα,φα) with φα given by (2.2) is an element of the manifold of minimizers MGα of the energy functional Gα2.5 MGα:={(ψ,φ)∈H1(R3)×Lε2(R3)|Gα(ψ,φ)=eα}.

The energy functional Eα is symmetric with respect to translations and changes of the phase of the wave function. Thus for any minimizer ψα of Eα (i.e. ψα∈MEα) it follows Θ(ψα)⊆MEα where2.6 Θ(ψα):={eiωψαy=eiωψα(·-y)|y∈R3,ω∈[0,2π)}.

For the (non-regularized) Pekar functional corresponding to (1.4), the existence of a unique pair of ground states (ψPekar,φPekar) up to phases and translations was proven [21] for all α>0. For the regularized model we prove the existence of a ground state for all α>0.

Proposition 2.1

(Existence) Let ε,v satisfy Assumption 1.1. For all α>0 there exists a pair of minimizers (ψα,φα)∈MGα with φα=-ασψα and 0<ψα∈C∞(R3) satisfying the Euler-Lagrange equation2.7 hαφα-μψαψα=0,withμψα:=⟨ψα|hαφα|ψα⟩.

We remark that the ground state’s uniqueness for the regularized model is in general not known. For technical reasons, we can not prove uniqueness up to translations and phases for large α>α0. For that, a refined approximation of the ground state than the one in Proposition 2.2 is needed to conclude ground state’s uniqueness up to translations and phase by the local coercivity estimates in Corollary 3.1 for large α>α0.

Note that the first part of Theorems 2 and 3 immediately follow from Proposition 2.1 that is proven in Sect. 3.4.

The ground state ψα’s properties for large coupling constants α>α0 results from the asymptotic behavior of the energy functional Gα. In fact in the strong coupling limit α→∞ the ground state energy eα=Gα(ψα) is well described through the harmonic oscillator2.8 hosc:=-Δ2m+mω2x22,with frequencyω2=α‖∇g‖223m.

Furthermore its well known ground state2.9 ψosc(x):=mωπ3/4e-mωx2/2

approximates the true ground state of Gα as the following Lemma shows.

Proposition 2.2

(Approximation of the ground state) Let ε,v satisfy Assumption 1.1. There exists α0>0 and constants C1,C2>0 (independent of α) such that2.10 C1α1/3≤eα+α‖g‖22-3α‖∇g‖222m≤C2for allα≥α0.

Furthermore let ψα∈MEα. There exists C3>0 (independent of α) such that for all α≥α02.11 distL2Θ(ψα),ψosc≤C3α-1/20,distH1Θ(ψα),ψosc≤C3α9/40.

Proposition 2.2 is proven in Sect. 3.4.

Here we introduced the norm2.12 distL2Θ(ψα),ψosc:=infy′,θ′‖eiθ′ψα′-ψosc‖2

(and similarly for the H1-norm) quantifying the distance of an element ψα of the manifold of minimizers to the harmonic oscillator’s ground state.

We remark that the rate of convergence of (2.11) depends for technical reasons on Assumption 1.1 namely the regularity of the function g.

Traveling waves

The dynamical equations corresponding to the energy functional Gα in (1.7) are given for (ψt,φt)∈H1(R3)×Lε2(R3) by the system of coupled partial differential equations (1.2). The dynamical equations are well-posed as the following Lemma shows.

Lemma 2.1

Let ε,v satisfy Assumption 1.1. For any ψ0,φ0∈H1(R3)×Lε2(R3) there exists a unique global solution of (1.2). Furthermore,2.13 Gα(ψ0,φ0)=Gα(ψt,φt),and‖ψt‖2=‖ψ0‖2

and there exists C>0 such that for (ψ0,φ0) with G(ψ0,φ0)≤Cα we have for all t∈R2.14 ‖∇ψt‖2≤Cαand‖φt‖Lε2≤Cα.

The proof of the Lemma follows similarly to [5, Lemma 2.1] considering the non-regularized Landau-Pekar equations. The arguments presented in [5] apply for the regularized case, too, so that we refer for the proof of Lemma 2.1 to [5, Lemma 2.1].

A traveling wave of velocity v∈R3 is a solution of (1.2) with initial data (ψv,φv)∈H1(R3)×Lε2(R3), ‖ψv‖2=1 satisfying (1.5). The existence of subsonic traveling waves is given by the following Proposition.

Proposition 2.3

Let ε,v satisfy Assumptions 1.1 and 1.2. Furthermore assume that |v|<vcrit. There exists a traveling wave solution of the form (1.5).

Furthermore assume that G(ψv,φv)-eα≤κ for sufficiently small κ>0 (independent of α) and ev≥-eα+v2/4. Assume that the ground state ψα of Eα is unique up to translations and rotations. Then there exists α0>0 and a constant C>0 (independent of α,v) such that 2.15 distL2Θ(ψα),ψv≤C|v|

for all α≥α0 and |v|≤1.

Note that Theorem 1 follows immediately from Proposition 2.3(a). The proof of Proposition 2.3 is given in Sect. 2.2.

Furthermore note that a similar approximation as in (2.15) holds for the field φv, too. For this we remark that instead of minimizing w.r.t. to the field φ in (2.1) first (as explained in Sect. 2.1) we can take the infimum w.r.t. to the wave function ψ first, too yielding the functional2.16 Fα(φ)=infψGα(ψ,φ),MFα={φ∈Lε2(R3)|Fα(φα)=eα}.

We remark that by the energy functional’s symmetries for any φα∈MFα and2.17 Ω(φα):={eiz·φα|z∈R3}

it follows that Ω(φα)⊆MFα. Then under the same assumption as in Proposition 2.3 there exists C>0 (independent of α) such that2.18 distLε2Ω(φα),φv≤Cα|v|.

We remark that Proposition 2.3(a) shows that subsonic traveling waves exist for all α>0. However the approximations (2.15), (2.18) of the second part of the Theorem holds for sufficiently large α≥α0 only. The restriction to sufficiently large α>0 in part (b) ensures the validity of the global coercivity estimates (see Corollary 3.2) that are proven for sufficiently large α>α0 only. Furthermore we notice that for v=0 the pair of ground states (ψα,φα) provide a traveling wave solution with ev=eα. In particular the assumption on the phase from part (b), made for technical reasons only, is satisfied for v=0.

Properties of the energy functional Eα

In this section, we prove Propositions 2.1 and 2.2 on the properties of the energy functional Gα.

The proof of Proposition 2.1 relies on a comparison of properties of Eα with the properties of hosc in the limit α→∞. Then existence and uniqueness (up to translations and changes of the phase) for pairs of minimizers (ψα,φα) of Gα follow with the choice φα=-ασψα.

First, in Lemma 3.1, we prove the existence of a minimizer ψα for all α. Next we show the ground state (energy) is well approximated through the harmonic oscillator (Lemma 3.2). This approximations allows to show that the Hessian modulo its zero modes of Eα is asymptotically for α→∞ characterized by the harmonic oscillator, and thus positive (Lemma 3.3). This fact has several consequences: We infer first local (Corollary 3.1) and later global coercivity estimates (Corollary 3.2) for sufficiently large α≥α0. For the latter we assume that the ground state ψα of MEα is unique up to translations and phase. Furthermore we obtain that the ground state energy eα is separated from the first excited eigenvalue by a gap of order α (Corollary 3.3).

We remark that the strategy for the proofs in this section follow [6, Section 3] considering the non-regularized Pekar functional on a Torus of length L. For sufficiently large L the uniqueness of the ground state and coercivity estimates are proven based on a comparison with the non-regular Pekar functional defined on the full space for which these properties are well known.

Existence

First we show the existence of minimizers of Eα for all α>0 in the subsequent Lemma.

Lemma 3.1

Let ε,v satisfy Assumption 1.1. For all α>0, there exists a minimizer 0<ψα∈C∞(R3) of the functional Eα satisfying the Euler-Lagrange equation3.1 hαφα-μψαψα=0,withμψα:=⟨ψα|hαφα|ψα⟩.

Proof

Since h=g∗g, we have ‖h‖∞≤C‖g‖22 and3.2 ⟨ψ|h∗|ψ|2|ψ⟩≤C‖g‖22‖ψ‖24

so that by Assumption 1.1 there exists C>0 such that3.3 Eα(ψ)≥12m‖∇ψ‖22-Cα.

From (3.3) we infer on one hand that eα≥-Cα for all α. On the other hand, in order to prove the existence of a minimizer, we remark that (3.3) shows that any minimizing sequence ψnn∈N is bounded in H1, uniformly in n∈N. For this reason the sequence by [22, Lemma 6] resp. [23, Theorem 8.10], there exists a sequence ynn∈N∈R3 such that the translated sequence ψnnn∈N has a sub-sequence ψnjnj∈N:=ψnjnjnj∈N that converges weakly in H1(R3) to a non-zero function. It follows from the Sobolev inequality that this sub-sequence converges strongly in Lp for 2≤p≤6 to a non-zero limit. The limiting function ψα∈H1 is again L2-normalized and, moreover, satisfies3.4 ⟨ψα|-Δ2m|ψα⟩≤limnj→∞⟨ψnj|-Δ2m|ψnj⟩

by semi-lower continuity of the H1-norm. Since3.5 ⟨ψnj|h∗|ψnj|2|ψnj⟩-⟨ψα|h∗|ψα|2|ψα⟩≤ψnjh∗|ψnj|2ψα-ψnj+ψαh∗|ψn|2ψα-ψnj+ψnjh∗|ψα|2ψα-ψnj+ψαh∗|ψα|2ψα-ψnj

and ‖h∗|ψ1|2ψ2‖2≤C‖ψ1‖22‖ψ2‖2 for any ψ1,ψ2∈L2, we find3.6 ⟨ψnj|h∗|ψnj|2|ψnj⟩-⟨ψα|h∗|ψα|2|ψα⟩≤C‖ψα-ψnj‖2→0

as n→∞. Therefore,3.7 Eα(ψα)≤lim infnj→∞Eα(ψnj)=eα

and with Eα(ψα)≥lim infnj→∞Eα(ψnj)=eα (as (ψnj)nj∈N is a minimizing sequence), we conclude that Eα(ψα)=eα and, thus, ψα is a minimizer. By invariance of Eα w.r.t. to translations and phase, any element of Θ(ψα) defined in (2.6) is a minimizer, too. The positivity and regularity properties of ψα follows by standard bootstrap arguments (see for example [6, Lemma 3.3]). □

Approximation

Next we prove that in the strong coupling limit the spectrum of Eα is well approximated by the harmonic oscillator hosc’s spectrum. The idea is to use the Taylor expansion of the potential given by Assumption 1.1 through3.8 h(x)=∫v2(k)ε(k)eik·xdk=∫v2(k)ε(k)cos(k·x)dk

to show its asymptotic quadratic behavior. The ground state energy of hosc (as defined in (2.8)) is well known3.9 eosc=3α‖∇g‖222m

and separated from the rest of the spectrum by a gap of order α. For this we compare the Hessian of Eα with3.10 Hosc:=inff∈H1(R3),‖f‖2=1f∈span{ψosc}⊥⟨f|Hosc|f⟩,withHosc=hosc-eosc

that is known to be positive, and thus, yielding coercivity estimates of the form3.11 ⟨f|hosc|f⟩-eosc≥Cα1/2infθ∈(0,2π]‖eiθψosc-f‖22⟨f|hosc|f⟩-eosc≥Cα1/4infθ∈(0,2π]‖eiθψosc-f‖H1(R3)2.

Furthermore we compare the deviation of the ground state of ψα with the one of the harmonic oscillator that is known to be ‖x2ψosc‖2=Cα-1/2 for some C>0.

Lemma 3.2

Let ε,v satisfy Assumption 1.1. Then there exists C1,C2,C3>0 (independent of α) such that 3.12 C1α1/3≤eα+α‖g‖22-3α‖∇g‖222m≤C2

and 3.13 |μψα-μψosc|≤C3α5/12

where we introduced the notation μψosc=⟨ψosc|-Δ+2α(h∗|ψosc|2)|ψosc⟩. Let ψα∈MEα such that 3.14 distL2Θ(ψα),ψosc=‖ψosc-ψα‖2.

Then there exists α0>0 and C>0 (independent of α) such that for all α≥α0 we have 3.15 ‖x2ψα‖2≤Cα-1/2

Let ψα∈MEα. Then there exists α0>0 and C>0 (independent of α) such that for all α≥α0 we have 3.16 distL2Θ(ψα),ψosc≤Cα-1/20.

Let ψα∈MEα such that 3.17 distL2(Θ(ψα),ψosc)=‖ψα-ψosc‖2.

Then there exists α0>0 and C>0 (independent of α) such that for sufficiently large α≥α0 we have 3.18 ‖x2ψα-ψosc‖2≤Cα-11/20.

Under the same assumptions as in part (c) there exists α0>0 and C1,C2,C3>0 (independent of α) such that for sufficiently large α≥α0 we have 3.19 C1α1/4≤‖∇ψα‖2≤C2α1/4

and furthermore 3.20 ‖∇ψα-ψosc‖2≤C3α9/40.

Proof

First we remark that in the following proof we denote with C>0 a constant independent of α.

Proof of (a): For the upper bound of the ground state eα we pick the harmonic oscillator’s ground state ψosc defined in (2.9) as trial state. Its energy serves as an upper bound for the ground state energy3.21 eα≤Eαψosc

and can be explicitly computed with by the potential’s (3.8) Taylor expansion, Assumption 1.1 and cos(x)≥1-x223.22 eα≤-α‖g‖22+eosc+C

for a constant C>0 (independent of α).

For the lower bound we use the IMS localization technique to show that it suffices to consider the problem on a ball of radius R, where we can use the potential’s Taylor expansion. To this end, let ψα denote a minimizer realizing3.23 eα=infψ∈H1(R3)Eα(ψ)=Eα(ψα)

and χ∈C∞(R3) a function with support on the ball B1 with radius one such that ‖χ‖=1 and χ(0)=1. We define the rescaled function χR with ‖χR‖2=1 supported on BR and denote with χR,z=χR(·-z) its shift. The idea is to choose R dependent on α. However for simplicity we neglect the dependence of R on α in the notation. We observe that the L2-normalized function3.24 ψαR,z=χR,zψα/‖χR,zψα‖2

satisfies3.25 |ψα(x)|2=∫BR|χR,zψα|2(x)dz=∫BR|ψαR,z(x)|2‖χR,zψα‖22dz

and thus, by completing the square and standard techniques of IMS localization3.26 Eα(ψα)+‖∇χR‖22=∫BREα(ψαR,z)+α‖vε-1/2(ϱ^ψα-ϱ^ψαR,z)‖22‖χR,zψα‖22dz.

Since χR,zψα‖22dz denotes a probability measure, there exists z∈BR such that3.27 Eα(ψα)+‖∇χR‖22≥Eα(ψαR,z)+α‖vε-1/2(ϱ^ψα-ϱ^ψαR,z)‖22.

By scaling, we furthermore find ‖∇χR‖22≤CR-2 yielding3.28 Eα(ψα)≥Eα(ψαR,z)+α‖vε-1/2(ϱ^ψα-ϱ^ψαR,z)‖22-CR-2.

We use (3.28) to prove both, the energy’s lower bound and the approximation of the ground state. We start with the lower bound on the ground state energy first. For this, we observe that (3.28) implies3.29 Eα(ψα)≥Eα(ψαR,z)-CR-2,

i.e. it suffices to compute the energy Eα for function ψαR,z supported on BR, where we can use the Taylor expansion of h3.30 Eα(ψαR,z)=⟨ψαR,z|-Δ2m+α(h∗|ψαR,z|2)|ψαR,z⟩≥-α‖g‖22+⟨ψαR,z|-Δ2m+α‖∇g‖22(x2∗|ψαR,z|2)|ψαR,z⟩-CαR4.

We observe that (3.27) is invariant w.r.t. translations and changes of phase of ψα and ψR,z and thus, we can furthermore restrict to ψαR,z such that3.31 ⟨ψαR,z|x|ψαR,z⟩=0

for which we find3.32 Eα(ψα)≥⟨ψαR,z|hosc|ψαR,z⟩-α‖g‖22-CαR4-CR-2.

where hosc denotes the harmonic oscillator hosc=-Δ2m+mω22x2. By definition, ψαR,z is L2-normalized and thus a competitor for the ground state of hosc, i.e.3.33 Eα(ψα)≥infψ∈H1⟨ψ|hosc|ψ⟩-α‖g‖22-CαR4-CR-2≥eosc-α‖g‖22-CαR4-CR-2.

Optimizing w.r.t. to the parameter R (yielding R=α-1/6), we arrive at3.34 eα≥eosc-α‖g‖22-Cα1/3

proving part (a).

Properties of ψα,ψαR,z: As a preliminary step to prove the remaining parts of this Proposition we prove useful properties of the ground state ψα and ψαR,z (constructed in (3.24), satisfying (3.27) and by translational invariance of the problem (3.31)). We observe that cos(x)≤1 (and thus h≤‖g‖22) implies3.35 ⟨ψαR,z|-Δ2m|ψαR,z⟩=Eα(ψαR,z)+α⟨ψαR,z|(h∗|ψαR,z|2)|ψαR,z⟩≤Eα(ψαR,z)+α‖g‖22.

With(3.28) and the ground state energy’s approximation (part (a)) we obtain3.36 ‖ψαR,z‖H12≤Cα,and similarly‖ψα‖H12≤Cα,

and in the same way3.37 α‖v/ε1/2ϱ^ψα‖22,α‖v/ε1/2ϱ^ψαR,z‖22≤Cα.

With the upper bound (3.34) we furthermore deduce from (3.35) resp. the Euler-Lagrange equation3.38 ‖Δψα‖22≤Cα

for all α≥α0. However for ψαR,z we observe first that (3.28) resp. (3.32) together with the harmonic oscillator’s coercivity property (3.11) show for R=α-1/63.39 αinfθ‖ψαR,z-eiθψosc‖22≤⟨ψαR,z|hosc|ψαR,z⟩-eosc≤Eα(ψαR,z)-eosc+α‖g‖22+Cα1/3.

With the upper bound on the energy (3.22) we find3.40 infθ‖ψαR,z-eiθψosc‖22≤Cα-1/6,infθ‖ψαR,z-eiθψosc‖H12≤Cα1/12.

In particular for sufficiently large α≥α0 we have3.41 ‖∇ψαR,z‖2≤Cα-1/4.

Approximation of Lagrange multipliers: We prove the approximation of the Lagrange multiplier μψα with μψosc using the previous results. In particular by translational invariance of the problem we choose ψα such that ψαR,z (as constructed in (3.24)) satisfies (3.27) and (3.31)). We write3.42 μψα-μψosc=Eα(ψα)-Eα(ψosc)-α‖v/ε1/2ϱψα‖22+α‖v/ε1/2ϱψosc‖22

yielding with (3.27) and (3.37) to3.43 |μψα-μψosc|≤|Eα(ψα)-Eα(ψosc)|+α‖v/ε1/2ϱψosc‖2+‖v/ε1/2ϱψα‖2‖v/ε1/2ϱψα-ϱψosc‖2≤Cα1/3+α‖v/ε1/2ϱψα-ϱψosc‖2.

Since3.44 ‖v/ε1/2(ϱψα-ϱψosc)‖2≤‖v/ε1/2(ϱψαR,z-ϱψosc)‖2+‖v/ε1/2(ϱψαR,z-ϱψosc)‖2

we find with (3.27), ‖v/ε1/2(ϱψαR,z-ϱψosc)‖2≤C‖ψosc-ψαR,z‖2 and (3.40)3.45 ‖v/ε1/2ϱψα-ϱψosc‖2≤Cα-1/3+Cα-1/12.

Thus we obtain from (3.43) for sufficiently large α>α03.46 |μψα-μψosc|≤Cα5/12.

Scaling of the ground state: To show the ground state’s scaling (3.15) for ψα satisfying (3.14) we observe that by the Euler-Lagrange equation we have3.47 -Δ-μψαψα=2αh∗|ψα|2ψα.

The idea is now to use the properties of the resolvent -Δ-μψα-1 (that is well defined sind from (3.46) we have μψα≥-Cα for sufficiently large α≥α0) to prove the desired bound. The resolvent’s Green’s function is given in terms of the inverse Fourier transform F-1 of3.48 Gμψα(z)=F-1(p2/(2m)-μψα)-1(z)

and can by functional calculus explicitly computed. In fact we have3.49 F-12mΔ-μψα-1(p)=∫0∞e-t(12mp2-μψα)dt

which leads with (3.48) to3.50 Gμψα(z)=12π3/2∫R3∫0∞eiz·petμψαe-t2mp2dtdp

and we arrive with Fubini’s theorem at3.51 Gμψα(z)=(2m)3/2∫0∞etμψαe-mz22tt3/2dt=(2m)3/2π|z|e--4mμψα|z|

for |z|≠0. We plug this identity into (3.47) and find using assumption (3.14), and the notation F:=2αh∗|ψα|2ψα (i.e. F denotes the right hand side of (3.47))3.52 ψα(x)=∫Gμψα(x-y)F(y)dy.

Thus the weighted L2-norm, we aim to find an upper bound for, becomes3.53 ‖x3ψα‖22=∫x6Gμψα(x-y)Gμψα(x-z)F(z)F(y)dxdydz

that we can estimate by Cauchy Schwarz with3.54 ‖x3ψα‖22≤∫x6|Gμψα(x-y)|2|F(y)|2dxdy=(2m)3π∫x6|x-y|2e-2-4mμψα|x-y||F(y)|2dydx.

We split the integral into several regions to find the desired bound: First we consider the case |x|>2|y| for which we have |x-y|≥|x|-|y|≥|x|2. By substitution we can therefore estimate the integral in this region by3.55 ∫|x|>2|y|x6|x-y|2e-2-4mμψα|x-y||F(y)|2dydx≤∫x4e-2-4mμψα|x||F(y)|2dydx≤Cα5/2‖F‖22

where we used -μψα≥Cα for sufficiently large α≥α0. Now let |x|<2|y| and |x-y|>|y|2, then we have |x||x-y|≤4 and it follows3.56 ∫|x|<2|y||x-y|≥|y|/2x6|x-y|2e-2-4mμψα|x-y||F(y)|2dydx≤4∫x4e--4mμψα|y||F(y)|2dydx≤Cα5/2‖F‖22.

with similar arguments as before. Finally for |x|<2|y| and |x-y|≤|y|2 it follows |x|≥|y|-|x-y|≥|y|/2. Hence3.57 ∫|x|<2|y||x-y|<|y|/2x6|x-y|2e-2-4mμψα|x-y||F(y)|2dydx≤2∫|y|/2<|x|<2|y|y6|x-y|2e--4mμψα|x-y||F(y)|2dydx≤C(-μψα)1/2∫y6e-C-4mμψα|y||F(y)|2dy≤Cα5/2‖F‖22

where we used that |y|6e-C-4mμψα|y|≤C~α3. Summarizing the estimates we conclude by3.58 ‖x3ψα‖2≤Cα-5/4‖F‖2

where F denotes the r.h.s. of (3.47). Since ‖h‖∞≤C,ψα is a L2-normalized function we find that3.59 ‖x3ψα‖2≤Cα-3/4.

for sufficiently large α≥α0. In particular for n=3 we find ‖x3ψα‖2≤Cα-3/4 and thus, in particular,3.60 ‖x2ψα‖22≤Cα-1/2.

Proof of (b): In order to prove the ground state ψα’s approximation we observe that by the previous discussion it is enough to consider the problem on the ball BR1(0) with R1=α-1/5. In fact (3.15) shows3.61 infθ‖ψosc-eiθψα‖2≤‖ψosc-eiθψα‖L2(BR1(0))+Cα-1/20.

We consider ψα and ψαR,z (constructed in (3.24)), satisfying (3.27) and by translational invariance of the problem (3.31)). With these notations we in particular have from (3.40)3.62 infθ‖ψosc-eiθψα‖L2(BR1(0))≤‖ψα-ψαR,z‖L2(BR1(0))+infθ‖ψosc-eiθψαR,z‖L2(BR1(0))≤‖ψα-ψαR,z‖L2(BR1(0))+Cα-1/12

and thus it remains to show ψαR,z is close to ψα. To this end we control the L2-norm of the difference ϱψα-ϱψαR,z first and deduce as a second step from this estimates for L2-norm of ψαR,z-ψα. For this we write the L2-norm of ϱψα-ϱψαR,z in momentum space. We shall show that for low momenta, we control the norm with Assumption 1.1 and (3.27) while for high momenta the L2-norm is small by regularity properties of ϱψα,ϱψαR,z. For the latter one we observe that from (3.38) we have3.63 ‖k2ϱ^ψα‖∞≤C‖Δψα‖2‖ψα‖2+C‖∇ψα‖22≤Cα1/2

and thus there exists C>0 such that3.64 |ϱ^ψα(k)|≤Cα1/2|k|2,

To obtain a similar bound for ϱ^ψαR,z we first have to derive a bound for the H2-norm of ψαR,z. For this we remark that by definition (3.24) we have3.65 ‖χR,zψα‖2ΔψαR,z=χR,z(Δψα)+2(∇χR,z)(∇ψα)+(ΔχR,z)ψα.

With the Taylor expansion of χR,z around z we find that ‖χR,zψα‖2≥‖ψα‖2+R-1‖x(∇χR,z(w))ψα‖2 for w∈BR(z) and thus with the ground state’s scaling properties (3.15) we have ‖χR,zψα‖2≥C for sufficiently large α≥α0. Hence we find with (3.19), (3.38) and the scaling of χR,z at ‖ΔψαR,z‖2≤Cα and thus3.66 |ϱ^ψαR,z(k)|≤Cα1/2|k|2.

From (3.64) and (3.66) we find that for high momenta, i.e. all k∈BR2c={k∈R3:|k|>R2} we have3.67 ‖ϱ^ψα-ϱ^ψαR,z‖L2(BR2c)≤Cα1/2R21/2.

We remark that we choose R2:=R2(α)=α1/5, however, for simplicity (as for R), we neglect the dependence of α in its notation. For low momenta, i.e. k∈BR2 we use that by Assumption (1.1) we have |g^(k)|≥(1+|k|)-9/2 and thus3.68 ‖ϱ^ψα-ϱ^ψαR,z‖L2(BR2)≤C(1+|R2|)9/2‖vε-1/2(ϱ^ψα-ϱ^ψαR,z)‖L2(BR2)≤C(1+|R2|)9/2‖vε-1/2(ϱ^ψα-ϱ^ψαR,z)‖2

and we find with (3.27) (assuming R2≤α1/5)3.69 ‖ϱ^ψα-ϱ^ψαR,z‖2≤Cα-1/3(1+|R2|)9/2.

Optimizing with respect to R2 leads to R2=α1/5 and thus for sufficiently large α≥α0 to3.70 ‖ϱ^ψα-ϱ^ψαR,z‖2≤Cα2/5.

In particular it follows as ψαR,z and ψα have the same phase3.71 ‖ψα-ψαR,z‖44=‖|ψα|-|ψαR,z|‖44≤‖ϱ^ψα-ϱ^ψαR,z‖22≤Cα4/5.

We recall that we need to estimate the L2-norm difference of ψα,ψαR,z on the ball BR1, i.e. we have by Cauchy Schwarz’s inequality with R1=α-1/53.72 ‖ψα-ψαR,z‖L2(BR1)2≤CR13/2α2/5≤Cα-1/5

and we arrive with (3.61) at3.73 infθ‖eiθψα-ψosc‖22≤Cα-1/10.

Proof of part (c): In order to prove H1-norm convergence of ψα to ψosc we observe that from the Euler-Lagrange equations of ψα resp. ψosc, Assumption 1.1 and the scaling properties of ψosc3.74 -12mΔψα-ψosc=αh∗|ψα|2+μψαψα-αh∗|ψosc|2+μψoscψosc+Cα(|·|4∗|ψosc|2)+1ψosc

for a constant C>0 independent of α. We recall that both ψα and ψosc are L2-normalized functions. Hence introducing the notation h~=h-‖g‖22 and3.75 μ~ψ=⟨ψ|-Δ2m-α(h~∗|ψ|2)|ψ⟩

for any ψ∈H1(R3) we arrive at3.76 -12mΔψα-ψosc=αh~∗ψαψα-ψosc¯ψα+αh~∗ψα-ψoscψosc¯ψα+αh~∗|ψosc|2ψosc-ψα-μ~ψα-μ~ψoscψα+μ~ψαψosc-ψα+μ~ψoscψα-ψosc+Cα(|·|4∗|ψosc|2)ψosc

From Assumption 1.1 we have |h~(x)|≤Cx2 and thus together with the ground state’s scaling properties (part (c)) we find |μ~ψα|≤Cα1/2, |μ~ψosc|≤Cα1/2 and3.77 |μ~ψα-μ~ψosc|=|μψα-μψosc|≤Cα5/12

from part (part (a)). Therefore we find with the approximation of the ground state (part (b)) and the ground state’s scaling properties (part (c)) that3.78 |ψosc-Δψα-ψosc|,|ψα-Δψα-ψosc|≤Cα9/20

finally yielding3.79 ‖∇ψα-∇ψosc‖22≤Cα9/40

and the desired lower and upper bound in (3.19).

Proof of (d): To prove convergence of ψα to ψosc for ψα satisfying (3.14) in the weighted L2-norm, too, we proceed similarly as in part (d). From the Euler-Lagrange equation of ψα we get3.80 12mΔ-μψαψα-ψosc=2αh∗|ψα|2ψα-2αh∗|ψosc|2+(μψosc-μψα)ψosc=2αh∗ψαψα-ψosc¯ψα+2αh∗ψα-ψoscψosc¯ψα+2αh∗|ψosc|2ψosc-ψα-μψα-μψoscψosc+Cα(|·|4∗|ψosc|2)ψosc.

By assumption resp. part (a) we have3.81 distL2ψosc,Θ(ψα)=‖ψα-ψosc‖2≤Cα-1/20

and furthermore we have from (3.58) (using assumption (3.14)) the estimate3.82 ‖x2(ψα-ψosc)‖2≤Cα-1/2‖F‖2.

where F denotes the r.h.s. of (3.80). It follows from the approximation of the Lagrange multiplier (part (a)) and the ground state (part (b)) that ‖F‖2≤Cα-1/20 and we finally arrive at the desired bound of part (c). □

Properties of the Hessian

The Hessian Hα of Eα is defined for any ψα∈MEα by3.83 Hα:=limε→01δ2Eαψα+δf‖ψα+δf‖2-eαfor allf∈H1(R3).

We can explicitly compute the Hessian and find3.84 Hα:=⟨Imf|Hα|Imf⟩+⟨Ref|QαHα-4XψαQα|Ref⟩

where Qα=1-Pα=1-ψαψα and3.85 Hα=hαφα-μψα,andXψα(x;y)=ψα(x)h(x-y)ψα(y).

Positivity of the Hessian

We compare the Hessian’s components3.86 Hα(1):=infψα∈MEαinff∈H1(R3),‖f‖2=1f∈span{ψα}⊥⟨f|Hα|f⟩,Hα(2):=infψα∈MEαinff∈H1(R3),‖f‖2=1f∈span{ψα,∂1ψα,∂2ψα,∂3ψα}⊥⟨f|Hα-4Xψα|f⟩

with Hosc (defined in (3.10)) known to satisfy Hosc≥Cα for a constant C>0 independent of α. We remark that by definition the Hessian Hα is defined modulo its zero modes namely the ground state ψα for the first component resp. ψα and its partial derivatives ∂jψα for i=1,2,3 for the second component.

Lemma 3.3

Let ε,v satisfy Assumption 1.1. Then, there exists α0>0 and C>0 (independent of α) such that3.87 Hα(i)≥Cα1/2,for allα≥α0.

Proof

We present the proof of the Hessian’s component Hα(2). The statement for H(1) then follows with similar arguments.

For any ψα∈MEα and f∈H1(R3) we define the projection Qα′=1-Pα′,Pα′=ψαψα+∑j=13∂jψα∂jψα/‖∂jψα‖22 and furthermore the L2-normalized function3.88 gα:=Qα′f‖Qα′f‖2.

Thus in the following we consider the expectation value3.89 ⟨gα|Hα-4Xψα|gα⟩=⟨gα|Hα-4X~ψα|gα⟩

where we introduced the notation3.90 X~ψα(x;y)=ψα(x)h~(x-y)ψα(y)

with h~=h-‖g‖22 and used that the zero-th order term of the expansion of h in the above expectation value vanishes as Qα′f is orthogonal to ψα. By translational invariance of the problem we restrict to ψα such that distL2(Θ(ψα),ψosc)=‖ψosc-ψα‖2 (so that Lemma 3.2(c), (d) apply). We recall that we want to compare Hα(i) with Hosc that is of order α. Thus any term we can show to be o(α) will be dominated in the end by Hosc and thus will be considered to be sub-leading for sufficiently large α≥α0.

We shall first show that gα is approximately orthogonal to the harmonic oscillator’s ground and first excited state Posc′=ψoscψosc+∑j=13∂jψosc∂jψosc/‖∂jψosc‖22, i.e. that it is enough to consider3.91 ⟨Qosc′gα|Hα-4X~ψα|Qosc′gα⟩.

This follows from the observation that the difference is given by3.92 ⟨gα|Hα-4X~ψα|gα⟩-⟨Qosc′gα|Hα-4X~ψα|Qosc′gα⟩=2RePosc′gαHα-4X~ψαgα+Posc′gαHα-4X~ψαPosc′gα.

On the one hand, since3.93 ‖PoscQαgα‖2≤C‖ψα-ψosc‖2‖gα‖2≤Cα-1/20‖gα‖2

and similarly denoting Posc(j)=:∂jψosc∂jψosc/‖∂jψosc‖22 and Pα(j)=:∂jψα∂jψα/‖∂jψα‖22,3.94 ‖Posc(j)Qα(j)gα‖2≤C‖ψosc‖2-1‖∂jψosc-ψα‖2‖gα‖2≤Cα-1/40

from Lemma 3.2(d) for sufficiently large α≥α0 and ‖∇ψosc‖≥cα1/4 for some c>0. Thus we arrive at3.95 ‖Posc′gα‖2≤Cα-1/40.

On the other hand we have3.96 H=hαφosc-μψosc+(μψα-μψosc)

so that with Lemma 3.2(d) we have ‖HαPosc′f‖2≤Cα1/2 and we arrive at3.97 ⟨gα|Hα-4X~ψα|gα⟩-⟨Qosc′gα|Hα-4X~ψα|Qosc′gα⟩≥-Cα-19/40.

As a next step, we shall replace Qoscgα with the L2-normalized function3.98 gosc:=Qosc′Qα′f‖Qosc′Qα′f‖2

For this we first observe that3.99 Qα′gα=‖Qosc′Qα′f‖2‖Qα′f‖2gosc

and thus, we need to control the normalization constants’ ratio. For this we use (3.95) and find that3.100 ‖Qα′f‖22=‖Qosc′Qα′f‖22+‖Posc′Qα′f‖22≤‖Qosc′Qα′f‖22+Cα-1/12‖Qα′f‖2

for a constant C>0 independent in α. In particular we obtain3.101 ‖Qα′f‖2≥C(1-α-1/40)-1‖Qosc′Qα′f‖2

for sufficiently large α≥α0. Thus from (3.97) and (3.98) we get3.102 ⟨gα|Hα-4X~ψα|gα⟩≥C(1-α-1/40)-2⟨gosc|Hα-4X~ψα|gosc⟩-Cα19/40.

Next we show that the operator X~ψα contributes sub-leading (i.e. o(α)) only. For this we write3.103 X~ψα-X~ψosc=ψα(x)-ψosc(x)h~(x-y)ψα(y)+ψosc(x)h~(x-y)ψα(y)-ψosc(y).

With |h~(x)|≤Cx2 we find from Lemma 3.2(c) and ‖gosc‖2=1 that3.104 ⟨gosc|X~ψα|gosc⟩≥⟨gosc|X~ψosc|gosc⟩-Cα9/20.

We recall that gosc is orthogonal to ψosc and its partial derivatives. In particular, as ∇ψosc=xψosc, the function gosc is orthogonal to xψosc, too. Therefore not only the zero-th but also the first-order term of the Taylor expansion of h in X~ψosc evaluated in gosc vanishes, i.e.3.105 ⟨gosc|X~ψosc|gosc⟩=⟨gosc|X~~ψosc|gosc⟩

where X~~ψosc(x,y)=αψosc(x)(h(x-y)-‖g‖22-‖∇g‖22(x-y)2)ψosc(y). Since |h~(x)-‖∇g‖22x2|≤Cx4 by Assumption 1.1 we find with the harmonic oscillators scaling properties that ⟨gosc|X~~ψosc|gosc⟩≥-C, and thus from (3.102)3.106 ⟨gα|Hα-4X~ψα|gα⟩≥C(1-α-1/40)-2⟨gosc|Hα|gosc⟩-Cα19/40

for sufficiently large α≥α0. Now it remains to compare the r.h.s. with the harmonic oscillator. For this we split the operator Hα into one part that is localized on a ball BR with R=α-1/6 (that we shall show is bounded from below by Hosc that is O(α)) and a part outside BRc (that we will show is bounded from below by a positive constant of O(α), i.e. trivially satisfying the claim for sufficiently large α≥α0).

For the localization we consider a partition of unity 0≤η1,η2≤1 with ηi∈C0∞(R3) and3.107 η1(x)=1x∈B10x∈B2c,η2=1-|η1|2.

and define the rescaled version ηRi(x)=ηi(x/R) and the L2-normalized function3.108 gRi=ηRigosc/‖ηRigosc‖2.

With standard arguments of IMS localization we find3.109 ⟨gosc|Hα|gosc⟩=∑i=12‖ηRigosc‖22⟨gRi|Hα|gRi⟩-∑i=12⟨gosc||∇ηRi|2|gosc⟩.

By scaling the last summand is of order R-2=α1/3 yielding3.110 ⟨gosc|Hα|gosc⟩≥∑i=12‖ηRig‖22⟨gRi|Hα|gRi⟩-Cα1/3

and it remains to estimate the expectation value ⟨fRi|Hα|fRi⟩ for i=1,2.

We start with the expectation value w.r.t gR2 supported on BRc. Since cos(k·x)≤1, we find3.111 ⟨gR2|Hα|gR2⟩≥-μψα-⟨gR2|αVαφα|gR2⟩≥2α‖g‖22+⟨gR2|αVαφα|gR2⟩.

To show that the remaining term contributes sub-leading (i.e. o(α)) only, we need to control the L∞- norm of Vαφα on BRc, i.e.3.112 α|Vαφα(x)|≤Cα∫|h(x-y)||ψα(y)|2dy+Cα5/12

for |x|≥α-1/6. We split the integral in BR~ and BR~c where now we choose R~=α-1/5. For y∈BR~c we find by the scaling properties of ψα that3.113 ‖ψα‖L2(BR~c)2≤Cα2/5‖xψα‖L2(BR~c)2≤α-1/10

and we arrive with ‖h‖L∞(R3)≤C for |x|≥α-1/6 at3.114 α|Vαφα(x)|≤Cα∫|y|≤α-1/5|h(x-y)||ψα(y)|2dy+Cα9/10.

Now let |y|≤α-1/5 and |x|≥α-1/6. Then we have |x-y|≥|x|-|y|≥Cα-1/6 for sufficiently large α≥α0. Thus, with ‖h‖L∞(R3)≤C3.115 α|Vαφα(x)|≤Cα1+1/6∫|y|≤α-1/5|x-y||ψα(y)|2dy+Cα5/12

for |x|≥α-1/6. With Cauchy Schwarz inequality and ‖ψα‖4≤C‖ψα‖H1≤Cα1/4 (from Lemma 3.2(d)) we find3.116 α|Vαφα(x)|≤Cα1+1/6+1/4-1/2+Cα5/12≤Cα11/12.

Hence we deduce from (3.111)3.117 ⟨gR2|Hα|gR2⟩≥2α‖g‖22-C2α11/12≥C1α

for constant C1,C2 independent of α and sufficiently large α≥α0.

We recall that the goal for the expectation value3.118 ⟨gR1|Hα|gR1⟩

with fR1 supported on BR is a comparison with the Hessian of the harmonic oscillator Hosc that is O(α). For this we observe that fR1 is almost orthogonal to ψosc and its partial derivatives as3.119 ‖PoscgR1‖2≤C‖ψosc‖L2(BRc)≤Cα-1/4

by Lemma 3.2(c)) and similarly for the partial derivatives. Thus (with similar arguments as in the beginning of this proof (see Eq. (3.91) and subsequent)) instead of gR1 we consider in the following the L2-normalized function3.120 g~R1:=Qosc′gR1‖Qosc′gR1‖2

paying a price sub-leading in α (i.e. o(α) and given by3.121 gR1HαfR1≥(1-α-1/12)2g~R1Hαg~R1-Cα1/3.

We use the Taylor expansion of h, ⟨ψα|x|ψα⟩=0 and Lemma 3.2(d) and find3.122 ⟨g~R1|Hα|g~R1⟩=⟨g~R1|hαφα-μψα|g~R1⟩≥⟨f~R1|hosc-ψαhoscψα|f~R1⟩-Cα1/3

and thus (since ψαhoscψα≥ψoschoscψosc)3.123 ⟨g~R1|Hα|g~R1⟩≥⟨g~R1|Hosc|g~R1⟩-Cα1/3.

By construction g~R1 is a L2-normalized function and orthogonal to the harmonic oscillator’s ground state and its partial derivatives. Thus g~R1 is a competitor for a minimizer of the harmonic oscillator’s Hessian and we conclude that3.124 ⟨gR1|Hα|gR1⟩≥(1-α-1/40)2Hosc-Cα19/40.

Since H(2) is a convex combination of (3.117) and (3.124), we find3.125 H(2)≥C(1-α-1/40)4Hosc-Cα19/40

and conclude that there exists α≥α0 such that H(2)≥Cα1/2 for all α≥α0. □

The Hessian’s positivity in the strong coupling limit α→∞ leads to local coercivity estimates summarized in the following Corollary.

Corollary 3.1

There exists α0>0 and κ,C>0 (independent of α) such that for all α>α0 and ψα∈MEα, any L2-normalized ψ∈H1(R3) and φ∈L2(R3) with3.126 distL2Θ(ψα),ψ≤κα-1/2

we have3.127 Gα(ψ,φ)-eα≥CαdistL2Θ(ψα),ψ2,

3.128 Gα(ψ,φ)-eα≥CαdistLε2Ω(φα),φ2.

The proof is based on an expansion of Gα around the ground state energy eα. In the following, we provide an expansion of G(ψ,φ) which will be useful for later proofs. For this,let δ1=ψ-ψα and δ2=φ-φα3.129 Gα(ψ,φ)-eα=Gα(ψ,φ)-⟨ψα|hαφα|ψα⟩-‖ε1/2φα‖22=2Reδ1hαφαψα+α⟨ψα|Vδ2|ψα⟩+2αReε1/2φαε1/2δ2+⟨δ1|hαφα|δ1⟩+2αReψαVδ2δ1+‖ε1/2δ2‖22+O(α‖δ1‖H12‖δ2‖2).

We observe that the sum of the second and third term vanish by the definition of the potential (1.3) and φα=-ασψα. For the last two terms of the r.h.s., we complete the square3.130 2αReψαVδ2δ1+‖ε1/2Reδ2‖22=‖ε1/2Reδ2+2(2π)3/2αvε-1/2(Reδ1)ψα^‖22-4α⟨Reδ1|Xψα|Reδ1⟩

where Xψα is defined in (3.85) so that we arrive at3.131 Gα(ψ,φ)-eα=⟨Imδ1|hαφα|Imδ1⟩+‖ε1/2Imδ2‖22+2Reδ1hαφαψα+⟨Reδ1|hαφα-Xψα|Reδ1⟩+‖ε1/2Reδ2+2(2π)3/2αvε-1/2(Reδ1)ψ0^‖22+O(α‖δ1‖22‖δ2‖2).

The Euler-Lagrange equation of ψα together with the notation (3.85), (3.85) and the observation that by L2-normalization of ψα and ψ3.132 1=‖ψ‖22=‖ψα+δ1‖22=1+‖δ1‖22+2Reδ1ψα

and therefore3.133 2Reδ1ψ=-‖δ1‖22

we find with (3.85)3.134 Gα(ψ,φ)-eα=⟨QαImδ1|Hα|QαImδ1⟩+‖ε1/2Imδ2‖22+⟨Reδ1|QαHα-4XψαQα|Reδ1⟩+‖ε1/2Reδ2+2(2π)3/2αvε-1/2(Reδ1)ψα^‖22+O(α‖δ1‖22‖δ2‖2)+O(α‖δ1‖23)+O(α‖δ1‖24).

Proof of Corollary 3.1

In order to prove (3.127) first, we remark that it suffices to consider ψ∈H1(R3) such that3.135 Imeiθψαyψ=0,andReeiθψαy∇ψ=0

hold. In particular we assume w.l.o.g. that θ=0 and y=0.

Furthermore, completing the square we get3.136 Gα(ψ,φ)-eα≥Eα(ψ)-eα

and thus it suffices to consider the case δ2=α(σψ-σψα). It follows from (3.134) and (3.135)3.137 Eα(ψ)-eα≥⟨Imδ1|Hα|Imδ1⟩+⟨Reδ1|Qα′Hα-4XψαQα′|Reδ1⟩+O(α‖δ1‖23).

We recover back the Hessian of Eα which by Lemma 3.3 is positive for sufficiently large α≥α0. Moreover, it follows from Lemma 3.3 that there exists C1>0 (independent of α) such that with ‖δ1‖2≤δα-1/2 for sufficiently small δ>0 by assumption, we have3.138 Eα(ψ)-eα≥C1α‖δ1‖22.

Moreover, since cos(k·x)≤1 by Lemma 3.2 there exists a constants κ1,κ2>0 such that3.139 hαφα-eα≥-κ1Δ-κ1α.

Interpolating between (3.138) and (3.139), there exists a constant C2>0 such that for α≥α03.140 Eα(ψ)-eα≥C2α1/4‖δ1‖H12.

By translational and rotational invariance of the energy, we conclude3.141 Eα(ψ)-eα≥C2α1/4distH1Θ(ψα),ψ.

Second we prove (3.128): Completing the square leads to3.142 Gα(ψ,φ)-eα=Eα(ψ)-eα+‖Reφ+ασψ‖Lε22+‖Imφ‖Lε2

so that we find from (3.138) that there exists y∈R3 and κ1>0 such that3.143 Gα(ψ,φ)-eα≥ακ1‖ψ-ψαy‖22+‖Reφ+ασψ‖Lε22+‖Imφ‖Lε22.

By regularity of ε,v, there exists κ3>0 such that3.144 G(ψ,φ)-eα≥κ3α‖σψ-σψαy‖Lε22+‖φ+ασψ‖Lε22

and we find by completing the square3.145 G(ψ,φ)-eα≥‖(1+κ3/α)1/2σψ-σψαy-(1+κ3/α)-1/2αφ+ασψαy‖Lε22+κ3α+κ3‖φ+ασψαy‖Lε22+‖Imφ‖Lε22≥κ4α‖φ+ασψαy‖Lε22+‖Imφ‖Lε22

for a constant κ4>0. We conclude that3.146 G(ψ,φ)-eα≥κ4αdistLε2Ω(φα),Reφ.

□

Global coercivity estimates

The Hessian’s positivity shows the validity of global coercivity estimates of the energy. For this, we additionally have to assume that the ground state ψα is unique up to translations and phase, i.e. that MEα=Θ(ψα). We remark that to prove the ground states uniqueness up to translations and phase by the local coercivity estimates Corollary 3.1, one needs an improved approximation of the ground state than in comparison to Proposition 2.2.

Corollary 3.2

Assume that the ground state ψα of Eα is unique up to translations and rotations. There exists universal constants α0≥0 and C>0 (independent of α) such that for any L2-normalized ψ∈H1(R3) and φ∈L2(R3) with we have for all α≥α03.147 Gα(ψ,φ)-eα≥CαdistL2Θ(ψα),ψ2,

3.148 Gα(ψ,φ)-eα≥CαdistLε2Ω(φα),φ2.

The proof follows the arguments presented in [24, Lemma 2.6].

Proof

We first prove the global bound (3.147). Then the second bound (3.148) follows similarly to the proof of Corollary 3.1.

In order to prove (3.147) we remark (similarly to the proof of Corollary 3.1) that is suffices to consider ψ∈H1(R3) such that3.149 Imeiθψαyψ=0,andReeiθψαy∇ψ=0.

W.l.o.g. we assume y=0 and θ=0. By contradiction we assume that there does not exist a universal constant C>0 such that (3.147) holds. Then there exists a sequence of functions ψn∈L2(R3) with ‖ψn‖L2(R3)=1 such that3.150 Eα(ψn)≤eα+1n‖ψn-ψα‖H12≤2n‖ψn‖H1-Cα.

It follows that Eα(ψn)≥12‖∇ψn‖22-Cα. Therefore ψn is uniformly bounded in H1 and moreover a minimizing sequence. With similar arguments as in the proof of Lemma 3.1ψn converges to an element of the set of minimizes Θ(ψα) given by (3.149) through ψα. This is a contradiction since Corollary 3.1 shows that locally coercivity estimates hold true. □

Another consequence of the Hessian’s approximate behavior is the following property.

Corollary 3.3

There exists α0 and a constant C>0 (independent of α) such that for all α≥α0, we have Hα-eα>Cα.

Proof

The existence of a spectral gap of Hα of order α follows immediately from the global coervitiy estimates in Corollary 3.2. □

Proof of Propositions 2.1, 2.2

In this section we prove Proposition 2.1, 2.2 based on the results proven before.

Proof of Proposition 2.1

The proposition follows immediately from Lemma 3.1. □

Proof of Proposition 2.2

The proposition follows from Lemma 3.2 and Corollary 3.2. □

Proof for traveling waves

In this section, we prove Proposition 2.3 on existence of subsonic traveling waves of the regularized Landau-Pekar equations.

For this, we remark that it follows from the regularized polaron’s dynamics that the traveling wave (1.5) satisfies4.1 -iv·∇ψv=hφv+evψv,-ε-1v·kφv=φv+ασψv.

Proof of Proposition 2.3

Proof of (a): First, we prove the existence of traveling waves for sufficiently small velocities. Traveling wave solutions of (1.5) are stationary points of the action functional Iv given by4.2 Jv(ψ,φ):=⟨ψ|hφ|ψ⟩+‖ε1/2φ‖22+ev‖ψ‖22-v·⟨ψ|i∇|ψ⟩+⟨φ|p|φ⟩.

In the following we show that there exists a minimizer (ψv,φv) of Jv, and thus a traveling wave solution. Since4.3 |v||⟨ψ|i∇|ψ⟩|≤12‖∇ψ‖22+2v2‖ψ‖22,|v||⟨φ|ik|φ⟩|≤|v|‖|p|1/2φ‖22

and for arbitrary δ>04.4 ⟨ψ|Vφ|ψ⟩≤C‖ε1/2φ‖2‖ψ‖22≤δ‖ε1/2Reφ‖22+Cδα‖ψ‖24,

the action functional is bounded from below by4.5 Jv(ψ,φ)≥12‖∇ψ‖22+1-δ‖ε1/2φ‖22-|v|‖|p|1/2φ‖22-Cδα-Cv2

We remark that in the last step we used the L2-normalization of ψ. By Assumption 1.2, we have ε(k)≥vcrit|k| and thus,4.6 Jv(ψ,φ)≥12‖∇ψ‖22+vcrit(1-δ)-v‖|p|1/2φ‖22-Cδα-Cv2.

For |v|≤vcrit-δ, it follows that any minimizing sequence (ψn,φn)n∈N of Iv is uniformly bounded in H1(R3)×L|·|2(R3). Any minimizing sequence, thus, uniformly bounded and weakly converging in H1(R3)×Lε(R3) to a limiting functional that is possibly zero (by translational invariance of the action). With similar arguments as after Eq. (3.3), there exists a sequence (ψnn,eipnφn)n∈N that converges strongly in L2(R3)×Lε2(R3) to a pair of non-zero limiting functions. By semi-lower continuity of the H1- and the Lε-norm, and4.7 |⟨ψn|Vφn|ψn⟩-⟨ψv|Vφv|ψv⟩|≤C‖ψn-ψv‖2+C‖φn-φv‖Lε2

we conclude that the action functional Jv attains its infimum for (ψv,φv) which is a non-zero traveling wave solution (4.1).

Proof of scaling properties: As a preliminary step towards proving Proposition 2.3(b) we shall first prove that4.8 ‖x2ψv‖2≤Cα-1/2

i.e. that the traveling wave satisfies similar scaling properties as the harmonic oscillator. We proceed similarly as in the proof of Lemma 3.2(b). For this let H0:=-Δ/(2m)+iv·∇. Then the traveling wave equation (4.1) implies4.9 H0+evψv=Vαφvψv.

Since H0≥-v2/4 and ev≥-eα+v2/4, the resolvent H0+ev-1 is well defined and we can write4.10 ψv=H0+ev-1Vαφvψv.

The resolvent’s Green’s function is given in terms of the inverse Fourier transform F-1 of4.11 Gev(z)=F-1(p2/(2m)-v·p+ev)-1(z)

and can (by self-adjointness of H0 and functional calculus) explicitly computed. In fact by functional calculus we have4.12 H0+ev-1=∫0∞e-t(H0+ev)dt=∫0te-teve-tp2/(2m)-v·pdt

which leads with (4.11) to4.13 Gev(z)=12π3/2∫R3∫0teiz·pe-teve-tp2/(2m)-v·pdtdp

and we arrive with Fubini’s theorem at4.14 Gev(z)=(2m)3/2eiz·v∫0te-tev-v24e-mz22tt3/2dt=(2m)3/2πeiz·v|z|e-4mev-v24|z|

for |z|≠0. We plug this identity into (4.10) and find that4.15 ψv(x)=∫Gev(x-y)Vαφv(y)ψv(y).

Thus the weighted L2-norm, we aim to find an upper bound for, becomes4.16 ∫x6|ψv(x)|2dx=∫x6Gev(x-y)Vαφv(y)ψv(y)Gev(x-z)Vαφv(z)ψv(z)dxdydz

that we can estimate by Cauchy Schwarz with4.17 ∫x6|ψv(x)|2dx≤∫x2|Gev(x-y)|2|Vαφv(y)ψv(y)|2dxdy=π∫x6|x-y|2e-24mev-v24|x-y||Vαφv(y)ψv(y)|2dydx.

With4.18 ‖Vαφvψv‖22≤Cα‖φv‖Lε2‖ψv‖22

and ‖φv‖Lε≤Cα from Lemma 2.1, we can use a similar splitting of the above integral as in the proof of Lemma 3.2(b) to then conclude by ev-v2/4≥-eα≥Cα with (4.8).

Proof of (b): We observe that for v=0 a traveling wave solution is given by ψv=0=ψαy,φv=0=φαy with ev=μψαy for any y∈R. To prove (2.15) it follows (similarly to the proof of Corollary 3.1) that it suffices to consider the decomposition4.19 ψv=ψαy+δ1,φv=eiy·φα+δ2,andev=μψαy+μv

with Reδ1∇ψαy=0, Imδ1ψαy=0. In particular it follows from Corollary 3.2 and condition (i) that4.20 ‖δ1‖2≤κ1α-1/4,‖δ2‖Lε≤κ2α1/4andμv≤κ3α

for sufficiently small κ1,κ2,κ3>0 (independent of α). In the following we assume w.l.o.g. that y=0.

By definition of Hα (see (3.85)) and the decomposition (4.19), we can write the traveling wave equations (4.1) as4.21 -iv·∇ψα+δ1=VαReδ2+μvψα+Hα+VαReδ2+μvδ1

4.22 ε-1v·kφα+δ2=δ2+2(2π)3/2αvε-1ψαReδ1^+ασδ1

and it follows that the phase μv is given through the identity4.23 μv(1-12‖δ1‖22)=v·ψα∇Imδ1-ψαVαReδ2ψα-ψαVαReδ2Reδ1.

Plugging this identity back into (4.21), we get4.24 Hα-Aδ1=-QvVαReδ2ψv-v·∇ψv

where we introduced the notation Qv=1-ψvψv and the operator4.25 A=‖δ1‖222-‖δ1‖22v·ψα∇Imδ1-ψαVαReδ2Reδ1-ψαVαReδ2ψα.

With the decomposition’s properties (4.20) we find that4.26 ‖A‖≤C‖δ1‖22α‖δ2‖2+αv≤C(α1/4+v).

Thus by Corollary 3.3 there exists κ4>0 such that by assumption Qv(Hα+A)Qv≥κ4α, i.e. we can write4.27 δ1=QvHα-AVαReδ2ψv-v·∇ψv.

The second term of the r.h.s. leads with Proposition 3.2 to the desired bound. For the first term we observe that by definition of the potential and radialilty of v4.28 QvVαδ2ψv=QvVαδ2sψv

where δ2s denotes the symmetric part of δ2, i.e. δ2s(k)=δ2s(-k). We observe that splitting δ2 into its symmetric δ2s(k)=δ2s(-k) and anti-symmetric δ2a(k)=-δ2a(-k) we have from the traveling wave Eq. (4.22)4.29 δ2s=ε-2(k·v)21-ε-2(k·v)2φα-α1-ε-2(k·v)22(2π)3/2vε-1Reδ1ψα^+σδ1.

Here we used that ε-2(k·v)2≤v2/vcrit2<1 by Assumption 1.2. Hence4.30 QvVαReδ2ψv=2αQv∫eik·ε-2(k·v)21-ε-2(k·v)2φα(k)dkψv-αQv∫eik·h^(k)1-ε-2(k·v)22Reδ1ψα^(k)+ϱ^δ1(k)dkψv.

We observe that due to the projection Qv the first term of the Taylor expansion of cos(k·) vanishes. Thus with ε-2(k·v)2≤v2/vcrit2<1 and Reψαψα=-‖δ1‖22 we arrive at4.31 ‖Vαδ2sδ1‖2≤Cv2α‖x2ψv‖2+‖xψα‖2‖xψv‖2+Cα‖x2ψv‖2‖δ1‖22+‖δ1‖2‖xψα‖2‖xψv‖2+‖δ1‖2‖xδ1‖2‖xψv‖2.

From the scaling properties of the traveling wave (4.8) and the ground state (see Corollary 3.3) we conclude4.32 ‖Vαδ2sδ1‖2≤Cαv2+κ1α-1/4‖δ1‖2

yielding for |v|≤14.33 1-κ1α-1/4‖δ1‖2≤C|v|

and (2.18) follows from (4.22) resp. (4.23)4.34 ‖δ2‖≤Cα|v|resp.μv≤α3/4v2.

□

Proofs for definitions effective mass

In this section, we prove Theorems 2 and 3 on the definition of the effective mass.

We remark that the proofs presented in this Section follow ideas from [11] where the non-regularized Landau-Pekar equations have been considered. For the non-regularized Landau-Pekar equations traveling waves are conjectured to not exists. However assuming their existence an energy expansion in the vein of the proof of Theorem 2 was sketched. Furthermore a different approach for a definition of the mass through an energy-velocity expansion was presented. The proof of Theorem 3 given below uses ideas presented there.

Effective mass through traveling waves

We consider the definition of the effective mass through subsonic traveling waves first (whose existence follow from Proposition 2.3).

Proof of Theorem 2

Let α≥α0 large enough and |v|<vcrit. Then, by Proposition 2.3 and 2.1, there exists y,z∈R3 and θ∈(0,2π] such that5.1 ‖eiθψαy-ψv‖2≤Cv,‖eiz·φα-φv‖Lε2≤Cα|v|

with Imψveiθψα=0, Re∇ψveiθψαy=0 and eiz·φα∇φv=0 and ψα is uniquely given (up to translations and changes of phase). W.l.o.g. we assume in the following y=0,θ=0. Then, (3.145) shows (with similar arguments as used in [11]) that we it suffices to consider z=0, too. Thus, we decompose the traveling wave as5.2 ψv,φv=ψα+vξv,φα+vηv

with ‖ξv‖2≤C and ‖ηv‖Lε2≤Cα. Note that Proposition 2.3 moreover shows that ev=μψα+O(α3/4v2) and the linearisation of the traveling wave Eq. (4.1) read5.3 ∇ψαkφα=Hα(2π)3/2αψα∫dkv(k)eik·(2π)3/2α∫dkv(k)eik·ψαεξvηv.

In particular, it follows5.4 HαImξv=∇ψα

5.5 εImηv=kφα

5.6 HφαReξv+αψαVReηv=0

5.7 αVψαReξv+εReηv=0.

Combining (5.7) and (5.6), we find5.8 Hα-4XψαReξv=0.

As Hα is invertible on the span of ∂1ψα, we find from (3.131) with (5.4) and (5.5) for all |v|≤Cα-15.9 G(ψv,φv)=eα+m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22v22+O(α|v|3).

□

Effective mass through energy-momentum expansion

Here, we consider the definition of the effective mass through the energy-momentum expansion explained in Sect. 5.

Proof of Theorem 3

We first pick a trial state to show an upper bound on infIpG(ψ,φ) which we use later for the lower bound. For this, we choose α0>0 sufficiently large, such that by Proposition 2.1 (b), there exists a unique (up to translations and phases) pair of minimizers (ψα,φα) of Gα.

Proof of the upper bound of (a): It is easy to check that the trial states5.10 ψ0=(1-μp)λp-1ψα+iλp·Hα-1∇ψα,φ0(k)=φα(k)+i(1-μp)λp·k(ε(k))-1φα(k)

with the choice5.11 (1-μp)λp=pm+2(2π)3α3‖kvε-3/2ϱ^ψα‖22

and (1-μp)2+λp2=1 (i.e. μp=O(α-2p2)) satisfy constraint (1.15) (using that Hα-1∇ψα=mxψα) and that for large α≥α05.12 ‖ψα-ψ0‖2≤Cα-1pand‖φα-φ0‖2≤Cα-1/2p.

With these observations, we plug the trail states into the expansion of Gα in (3.134) and find5.13 Gα(ψ0,φ0)-eα≤(1-μp)2λp2⟨∇ψα|Hφα-1|∇ψα⟩+13‖ε-1/2kφα‖22+O(α-2p2)=(1-μp)2λp22m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22+O(α-2p2)=m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22-1p22+O(α-2p2).

Proof of the lower bound of (a): For p≤α1/4, it follows from the upper bound and Corollary 3.2 that for any element (ψ,φ)∈Ip, there exists y∈R3 and θ,ω∈(0,2π] such that5.14 α-1/4‖eiθψαy-ψ‖2=O(α-1/2p)=α1/4‖eiωφα-φ‖Lε2.

W.l.o.g. we assume y=0,θ=0 and, with (3.145) we consider furthermore ω=0. It follows from the expansion (3.134) of the energy Gα5.15 Gα(ψ,φ)-eα≥⟨Imδ1|Hosc|Imδ1⟩+‖ε1/2Imφ‖22+O(α-5/4p3).

Completing the square, we find with (5.11)5.16 Gα(ψ,φ)-eα≥⟨Imδ1-λpHφα-1∇ψα|Hφα|Imδ1-λpHφα-1∇ψα⟩+⟨Imφ-λpε-1kφα|ε|Imφ-λpε-1kφα⟩+2λp⟨∇ψα⟩Imδ1+⟨kφα⟩φ-λp2⟨∇ψα|Hosc-1|∇ψα⟩-λp2⟨kφα|ε-1|kφα⟩+O(α-5/4p3).

For the lower bound, we can neglect the first two lines and obtain5.17 G(ψ,φ)-eα≥2λp(1-μp)⟨∇ψα⟩Imδ1+⟨kφα⟩Imφ-λp2(1-μp)2⟨∇ψα|Hosc-1|∇ψα⟩-λp2⟨kφα|ε-1|kφα⟩+O(α-5/4p3).

For the first line, we use the constraint (ii)p, (5.14) together with Assumption 1.2 and the trivial bound ‖∇ψ0‖≤Cα.The second line, we compute explicitly and obtain5.18 G(ψ,φ)-eα≥m+2(2π)3α3‖kvε-3/2ϱ^ψα‖22-1p22+O(α-5/4p3).

Combining now the upper (5.18) and the lower bound (5.13), we arrive at Theorem 3.

Proof of (b): The corresponding Lagrange functional to the minimization problem is given by5.19 Lp(ψ,φ,λ,μ):=Gα(ψ,φ)-λ⟨ψ|i∇|ψ⟩+⟨φ|ik|φ⟩-p-μψψ.

Thus any minimizer satisfies the traveling wave equations with velocity λ, i.e.5.20 -iλ∂1ψp=hφp-μψp,λk1φp=εφp+(2π)3/2αvϱ^ψp.

By definition of the set Ip in (1.16), Proposition 2.3 shows that we can decompose5.21 ψp=ψα+λδ1,φp=φα+λδ2

with ‖δ1‖2≤Cα-1/4 and ‖ε1/2δ2‖2≤Cα1/4. The coercivity estimates from Corollary 3.2 together with the upper bound of part (a) then show5.22 λ‖δ1‖2≤Cα-3/4p,andλ‖ε1/2δ2‖2≤Cα-1/4p.

Together with the traveling waves equation it follows from constraint (ii)p for all p≤15.23 p=λmeff+O(α-1p2)=λmeff+O(α-1p2).

Furthermore from the first part of the theorem and Theorem 2, we conclude5.24 Gα(ψv′,φv′)=Ep+O(α-2p3).

where (ψv′,φv′) denotes a traveling wave with velocity v′=meff-1p+O(α-1p2). □

Acknowledgements

S.R. would like to thank Robert Seiringer for fruitful discussions, Krzysztof Myśliwy for helpful remarks and the reviewer for careful reading and useful comments. Funding from the European Union’s Horizon 2020 research and innovation program under the ERC grant agreement No. 694227 is gratefully acknowledged.

Funding

Open Access funding enabled and organized by Projekt DEAL.

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Declarations

Conflict of interest

The authors declare that they have no Conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Fröhlich H Theory of electrical breakdown in ionic crystals Proc. R. Soc. Lond. A 1937 160 901 230 241 10.1098/rspa.1937.0106
Fröhlich, H.: Theory of electrical breakdown in ionic crystals. Proc. R. Soc. Lond. A 160(901), 230–241 (1937)10.1098/rspa.1937.0106
2. Landau L Über die Bewegung der Elektronen im Kristallgitter Phys. Z. Sowjetunion 1933 3 664
Landau, L.: Über die Bewegung der Elektronen im Kristallgitter. Phys. Z. Sowjetunion 3, 664 (1933)
3. Pekar, S.: Zh. Eksp. Teor. Fiz. 16, 335 (1946); J. Phys. USSR. 10, 341 (1946)
4. Landau L Pekar S Effective mass of a Polaron J. Exp. Theor. Phys. 1948 18 419 423
Landau, L., Pekar, S.: Effective mass of a Polaron. J. Exp. Theor. Phys. 18, 419–423 (1948)
5. Frank R Zhou G Derivation of an effective evolution equation for a strongly coupled polaron Anal. PDE 2017 10 379 422 10.2140/apde.2017.10.379
Frank, R., Zhou, G.: Derivation of an effective evolution equation for a strongly coupled polaron. Anal. PDE 10, 379–422 (2017)10.2140/apde.2017.10.379
6. Feliciangeli D Seiringer R The strongly coupled Polaron on the torus: quantum corrections to the Pekar asymptotics Arch. Ration. Mech. Anal. 2021 242 1835 1906 10.1007/s00205-021-01715-7
Feliciangeli, D., Seiringer, R.: The strongly coupled Polaron on the torus: quantum corrections to the Pekar asymptotics. Arch. Ration. Mech. Anal. 242, 1835–1906 (2021)10.1007/s00205-021-01715-7
7. Griesemer M On the dynamics of Polarons in the strong-coupling limit Reviews in Mathematical Physics 2017 29 10 1750030 10.1142/S0129055X17500301
Griesemer, M.: On the dynamics of Polarons in the strong-coupling limit. Reviews in Mathematical Physics 29(10), 1750030 (2017)10.1142/S0129055X17500301
8. Leopold N Mitrouskas D Rademacher S Schlein B Seiringer R Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron Pure Appl. Anal. 2021 3 4 653 676 10.2140/paa.2021.3.653
Leopold, N., Mitrouskas, D., Rademacher, S., Schlein, B., Seiringer, R.: Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron. Pure Appl. Anal. 3(4), 653–676 (2021)10.2140/paa.2021.3.653
9. Leopold N Rademacher S Schlein B Seiringer R The Landau-Pekar equations: adiabatic theorem and accuracy Anal. & PDE 2021 14 2079 2100 10.2140/apde.2021.14.2079
Leopold, N., Rademacher, S., Schlein, B., Seiringer, R.: The Landau-Pekar equations: adiabatic theorem and accuracy. Anal. & PDE 14, 2079–2100 (2021)10.2140/apde.2021.14.2079
10. Mitrouskas D A note on the Fröhlich dynamics in the strong coupling limit Lett. Math. Phys. 2021 111 2 45 10.1007/s11005-021-01380-7
Mitrouskas, D.: A note on the Fröhlich dynamics in the strong coupling limit. Lett. Math. Phys. 111(2), 45 (2021)10.1007/s11005-021-01380-7
11. Feliciangeli D Rademacher S Seiringer R The effective mass problem for the Landau-Pekar equations J. Phys. A: Math. Theor. 2022 55 015201 10.1088/1751-8121/ac3947
Feliciangeli, D., Rademacher, S., Seiringer, R.: The effective mass problem for the Landau-Pekar equations. J. Phys. A: Math. Theor. 55, 015201 (2022)10.1088/1751-8121/ac3947
12. Brooks, M., Seiringer, R.: The Fröhlich Polaron at strong coupling–part II: energy-momentum relation and effective mass. Preprint: arXiv:2211.03353
13. Lieb E Seiringer R Divergence of the effective mass of a Polaron in the strong coupling limit J. Stat. Phys. 2020 180 23 33 10.1007/s10955-019-02322-3 32801392
Lieb, E., Seiringer, R.: Divergence of the effective mass of a Polaron in the strong coupling limit. J. Stat. Phys. 180, 23–33 (2020)32801392 10.1007/s10955-019-02322-3
14. Mitrouskas, D., Myśliwy, K., Seiringer, R.: Optimal parabolic upper bound for the energy-momentum relation of a strongly coupled polaron, Preprint: arXiv:2203.02454 (2022)
15. Bazaes, R., Mukherjee, C., Varadhan, S.R.S.: Effective mass of the Fröhlich Polaron and the Landau-Pekar-Spohn conjecture. Preprint: arXiv:2307.13058
16. Beetz V Polzer S Effective mass of the Polaron: a lower bound Commun. Math. Phys. 2023 399 173 188 10.1007/s00220-022-04553-0
Beetz, V., Polzer, S.: Effective mass of the Polaron: a lower bound. Commun. Math. Phys. 399, 173–188 (2023)10.1007/s00220-022-04553-0
17. Spohn H Effective mass of the Polaron: a functional integral approach Ann. Phys. 1987 175 278 318 10.1016/0003-4916(87)90211-9
Spohn, H.: Effective mass of the Polaron: a functional integral approach. Ann. Phys. 175, 278–318 (1987)10.1016/0003-4916(87)90211-9
18. Myśliwy K Seiringer R Polaron models with regular interactions at strong coupling J. Stat. Phys. 2022 186 1 5 10.1007/s10955-021-02851-w
Myśliwy, K., Seiringer, R.: Polaron models with regular interactions at strong coupling. J. Stat. Phys. 186(1), 5 (2022)10.1007/s10955-021-02851-w
19. Béthuel, F., Gravejat, P., Saut, J.-C.: Existence and properties of travelling waves for the Gross-Pitaevskii equation, Stationary and time dependent Gross-Pitaevskii equations. Contemp. Math. 473, Amer. Math. Soc. Providence, RI, pp. 55–103 (2008)
20. Fröhlich J Jonsson BLG Lenzmann E Boson stars as solitary waves Commun. Math. Phys. 2007 274 1 30 10.1007/s00220-007-0272-9
Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Boson stars as solitary waves. Commun. Math. Phys. 274, 1–30 (2007)10.1007/s00220-007-0272-9
21. Lieb EH Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation Stud. Appl. Math. 1977 57 93 105 10.1002/sapm197757293
Lieb, E.H.: Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57, 93–105 (1977)10.1002/sapm197757293
22. Lieb EH On the lowest eigenvalue of the Laplacian for the intersection of two domains Invent. Math. 1983 74 441 48 10.1007/BF01394245
Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74, 441–48 (1983)10.1007/BF01394245
23. Lieb EH Loss M Analysis 2001 Providence American Mathematical Society
Lieb, E.H., Loss, M.: Analysis. American Mathematical Society, Providence (2001)
24. Feliciangeli D Rademacher S Seiringer R Persistence of the spectral gap for the Landau-Pekar equations Lett. Math. Phys. 2021 111 1 19 10.1007/s11005-020-01350-5
Feliciangeli, D., Rademacher, S., Seiringer, R.: Persistence of the spectral gap for the Landau-Pekar equations. Lett. Math. Phys. 111, 1–19 (2021)10.1007/s11005-020-01350-5
25. Donsker M Varadhan S Asymptotics for the Polaron Comm. Pure Appl. Math. 1983 36 505 528 10.1002/cpa.3160360408
Donsker, M., Varadhan, S.: Asymptotics for the Polaron. Comm. Pure Appl. Math. 36, 505–528 (1983)10.1002/cpa.3160360408
26. Frank R Schlein B Dynamics of a strongly coupled polaron Lett. Math. Phys. 2014 104 911 929 10.1007/s11005-014-0700-7
Frank, R., Schlein, B.: Dynamics of a strongly coupled polaron. Lett. Math. Phys. 104, 911–929 (2014)10.1007/s11005-014-0700-7
27. Frank R Seiringer R Quantum corrections to the Pekar asymptotics of a strongly coupled polaron Commun. Pure Appl. Math. 2021 74 544 588 10.1002/cpa.21944
Frank, R., Seiringer, R.: Quantum corrections to the Pekar asymptotics of a strongly coupled polaron. Commun. Pure Appl. Math. 74, 544–588 (2021)10.1002/cpa.21944
28. Lieb E Thomas L Exact ground state energy of the strong-coupling Polaron Commun. Math. Phys. 1997 183 519 10.1007/s002200050040
Lieb, E., Thomas, L.: Exact ground state energy of the strong-coupling Polaron. Commun. Math. Phys. 183, 519 (1997)10.1007/s002200050040
29. Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima : asymptotic expansions. Annales de l’I. H. P., section A, tome. 38(3), 295–308 (1983)
