Persistence of the spectral gap for the Landau–Pekar equations

The Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.


Introduction and main results
The Landau-Pekar equations [5] provide an effective description of the dynamics for a strongly coupled polaron, modeling an electron moving in an ionic crystal. The strength of the interaction of the electron with its self-induced polarization field is described by a coupling parameter α > 0. In this system of coupled differential equations, the time evolution of the electron wave function ψ t ∈ H 1 (R 3 ) is governed by a Schrödinger equation with respect to an effective Hamiltonian h ϕ t depending on the polarization field ϕ t ∈ L 2 (R 3 ), which evolves according to a classical field equation. Motivated by the recent work in [7,8,10], we are interested in initial data for which the Hamiltonian h ϕ t possesses a uniform spectral gap (independent of t and α) above the infimum of its spectrum.
In the following we consider solutions (ψ t , ϕ t ) to the Landau-Pekar equations (1) with initial data (ψ 0 , ϕ 0 ) such that its energy G(ψ 0 , ϕ 0 ) is sufficiently close to e P , and show that for such initial data the Hamiltonian h ϕ t possesses a uniform spectral gap above the infimum of its spectrum for all times t ∈ R and any coupling constant α > 0. This is the content of the following Theorem.

Theorem 1 For any
Theorem 1 is proved in Sect. 3. It provides a class of initial data for the Landau-Pekar equations for which the Hamiltonian h ϕ t has a uniform spectral gap for all times t ∈ R. The existence of initial data with this particular property is of relevance for recent work [7,8,10] on the adiabatic theorem for the Landau-Pekar equations, and on their derivation from the Fröhlich model (where the polarization is described as a quantum field instead). For this particular initial data, the results obtained there can then be extended in the following way: Adiabatic theorem. Due to the separation of time scales in (1), the Landau-Pekar equations decouple adiabatically for large α (see [8] or also [2] for an analogous onedimensional model). To be more precise, in [8] the initial phonon state function is assumed to satisfy which implies that h ϕ 0 has a spectral gap and that there exists a unique positive and normalized ground state ψ ϕ 0 of h ϕ 0 . Under this assumption, denoting by (ψ t , ϕ t ) the solution of the Landau-Pekar equations (1) with initial data (ψ ϕ 0 , ϕ 0 ), [ where ψ ϕ t denotes the unique positive and normalized ground state of h ϕ t . The restriction on |t| in (9) is due to the need of ensuring that the spectral gap of the effective Hamiltonian h ϕ t does not become too small for initial data satisfying (8), which is only proven (in [8, Lemma II.1]) for times |t| ≤ T α 2 . Nevertheless, assuming that there exists Λ > 0 such that Λ(ϕ t ) > Λ for all times t ∈ R, the adiabatic theorem in [8, Thm. II.1] allows to approximate ψ t by e −i t 0 ds e(ϕ s ) ψ ϕ t for all times |t| α 4 . This raises the question about initial data for which the existence of a spectral gap of order one holds true for longer times, and Theorem 1 answers this question. In fact, by suitably adjusting the phase factor, we can prove the following stronger result.
Effective dynamics for the Fröhlich Hamiltonian. As already mentioned, the Landau-Pekar equations provide an effective description of the dynamics for a strongly coupled polaron. Its true dynamics is described by the Fröhlich Hamiltonian [4] H α acting on L 2 (R 3 ) ⊗ F, the tensor product of the Hilbert space L 2 (R 3 ) for the electron and the bosonic Fock space F for the phonons. We refer to [7,8] for a detailed definition. Pekar product states of the form  [7] that in order to obtain a norm approximation valid for times of order α 2 , one needs to implement correlations among phonons, which are captured by a suitable Bogoliubov dynamics acting on the Fock space of the phonons only. In fact, considering initial data satisfying (8), [7,Theorem I.3] proves that there exist constants C, T > 0 (depending on ϕ 0 ) such that where ω(s) = α 2 Im ϕ s , ∂ s ϕ s + ϕ s 2 2 and Υ t is the solution of the dynamics of a suitable Bogoliubov Hamiltonian on F (see [7,Definition I.2] for a precise definition). As for the adiabatic theorem discussed above, the restriction to times |t| ≤ T α 2 results from the need of a spectral gap of h ϕ t of order one (compare with [7, Remark I.4]), which under the sole assumption (8) is guaranteed by [8, Lemma II.1] only for |t| ≤ T α 2 . Theorem 1 now provides a class of initial data for which the above norm approximation holds true for all times of order α 2 , in the following sense.
for sufficiently small ε > 0. Then, h ϕ 0 has a ground state ψ ϕ 0 . Let (ψ t , ϕ t ) be the solution to the Landau-Pekar equations (1) with initial data (ψ ϕ 0 , ϕ 0 ). Then, there exists a C > 0 (independent of ϕ 0 and α) such that Again, the smallness condition on ε in Corollary 2 can be made explicit in terms of properties of ϕ P . Corollary 2 is an immediate consequence of Theorem 1 and the method of proof in [7], as explained in [7,Remark I.4].

Properties of the spectral gap and the Pekar functionals
Throughout the paper, we use the symbol C for generic constants, and their value might change from one occurrence to the next.

Preliminary Lemmas
We begin by stating some preliminary Lemmas we shall need throughout the following discussion.
for all α > 0 and all t ∈ R.
For (19), we note that σ ψ = σ e iθ ψ for arbitrary θ ∈ R. Hence, it is enough to prove the result for θ = 0. We write the difference For the first term, we write The Hardy-Littlewood-Sobolev inequality implies that and we obtain with the Sobolev inequality that The second term of (22) can be bounded in a similar way, and we obtain the desired estimate.
We recall the definition of the reduced resolvent where q ψ ϕ = 1 − |ψ ϕ ψ ϕ |. In the following Lemma we collect useful estimates on R ϕ .
Proof The first identity for the norm of the reduced resolvent follows immediately from the definition of the spectral gap Λ(ϕ) in (6). For ψ ∈ L 2 (R 3 ) we have It follows from Lemma 2 that there exists C > 0 such that Since e(ϕ) < 0 this implies the desired estimate.

Perturbative properties of ground states and of the spectral gap
Since the essential spectrum of h ϕ is R + , the assumption e(ϕ) < 0 guarantees the existence of a ground state (denoted by ψ ϕ ) and of a spectral gap Λ(ϕ) > 0 of h ϕ . In the next two Lemmas we investigate the behavior of Λ(ϕ) and ψ ϕ under L 2 -perturbations of ϕ.

Pekar functionals
Recall the definition of the Pekar Functionals G, E and F in (3) and (4), and note that As was shown in [9], E admits a unique strictly positive and radially symmetric minimizer, which is smooth and will be denoted by ψ P . Moreover, the set of all minimizers of E coincides with This clearly implies that the set of minimizers of F coincides with In the following we prove quadratic lower bounds for the Pekar Functionals E and F. The key ingredients are the results obtained in [6]. In particular, these results allow to infer, using standard arguments, the following Lemma 6, which provides the quadratic lower bounds for E. (We spell out its proof for completeness in the Appendix; a very similar proof in a slightly different setting is also given in [3]). Based on the bound for E, it is then quite straightforward to obtain the quadratic lower bound for F in the subsequent Lemma 7.

Lemma 6 (Quadratic Bounds for E) There exists a positive constant κ such that, for any L
Lemma 7 (Quadratic Bounds for F) There exists a positive constant τ such that, for any ϕ ∈ L 2 (R 3 ), Proof Recalling that our claim trivially follows by showing that for any For any such ψ let y * ∈ R 3 and θ * ∈ [0, 2π) be such that and denote e iθ * ψ P ( · − y * ) by ψ * P . By using the previous Lemma 6, the fact that ψ and ψ * P are L 2 -normalized, (19) and completing the square, we obtain for, some positive κ * > 0, This completes the proof of (51), and hence of the Lemma, with τ = κ * /(1 + κ * ).

Remark 1
The two previous quadratic bounds on E and F clearly imply, together with (4), that, for any L 2 -normalized ψ ∈ H 1 (R 3 ) and any ϕ ∈ L 2 (R 3 ), having low energy guarantees closeness to the surfaces of minimizers Θ(ψ P ) and Ω(ϕ P ), i.e.
Finally, we exploit the previous estimate to obtain the following Lemma. It states that for couples (ψ, ϕ) which have low energy ψ is close to ψ ϕ , the ground state of h ϕ , and ϕ is close to −σ ψ ϕ , in the following sense.

Proof of the main results
The conservation of G along solutions of the Landau-Pekar equations allows to apply the tools developed in Sect. 2 to get results valid for all times. This will in particular allow us to prove the results stated in Sect. 1. When combined with energy conservation, Remark 1 shows that we can estimate the distance to the sets of Pekar minimizers of solutions of the Landau-Pekar equations only in terms of the energy of their initial data. Since Ω(ϕ P ) contains only real-valued functions this yields bounds on the L 2norm of the imaginary part of ϕ t . That is, there exists a C > 0 such that if (ψ t , ϕ t ) solves the Landau-Pekar equations (1) with initial data (ψ 0 , ϕ 0 ), then for all t ∈ R and α > 0. It is then straightforward to obtain a proof of Theorem 1.
Conservation of energy also allows to extend the validity of Lemma 8 for all times. If (ψ t , ϕ t ) solves (1) with initial data (ψ 0 , ϕ 0 ) satisfying G(ψ 0 , ϕ 0 ) ≤ e P + ε for a sufficiently small ε, then ψ ϕ t is well defined for all times and Moreover, Theorem 1 implies that for all times Λ(ϕ t ) ≥ Λ for a suitable Λ > 0. It thus follows from Lemmas 1 and 3 that for some C > 0 With these preparations, we are now ready to prove Corollary 1.

Proof of Corollary 1
The proof follows closely the ideas of the proof of [8, Theorem II.1], hence we allow ourselves to be a bit sketchy at some points and refer to [8] for more details. It follows from the Landau-Pekar equations (1) that Lemmas 1-3 imply, together with (66), that there exists C > 0 such that In the same way, by the triangle inequality, Lemma 2 and (68), there exists C > 0 such that Moreover, it follows from (see [8,Lemma IV.2]) and by the same arguments as above that Recall the definitions of ψ t and ν in (11). The same computations as in [8,Eqs. (58)-(65)], using and integration by parts, lead to The difference to the calculations in [8] are the additional terms (77b) and the second term in (77e) resulting from the phase ν. While (77b) is, as we show below, only a subleading error term, the phase in (77e) leads to a crucial cancellation. This cancellation allows to integrate by parts once more, and finally results in the improved estimate in Corollary 1.
We shall now estimate the various terms in (77). Since q t ψ t 2 ≤ ψ t − ψ ϕ t 2 , we find for the first term using (69) and (71) for arbitrary δ > 0. Moreover, we have |ν(s)| ≤ Cα −4 ε for all s ∈ R, and find for the second term For the third term, we integrate by parts using (76) once more, with the result that The first two terms can be bounded in the same way as (77a) and (77b). For the third term, note that the r.h.s. of the inner product depends on time s through ϕ s only, hence its time derivative leads to another factor of α −2 . With (70), (73) and (74) we compute its time derivative. From the time derivative of the reduced resolvent in (74), we obtain one term for which the projection p s hits ψ s on the l.h.s. of the inner product, in which case we can only bound p s ψ s 2 ≤ 1. For the remaining terms, we use q s ψ s 2 ≤ ψ s − ψ ϕ s 2 instead. With the same arguments as above and (72), we obtain for α 1 and ε 1. A Gronwall type argument finally yields the desired bound (12).
Using again (91) we see that