Persistence of the spectral gap for the Landau–Pekar equations

Abstract

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.

1 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 $\alpha >0$. In this system of coupled differential equations, the time evolution of the electron wave function $\psi _t \in H^1( {\mathbb {R}}^3)$ is governed by a Schrödinger equation with respect to an effective Hamiltonian $h_{\varphi _t}$ depending on the polarization field $\varphi _t \in L^2( {\mathbb {R}}^3)$, 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 $h_{\varphi _t}$ possesses a uniform spectral gap (independent of t and $\alpha$) above the infimum of its spectrum.

The Landau–Pekar equations are of the form

$$\begin{aligned} \begin{aligned} i \partial _t \psi _t&= h_{ \varphi _t } \psi _t \\ i \alpha ^2 \partial _t \varphi _t&= \varphi _t + \sigma _{\psi _t} \end{aligned} \end{aligned}$$

(1)

with

$$\begin{aligned}&h_{\varphi } = - \varDelta + V_\varphi , \quad V_\varphi (x) = 2 (2\pi )^{3/2} \mathrm{Re}\, [ ( - \varDelta )^{-1/2} \varphi ] (x),\nonumber \\&\quad \sigma _{\psi } (x) = (2\pi )^{3/2}\left[ (- \varDelta )^{-1/2} \vert \psi \vert ^2\right] (x) . \end{aligned}$$

(2)

For initial data $( \psi _0, \varphi _0) \in H^1( {\mathbb {R}}^3 ) \times L^2( {\mathbb {R}}^3)$, (1) is well-posed for all times $t \in {\mathbb {R}}$ (see [1] or Lemma 1 below).

For $(\psi ,\varphi )\in H^1({\mathbb {R}}^3)\times L^2({\mathbb {R}}^3)$ with $\Vert \psi \Vert _2=1$, the energy functional corresponding to the Landau–Pekar equations is defined as

$$\begin{aligned} {\mathcal {G}} ( \psi , \varphi ) = \langle \psi , h_\varphi \psi \rangle + \Vert \varphi \Vert _2^2. \end{aligned}$$

(3)

One readily checks that for solutions of (1), ${\mathcal {G}}(\psi _t,\varphi _t)$ is independent of t [1, Lemma 2.1], and the same holds for $\Vert \psi _t\Vert _2$. We also define

$$\begin{aligned} {\mathcal {E}}(\psi )=\inf _{\varphi \in L^2({\mathbb {R}}^3)} {\mathcal {G}}(\psi ,\varphi ), \quad \quad {\mathcal {F}}(\varphi )=\inf _{\begin{array}{c} \psi \in H^1({\mathbb {R}}^3) \\ \Vert \psi \Vert _2=1 \end{array}} {\mathcal {G}}(\psi ,\varphi ). \end{aligned}$$

(4)

These three functionals are known as Pekar functionals and we shall discuss some of their properties in Sect. 2. It follows from the work in [9] that there exist $( \psi _{\mathrm {P}}, \varphi _{\mathrm {P}}) \in H^1( {\mathbb {R}}^3) \times L^2( {\mathbb {R}}^3)$ with $\Vert \psi _{\mathrm {P}}\Vert _2=1$, called Pekar minimizers, realizing

$$\begin{aligned} \inf _{\psi ,\varphi } {\mathcal {G}}(\psi ,\varphi ) = {\mathcal {G}}( \psi _{\mathrm {P}}, \varphi _{\mathrm {P}})={\mathcal {E}}(\psi _{\mathrm {P}})={\mathcal {F}}(\varphi _{\mathrm {P}}) = e_{\mathrm {P}} < 0 \,, \end{aligned}$$

(5)

and $( \psi _{\mathrm {P}}, \varphi _{\mathrm {P}})$ is unique up to symmetries (i.e., translations and multiplication of $\psi _{\mathrm {P}}$ by a constant phase factor). We also note that the Hamiltonian $h_{\varphi _{\mathrm {P}}}$ has a spectral gap above its ground state energy, i.e., $\varLambda (\varphi _{\mathrm {P}}) >0$, where we denote for general $\varphi \in L^2({\mathbb {R}}^3)$

$$\begin{aligned} \varLambda ( \varphi ) = \inf _{\begin{array}{c} \lambda \in \mathrm {spec}( h_{\varphi }) \\ \lambda \not = e(\varphi ) \end{array}} \vert \lambda - e( \varphi ) \vert \quad \text {with} \quad e(\varphi ) = \text {inf spec }h_\varphi \,. \end{aligned}$$

(6)

In the following we consider solutions $(\psi _t, \varphi _t)$ to the Landau–Pekar equations (1) with initial data $( \psi _0, \varphi _0)$ such that its energy ${\mathcal {G}}(\psi _0,\varphi _0)$ is sufficiently close to $e_{\mathrm {P}}$, and show that for such initial data the Hamiltonian $h_{\varphi _t}$ possesses a uniform spectral gap above the infimum of its spectrum for all times $t \in {\mathbb {R}}$ and any coupling constant $\alpha >0$. This is the content of the following Theorem.

Theorem 1

For any $0< \varLambda < \varLambda ( \varphi _{\mathrm {P}})$ there exists $\varepsilon _\varLambda >0$ such that if $( \psi _t, \varphi _t )$ is the solution of the Landau–Pekar equations (1) with initial data $(\psi _0, \varphi _0) \in H^1( {\mathbb {R}}^3) \times L^2( {\mathbb {R}}^3)$ with $\Vert \psi _0 \Vert _2=1$ and ${\mathcal {G}} (\psi _0, \varphi _0) \le e_{\mathrm {P}} + \varepsilon _\varLambda$, then

$$\begin{aligned} \varLambda (\varphi _t )\ge \varLambda \quad \mathrm {for \, all} \quad t\in {\mathbb {R}}, \, \alpha >0. \end{aligned}$$

(7)

Theorem 1 is proved in Sect. 3. It provides a class of initial data for the Landau–Pekar equations for which the Hamiltonian $h_{\varphi _t}$ has a uniform spectral gap for all times $t \in {\mathbb {R}}$. 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 $\alpha$ (see [8] or also [2] for an analogous one-dimensional model). To be more precise, in [8] the initial phonon state function is assumed to satisfy

$$\begin{aligned} \varphi _0 \in L^2( {\mathbb {R}}^3) \quad \mathrm {with} \quad e( \varphi _0) = \text {inf spec }h_{\varphi _0} <0, \end{aligned}$$

(8)

which implies that $h_{\varphi _0}$ has a spectral gap and that there exists a unique positive and normalized ground state $\psi _{\varphi _0}$ of $h_{\varphi _0}$. Under this assumption, denoting by $( \psi _t, \varphi _t)$ the solution of the Landau–Pekar equations (1) with initial data $(\psi _{\varphi _0}, \varphi _0)$, [8, Thm. II.1 & Rem. II.3] proves that there exist constants $C,T>0$ (depending on $\varphi _0$) such that

$$\begin{aligned} \Vert \psi _t - e^{-i \int _0^t ds \, e( \varphi _s ) } \psi _{\varphi _t} \Vert _2^2 \le C \alpha ^{-4} \quad \text {for all} \quad |t| \le T \alpha ^2 , \end{aligned}$$

(9)

where $\psi _{\varphi _t}$ denotes the unique positive and normalized ground state of $h_{\varphi _t}$. The restriction on |t| in (9) is due to the need of ensuring that the spectral gap of the effective Hamiltonian $h_{\varphi _t}$ does not become too small for initial data satisfying (8), which is only proven (in [8, Lemma II.1]) for times $|t| \le T \alpha ^2$. Nevertheless, assuming that there exists $\varLambda >0$ such that $\varLambda ( \varphi _t ) > \varLambda$ for all times $t \in {\mathbb {R}}$, the adiabatic theorem in [8, Thm. II.1] allows to approximate $\psi _t$ by $e^{-i \int _0^t ds \, e( \varphi _s ) } \psi _{\varphi _t}$ for all times $|t| \ll \alpha ^4$. 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.

Corollary 1

Let $\varphi _0\in L^2({\mathbb {R}}^3)$ be such that

$$\begin{aligned} {\mathcal {F}}(\varphi _0) \le e_{\mathrm {P}} + \varepsilon \end{aligned}$$

(10)

for sufficiently small $\varepsilon >0$. Then, $h_{\varphi _0}$ has a ground state $\psi _{\varphi _0}$. Let $( \psi _t, \varphi _t)$ be the solution to the Landau–Pekar equations (1) with initial data $( \psi _{\varphi _0}, \varphi _0)$ and define

$$\begin{aligned} \nu (s) = - \alpha ^{-4} \langle \psi _{\varphi _s}, \, V_{\mathrm{Im}\, \varphi _s} R_{\varphi _s}^3 V_{\mathrm{Im}\, \varphi _s} \psi _{\varphi _s} \rangle \quad \text{ and } \quad {\widetilde{\psi }}_t=e^{i\int _0^t ds(e(\varphi _s)+\nu (s))}\psi _t, \end{aligned}$$

(11)

where $R_{\varphi _s} = q_s ( h_{\varphi _s} - e( \varphi _s))^{-1} q_s$ with $q_s = 1 - \vert \psi _{\varphi _s} \rangle \langle \psi _{\varphi _s} \vert$. Then, there exists a $C>0$ (independent of $\varphi _0$ and $\alpha$) such that

$$\begin{aligned} \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2 \le C \varepsilon \alpha ^{-4} \left( 1 + \alpha ^{-2} |t|\right) e^{C \alpha ^{-4} |t|}. \end{aligned}$$

(12)

Our proof in Sect. 3 shows that the smallness condition on $\varepsilon$ in Corollary 1 can be made explicit in terms of properties of $\varphi _{\mathrm {P}}$. It also shows that $\min _{\theta \in [0,2\pi )} \Vert e^{i\theta } {\psi }_t - \psi _{\varphi _t} \Vert _2^2 \le C \varepsilon$ for all times t, independently of $\alpha$. The bound (12) improves upon this for large $\alpha$ as long as $\alpha ^{-4} |t| e^{C \alpha ^{-4} |t|} \ll \alpha ^2$ and hence, in particular, for $|t| \lesssim \alpha ^4$.

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] $H_\alpha$ acting on $L^2( {\mathbb {R}}^3) \otimes {\mathcal {F}}$, the tensor product of the Hilbert space $L^2( {\mathbb {R}}^3)$ for the electron and the bosonic Fock space ${\mathcal {F}}$ for the phonons. We refer to [7, 8] for a detailed definition. Pekar product states of the form $\psi _t \otimes W( \alpha ^2 \varphi _t ) \varOmega$, with $(\psi _t,\varphi _t)$ a solution of the Landau–Pekar equations, W the Weyl operator and $\varOmega$ the Fock space vacuum, were proven in [8, Thm. II.2] to approximate the dynamics defined by the Fröhlich Hamiltonian $H_\alpha$ for times $|t| \ll \alpha ^2$. Recently, it was shown in [7] that in order to obtain a norm approximation valid for times of order $\alpha ^2$, 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 $\varphi _0$) such that

$$\begin{aligned}&\Vert e^{-iH_\alpha t} \psi _{\varphi _0} \otimes W( \alpha ^2 \varphi _0) \varOmega -e^{-i \int _0^t ds \, \omega (s)} \psi _t \otimes W( \alpha ^2 \varphi _t) \varUpsilon _t \Vert _{L^2( {\mathbb {R}}^3) \otimes {\mathcal {F}}} \le C \alpha ^{-1} \nonumber \\&\quad \mathrm {for \, all } \quad |t| \le T \alpha ^2 , \end{aligned}$$

(13)

where $\omega (s) = \alpha ^2 \mathrm{Im}\langle \varphi _s , \partial _s \varphi _s\rangle + \Vert \varphi _s\Vert _2^2$ and $\varUpsilon _t$ is the solution of the dynamics of a suitable Bogoliubov Hamiltonian on ${\mathcal {F}}$ (see [7, Definition I.2] for a precise definition). As for the adiabatic theorem discussed above, the restriction to times $|t| \le T \alpha ^2$ results from the need of a spectral gap of $h_{\varphi _t}$ of order one (compare with [7, Remark I.4]), which under the sole assumption (8) is guaranteed by [8, Lemma II.1] only for $|t| \le T\alpha ^2$. Theorem 1 now provides a class of initial data for which the above norm approximation holds true for all times of order $\alpha ^2$, in the following sense.

Corollary 2

Let $\varphi _0\in L^2({\mathbb {R}}^3)$ be such that

$$\begin{aligned} {\mathcal {F}}(\varphi _0) \le e_{\mathrm {P}} + \varepsilon \end{aligned}$$

(14)

for sufficiently small $\varepsilon >0$. Then, $h_{\varphi _0}$ has a ground state $\psi _{\varphi _0}$. Let $( \psi _t, \varphi _t)$ be the solution to the Landau–Pekar equations (1) with initial data $( \psi _{\varphi _0}, \varphi _0)$. Then, there exists a $C>0$ (independent of $\varphi _0$ and $\alpha$) such that

$$\begin{aligned}&\Vert e^{-iH_\alpha t} \psi _{\varphi _0} \otimes W( \alpha ^2 \varphi _0) \varOmega -e^{-i \int _0^t ds \, \omega (s)} \psi _t \otimes W( \alpha ^2 \varphi _t) \varUpsilon _t \Vert _{L^2( {\mathbb {R}}^3) \otimes {\mathcal {F}}} \le C \alpha ^{-1} e^{ C \alpha ^{-2} |t| } \,.\nonumber \\ \end{aligned}$$

(15)

Again, the smallness condition on $\varepsilon$ in Corollary 2 can be made explicit in terms of properties of $\varphi _{\mathrm {P}}$. Corollary 2 is an immediate consequence of Theorem 1 and the method of proof in [7], as explained in [7, Remark I.4].

2 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.

2.1 Preliminary Lemmas

We begin by stating some preliminary Lemmas we shall need throughout the following discussion.

Lemma 1

(Lemma 2.1 in [1]) For any $( \psi _0, \varphi _0 ) \in H^1( {\mathbb {R}}^3) \times L^2( {\mathbb {R}}^3)$, there is a unique global solution $( \psi _t, \varphi _t)$ of the Landau–Pekar equations (1). Moreover, $\Vert \psi _0 \Vert _2 = \Vert \psi _t \Vert _2$, ${\mathcal {G}} ( \psi _0, \varphi _0) = {\mathcal {G}} ( \psi _t, \varphi _t )$ for all $t \in {\mathbb {R}}$ and there exists a constant $C>0$ such that

$$\begin{aligned} \Vert \psi _t \Vert _{H^1( {\mathbb {R}}^3)} \le C, \quad \Vert \varphi _t \Vert _2 \le C \end{aligned}$$

(16)

for all $\alpha >0$ and all $t \in {\mathbb {R}}$.

The following Lemma collects some properties of $V_\varphi$ and $\sigma _\psi$ (see also [8, Lemma III.2] and [7, Lemma II.2]).

Lemma 2

There exists $C>0$ such that for every $\varphi \in L^2( {\mathbb {R}}^3)$ and $\psi \in H^1({\mathbb {R}}^3)$

$$\begin{aligned} \Vert V_{\varphi } \Vert _6 \le C \Vert \varphi \Vert _2, \quad \Vert V_\varphi \psi \Vert _2 \le C \Vert \varphi \Vert _2 \Vert \psi \Vert _{H^1( {\mathbb {R}}^3)} \end{aligned}$$

(17)

and with the additional assumption $\Vert \psi \Vert _2 =1$

$$\begin{aligned} \Vert \psi \Vert _{H^1( {\mathbb {R}}^3)}^2 \le 2 \langle \psi , h_{\varphi } \psi \rangle + C ( \Vert \varphi \Vert _2^2 +1 ). \end{aligned}$$

(18)

Moreover, there exists $C>0$ such that for all $\psi _1, \psi _2 \in H^1({\mathbb {R}}^3)$

$$\begin{aligned} \Vert \sigma _{\psi _1} - \sigma _{\psi _2} \Vert _2 \le C \left( \Vert \psi _1 \Vert _{2} + \Vert \psi _2 \Vert _{2} \right) \min _{\theta \in [0,2\pi )}\Vert e^{i \theta } \psi _1 - \psi _2 \Vert _{H^1( {\mathbb {R}}^3)}. \end{aligned}$$

(19)

Proof

The first two inequalities in (17) follow immediately from [8, Lemma III.2] and [7, Lemma II.2]. In order to prove (18), let $\varepsilon >0$, then

$$\begin{aligned} \Vert \psi \Vert _{H^1( {\mathbb {R}}^3)}^2= & {} \langle \psi , \, h_{\varphi _0} \psi \rangle - \langle \psi , \, V_{\varphi _0} \psi \rangle + 1\nonumber \\\le & {} \langle \psi , \, h_{\varphi _0} \psi \rangle + \varepsilon \Vert \psi \Vert _{H^1( {\mathbb {R}}^3)}^2 + C \varepsilon ^{-1} \Vert \varphi _0 \Vert _2^2+1. \end{aligned}$$

(20)

Hence, choosing $\varepsilon = 1/2$, we arrive at (18).

For (19), we note that $\sigma _{\psi } = \sigma _{e^{i \theta } \psi }$ for arbitrary $\theta \in {\mathbb {R}}$. Hence, it is enough to prove the result for $\theta =0$. We write the difference

$$\begin{aligned} {\widehat{\sigma }}_{\psi _1} (k) - {\widehat{\sigma }}_{\psi _2} (k)&= |k|^{-1} \left( \langle \psi _1, e^{-ik \,\cdot \,} \psi _1 \rangle - \langle \psi _2, e^{-ik \, \cdot \,} \psi _2 \rangle \right) \nonumber \\&= |k|^{-1} \left( \langle \psi _1 - \psi _2, e^{-ik \, \cdot \,} \psi _1 \rangle + \langle \psi _2, e^{-ik \, \cdot \, }\left( \psi _1 - \psi _2\right) \rangle \right) , \end{aligned}$$

(21)

where ${\widehat{\sigma }} _\psi (k) = (2 \pi )^{-3/2} \int dx \; e^{-ik \cdot x} \sigma _\psi (x)$ denotes the Fourier transform of $\sigma _\psi$. Thus,

$$\begin{aligned} \Vert \sigma _{\psi _1} - \sigma _{\psi _2} \Vert _2^2 \le 2 \int dk \frac{1}{|k|^2} \left( \vert \langle \psi _1 - \psi _2, e^{-ik\, \cdot \, } \psi _1 \rangle \vert ^2 + \vert \langle \psi _2, e^{-ik \,\cdot \, }\left( \psi _1 - \psi _2\right) \rangle \vert ^2 \right) .\nonumber \\ \end{aligned}$$

(22)

For the first term, we write

$$\begin{aligned}&\int \frac{dk}{|k|^2} \vert \langle \psi _1 - \psi _2, e^{-ik \,\cdot \,} \psi _1 \rangle \vert ^2\nonumber \\&\qquad = C \int \frac{dx\,dy}{|x-y|} ( \psi _1 - \psi _2) (x) \overline{(\psi _1 - \psi _2 ) (y)} \, \overline{\psi _1(x)} \psi _1 (y) . \end{aligned}$$

(23)

The Hardy–Littlewood–Sobolev inequality implies that

$$\begin{aligned}&\int \frac{dk}{|k|^2} \vert \langle \psi _1 - \psi _2, e^{-ik \,\cdot \,} \psi _1 \rangle \vert ^2\nonumber \\&\qquad \le C \Vert \psi _1 \overline{( \psi _1 - \psi _2)} \Vert _{6/5}^2 \le C \Vert \psi _1 - \psi _2 \Vert _3^2 \Vert \psi _1 \Vert _2^2, \end{aligned}$$

(24)

and we obtain with the Sobolev inequality that

$$\begin{aligned} \int \frac{dk}{|k|^2} \vert \langle \psi _1 - \psi _2, e^{-ik\, \cdot \, } \psi _1 \rangle \vert ^2 \le C \Vert \psi _1 - \psi _2 \Vert _{H^1( {\mathbb {R}}^3)}^2 \Vert \psi _1 \Vert _2^2. \end{aligned}$$

(25)

The second term of (22) can be bounded in a similar way, and we obtain the desired estimate. $\square$

We recall the definition of the reduced resolvent

$$\begin{aligned} R_\varphi = q_{\psi _{\varphi }} \left( h_{\varphi } - e( \varphi ) \right) ^{-1} q_{\psi _{\varphi }}, \end{aligned}$$

(26)

where $q_{\psi _\varphi } = 1- \vert \psi _{\varphi } \rangle \langle \psi _\varphi \vert$. In the following Lemma we collect useful estimates on $R_\varphi$.

Lemma 3

There exists $C>0$ such that

$$\begin{aligned} \Vert R_{\varphi } \Vert = \varLambda ( \varphi )^{-1}, \quad \Vert \left( - \varDelta + 1 \right) ^{1/2} R_{\varphi }^{1/2} \Vert \le C ( 1 + \Vert \varphi \Vert _2 \Vert R_{\varphi }^{1/2} \Vert ) \end{aligned}$$

(27)

for any $\varphi \in L^2( {\mathbb {R}}^3)$ with $e( \varphi ) < 0$.

Proof

The first identity for the norm of the reduced resolvent follows immediately from the definition of the spectral gap $\varLambda ( \varphi )$ in (6). For $\psi \in L^2( {\mathbb {R}}^3)$ we have

$$\begin{aligned} \Vert \left( - \varDelta +1 \right) ^{1/2} R_\varphi ^{1/2} \psi \Vert _2^2 = \langle \psi , \, R_\varphi ^{1(2} \left( - \varDelta +1 \right) R_\varphi ^{1/2} \psi \rangle \,. \end{aligned}$$

(28)

It follows from Lemma 2 that there exists $C>0$ such that

$$\begin{aligned} \Vert \left( - \varDelta +1 \right) ^{1/2} R_\varphi ^{1/2} \psi \Vert _2^2&\le C \, \langle \psi , R_{\varphi }^{1/2} \left( h_{\varphi } + C \Vert \varphi \Vert _2^2 \right) R_\varphi ^{1/2} \psi \rangle \nonumber \\&= C\, \Vert q_{\psi _{\varphi }} \psi \Vert _2^2 + C \left( C \Vert \varphi \Vert _2^2 + e( \varphi ) \right) \Vert R_\varphi ^{1/2} \psi \Vert _2^2. \end{aligned}$$

(29)

Since $e( \varphi ) <0$ this implies the desired estimate. $\square$

2.2 Perturbative properties of ground states and of the spectral gap

Since the essential spectrum of $h_{\varphi }$ is ${\mathbb {R}}_+$, the assumption $e(\varphi )<0$ guarantees the existence of a ground state (denoted by $\psi _{\varphi }$) and of a spectral gap $\varLambda (\varphi )>0$ of $h_{\varphi }$. In the next two Lemmas we investigate the behavior of $\varLambda (\varphi )$ and $\psi _{\varphi }$ under $L^2$-perturbations of $\varphi$.

Lemma 4

Let $\varphi _0$ satisfy (8), and let $0< \varLambda < \varLambda ( \varphi _0 )$. Then, there exists $\delta _\varLambda >0$ (depending, besides $\varLambda$, only on the spectrum of $h_{\varphi _0}$ and $\Vert \varphi _0\Vert _2$) such that

$$\begin{aligned} \varLambda (\varphi ) \ge \varLambda \quad { for \, all} \quad \varphi \in L^2( {\mathbb {R}}^3) \quad {with} \quad \Vert \varphi - \varphi _0 \Vert _2 \le \delta _\varLambda . \end{aligned}$$

(30)

Proof

By definition of the spectral gap

$$\begin{aligned} \varLambda ( \varphi ) = e_1 ( \varphi ) - e( \varphi ), \end{aligned}$$

(31)

where $e( \varphi )$ denotes the ground state energy of $h_\varphi$, and $e_1( \varphi )$ its first excited eigenvalue if it exists, or otherwise $e_1( \varphi ) =0$ (which is the bottom of the essential spectrum). By the min-max principle we can write

$$\begin{aligned} e_1( \varphi ) = \inf _{\begin{array}{c} A \subset L^2( {\mathbb {R}}^3) \\ \mathrm {dim} A =2 \end{array}} \sup _{\begin{array}{c} \psi \in A \\ \Vert \psi \Vert _2 =1 \end{array}} \langle \psi , h_\varphi \psi \rangle . \end{aligned}$$

(32)

For $\psi \in H^1( {\mathbb {R}}^3)$ with $\Vert \psi \Vert _2 =1$ we find with Lemma 2

$$\begin{aligned} \langle \psi , h_\varphi \psi \rangle&= \langle \psi , h_{\varphi _0} \psi \rangle + \langle \psi , \, V_{\varphi - \varphi _0} \psi \rangle \nonumber \\&\le \langle \psi , \, h_{\varphi _0} \psi \rangle + C\Vert \varphi - \varphi _0 \Vert _2 \Vert \psi \Vert _{H^1( {\mathbb {R}}^3)}^2. \end{aligned}$$

(33)

Thus, by (18), we have if $\Vert \varphi -\varphi _0\Vert _2\le \delta$

$$\begin{aligned} \langle \psi , \, h_{\varphi } \psi \rangle \le ( 1 + C \delta ) \langle \psi , \, h_{\varphi _0} \psi \rangle + C \delta (\Vert \varphi _0 \Vert _2^2+1) , \end{aligned}$$

(34)

and similarly

$$\begin{aligned} \langle \psi , \, h_{\varphi } \psi \rangle \ge ( 1 - C\delta ) \langle \psi , \, h_{\varphi _0} \psi \rangle - C \delta ( \Vert \varphi _0 \Vert _2^2+1). \end{aligned}$$

(35)

Since $e( \varphi _0 ) , e( \varphi _1) \le 0$, we therefore find

$$\begin{aligned} \varLambda ( \varphi )\ge & {} \varLambda ( \varphi _0) -C\delta \left( e( \varphi _0) + e_1( \varphi _0) + 2 (\Vert \varphi _0 \Vert _2^2+1) \right) \nonumber \\\ge & {} \varLambda ( \varphi _0 ) - 2C \delta ( \Vert \varphi _0 \Vert _2^2+1) > \varLambda \end{aligned}$$

(36)

for sufficiently small $\delta = \delta _\varLambda >0$. $\square$

Lemma 5

Let $\varphi _0$ satisfy (8), and let $\varphi \in L^2( {\mathbb {R}}^3)$ with

$$\begin{aligned} \Vert \varphi - \varphi _0 \Vert \le \delta _{\varphi _0} \end{aligned}$$

(37)

for sufficiently small $\delta _{\varphi _0} >0$. Then, there exists a unique positive and normalized ground state $\psi _{\varphi }$ of $h_\varphi$. Moreover, there exists $C>0$ (independent of $\varphi$) such that

$$\begin{aligned} \Vert \psi _{\varphi _0} - \psi _{\varphi } \Vert _{H^1 ( {\mathbb {R}}^3)} \le C \Vert \varphi - \varphi _0 \Vert _2 . \end{aligned}$$

(38)

Proof

We write

$$\begin{aligned} \psi _{\varphi } - \psi _{\varphi _0} = \int _0^1 d\mu \, \partial _\mu \psi _{\varphi _\mu }, \end{aligned}$$

(39)

with $\varphi _\mu = \varphi _0 + \mu ( \varphi - \varphi _0 )$. Note that $\psi _{\varphi _\mu }$ is well defined for all $\mu \in [0,1]$, since

$$\begin{aligned} \Vert \varphi _\mu - \varphi _0 \Vert _2 = \mu \Vert \varphi - \varphi _0 \Vert _2 \le \mu \delta _{\varphi _0} \le \delta _{\varphi _0} \end{aligned}$$

(40)

and therefore Lemma 4 guarantees the existence of a spectral gap

$$\begin{aligned} \varLambda ( \varphi _\mu ) \ge \varLambda > 0 \end{aligned}$$

(41)

for sufficiently small $\delta _{\varphi _0}$, uniformly in $\mu \in [0,1]$. First-order perturbation theory yields

$$\begin{aligned} \partial _\mu \psi _{\varphi _\mu } = R_{\varphi _\mu } V_{\varphi _0 - \varphi } \psi _{\varphi _\mu } \end{aligned}$$

(42)

and it follows from Lemma 2 that

$$\begin{aligned} \Vert \psi _{\varphi _0} - \psi _{\varphi } \Vert _{H^1( {\mathbb {R}}^3)}&\le \int _0^1 d\mu \, \Vert R_{\varphi _\mu } V_{\varphi - \varphi _0} \psi _{\varphi _\mu } \Vert _{H^1( {\mathbb {R}}^3)} \nonumber \\&\le C \int _0^1 d\mu \, \Vert \left( - \varDelta +1 \right) ^{1/2}R_{\varphi _\mu }^{1/2}\Vert ^2 \, \Vert \varphi - \varphi _0 \Vert _2 . \end{aligned}$$

(43)

Lemma 3 shows that

$$\begin{aligned} \Vert \left( - \varDelta +1 \right) ^{1/2}R_{\varphi _\mu }\Vert \le C \left( 1 + \Vert \varphi _\mu \Vert _2 \Vert R_{\varphi _\mu } \Vert \right) . \end{aligned}$$

(44)

Since $\Vert \varphi _\mu \Vert _2 \le \Vert \varphi _0 \Vert _2 + \mu \Vert \varphi - \varphi _0 \Vert _2 \le \Vert \varphi _0 \Vert _2 + \delta _{\varphi _0}$, the bound (41) implies that the right-hand side of (44) is bounded independently of $\mu$. Hence, the desired estimate (38) follows. $\square$

2.3 Pekar functionals

Recall the definition of the Pekar Functionals ${\mathcal {G}}$, ${\mathcal {E}}$ and ${\mathcal {F}}$ in (3) and (4), and note that

$$\begin{aligned} {\mathcal {G}}(\psi ,\varphi )={\mathcal {E}}(\psi )+\Vert \varphi +\sigma _{\psi }\Vert _2^2 \,. \end{aligned}$$

(45)

As was shown in [9], ${\mathcal {E}}$ admits a unique strictly positive and radially symmetric minimizer, which is smooth and will be denoted by $\psi _{\mathrm {P}}$. Moreover, the set of all minimizers of ${\mathcal {E}}$ coincides with

$$\begin{aligned} \varTheta (\psi _{\mathrm {P}})=\{e^{i\theta }\psi _{\mathrm {P}}(\, \cdot \, -y) \, | \, \theta \in [0,2\pi ),\, y\in {\mathbb {R}}^3\}. \end{aligned}$$

(46)

This clearly implies that the set of minimizers of ${\mathcal {F}}$ coincides with

$$\begin{aligned} \varOmega (\varphi _{\mathrm {P}})=\{\varphi _{\mathrm {P}}(\, \cdot \, -y) \, | \, y\in {\mathbb {R}}^3\} \quad \text {with} \quad \varphi _{\mathrm {P}}= -\sigma _{\psi _{\mathrm {P}}}. \end{aligned}$$

(47)

In the following we prove quadratic lower bounds for the Pekar Functionals ${\mathcal {E}}$ and ${\mathcal {F}}$. 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 ${\mathcal {E}}$. (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 ${\mathcal {E}}$, it is then quite straightforward to obtain the quadratic lower bound for ${\mathcal {F}}$ in the subsequent Lemma 7.

Lemma 6

(Quadratic Bounds for ${\mathcal {E}}$) There exists a positive constant $\kappa$ such that, for any $L^2$-normalized $\psi \in H^1({\mathbb {R}}^3)$,

$$\begin{aligned} {\mathcal {E}}(\psi )-e_{\mathrm {P}}\ge \kappa \min _{\begin{array}{c} y \in {\mathbb {R}}^3\\ \theta \in [0,2\pi ) \end{array}} \Vert \psi -e^{i\theta } \psi _{\mathrm {P}}(\,\cdot \, -y)\Vert _{H^1({\mathbb {R}}^3)}^2=\kappa \mathrm{dist}_{H^1({\mathbb {R}}^3)}^2(\psi , \varTheta (\psi _{\mathrm {P}})).\nonumber \\ \end{aligned}$$

(48)

Lemma 7

(Quadratic Bounds for ${\mathcal {F}}$) There exists a positive constant $\tau$ such that, for any $\varphi \in L^2({\mathbb {R}}^3)$,

$$\begin{aligned} {\mathcal {F}}(\varphi )-e_{\mathrm {P}}\ge \tau \min _{y\in {\mathbb {R}}^3} \Vert \varphi -\varphi _{\mathrm {P}}(\,\cdot \, -y)\Vert _2^2=\tau \mathrm{dist}_{L^2({\mathbb {R}}^3)}^2(\varphi ,\varOmega (\varphi _{\mathrm {P}})). \end{aligned}$$

(49)

Proof

Recalling that

$$\begin{aligned} {\mathcal {F}}(\varphi )=\inf _{\begin{array}{c} \Vert \psi \Vert _2=1\\ \psi \in H^1({\mathbb {R}}^3) \end{array}} {\mathcal {G}}(\psi ,\varphi ) \end{aligned}$$

(50)

our claim trivially follows by showing that for any $L^2$-normalized $\psi \in H^1({\mathbb {R}}^3)$ and $\varphi \in L^2({\mathbb {R}}^3)$

$$\begin{aligned} {\mathcal {G}}(\psi ,\varphi )-e_{\mathrm {P}}\ge \tau \, \hbox {dist}_{L^2({\mathbb {R}}^3)}^2(\varphi ,\varOmega (\varphi _{\mathrm {P}})). \end{aligned}$$

(51)

For any such $\psi$ let $y^*\in {\mathbb {R}}^3$ and $\theta ^* \in [0,2\pi )$ be such that

$$\begin{aligned} \Vert \psi - e^{i\theta ^*}\psi _{\mathrm {P}}(\,\cdot \, -y^*)\Vert ^2_{H^1({\mathbb {R}}^3)}=\mathrm{dist}_{H^1({\mathbb {R}}^3)}^2(\psi ,\varTheta (\psi _{\mathrm {P}})), \end{aligned}$$

(52)

and denote $e^{i\theta ^*}\psi _{\mathrm {P}}(\,\cdot \, -y^*)$ by $\psi _{\mathrm {P}}^*$. By using the previous Lemma 6, the fact that $\psi$ and $\psi _{\mathrm {P}}^*$ are $L^2$-normalized, (19) and completing the square, we obtain for, some positive $\kappa ^*>0$,

$$\begin{aligned} {\mathcal {G}}(\psi ,\varphi )-e_{\mathrm {P}}&={\mathcal {E}}(\psi )-e_{\mathrm {P}}+\Vert \varphi +\sigma _{\psi }\Vert ^2_2\ge \kappa \Vert \psi -\psi _{\mathrm {P}}^*\Vert _{H^1({\mathbb {R}}^3)}^2+\Vert \varphi +\sigma _{\psi }\Vert _2^2\nonumber \\&\ge \kappa ^*\Vert \sigma _{\psi }-\sigma _{\psi _{\mathrm {P}}^*}\Vert _2^2+\Vert \varphi +\sigma _{\psi }\Vert _2^2\nonumber \\&=\Vert (1+\kappa ^*)^{1/2} (\sigma _{\psi _{\mathrm {P}}^*}-\sigma _{\psi }) -(1+\kappa ^*)^{-1/2}(\varphi +\sigma _{\psi _{\mathrm {P}}^*})\Vert _2^2\nonumber \\&\quad + \frac{\kappa ^*}{1+\kappa ^*} \Vert \varphi +\sigma _{\psi _{\mathrm {P}}^*} \Vert _2^2 \nonumber \\&\ge \frac{\kappa ^*}{1+\kappa ^*} \Vert \varphi -\varphi _{\mathrm {P}}(\,\cdot \, -y^*)\Vert _2^2\ge \frac{\kappa ^*}{1+\kappa ^*} \mathrm{dist}_{L^2({\mathbb {R}}^3)}^2(\varphi ,\varOmega (\varphi _{\mathrm {P}})). \end{aligned}$$

(53)

This completes the proof of (51), and hence of the Lemma, with $\tau = \kappa ^*/(1+\kappa ^*)$. $\square$

Remark 1

The two previous quadratic bounds on ${\mathcal {E}}$ and ${\mathcal {F}}$ clearly imply, together with (4), that, for any $L^2$-normalized $\psi \in H^1({\mathbb {R}}^3)$ and any $\varphi \in L^2({\mathbb {R}}^3)$, having low energy guarantees closeness to the surfaces of minimizers $\varTheta (\psi _{\mathrm {P}})$ and $\varOmega (\varphi _{\mathrm {P}})$, i.e.

$$\begin{aligned} {\mathcal {G}}(\psi ,\varphi )\le e_{\mathrm {P}}+\varepsilon\Rightarrow & {} {\mathcal {E}}(\psi ),{\mathcal {F}}(\varphi )\le e_{\mathrm {P}}+\varepsilon \nonumber \\\Rightarrow & {} \hbox {dist}_{H^1}^2(\psi ,\varTheta (\psi _{\mathrm {P}})), \hbox {dist}_{L^2}^2(\varphi ,\varOmega (\varphi _{\mathrm {P}}))\le C\varepsilon . \end{aligned}$$

(54)

Finally, we exploit the previous estimate to obtain the following Lemma. It states that for couples $(\psi ,\varphi )$ which have low energy $\psi$ is close to $\psi _{\varphi }$, the ground state of $h_{\varphi }$, and $\varphi$ is close to $-\sigma _{\psi _{\varphi }}$, in the following sense.

Lemma 8

Let $\varepsilon >0$ be sufficiently small, $\psi \in H^1({\mathbb {R}}^3)$ be $L^2$-normalized, $\varphi \in L^2({\mathbb {R}}^3)$ and let $(\psi ,\varphi )$ be such that

$$\begin{aligned} {\mathcal {G}}(\psi ,\varphi ) \le e_{\mathrm {P}}+\varepsilon \,. \end{aligned}$$

(55)

Then, $h_\varphi$ has a positive ground state $\psi _{\varphi }$, and there exists $C>0$ (independent of $(\psi ,\varphi ))$ such that

$$\begin{aligned} \min _{\theta \in [0,2\pi )} \Vert \psi -e^{i\theta }\psi _{\varphi }\Vert _{H^1({\mathbb {R}}^3)}^2&\le C\varepsilon , \end{aligned}$$

(56)

$$\begin{aligned} \Vert \varphi +\sigma _{\psi _{\varphi }}\Vert ^2_2&\le C \varepsilon . \end{aligned}$$

(57)

Proof

Since ${\mathcal {F}}(\varphi )\le {\mathcal {G}}(\psi ,\varphi )$ for any $L^2$-normalized $\psi \in H^1({\mathbb {R}}^3)$, Lemma 7 implies that for any $\delta >0$ there exists $\varepsilon _{\delta }>0$ such that $\hbox {dist}_{L^2}(\varphi ,\varOmega (\varphi _{\mathrm {P}}))\le \delta$ whenever ${\mathcal {G}}(\psi ,\varphi )\le e_{\mathrm {P}}+\varepsilon _{\delta }$. Moreover, by Lemma 4, there exists ${\bar{\delta }}>0$ such that if $\hbox {dist}_{L^2}(\varphi ,\varOmega (\varphi _{\mathrm {P}}))\le \bar{\delta }$ then $\psi _{\varphi }$ exists. We then pick $\varepsilon =\varepsilon _{{\bar{\delta }}}$ and this guarantees that under the hypothesis of the Lemma $\psi _{\varphi }$ is well defined.

Using Lemmas 6 and 7, the assumption (55) implies that there exist $y_1$ and $y_2$ such that

$$\begin{aligned} \min _{\theta \in [0.2\pi )}\Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\,\cdot \, -y_1)\Vert _{H^1({\mathbb {R}}^3)}^2\le C\varepsilon ,\quad \Vert \varphi -\varphi _{\mathrm {P}}(\, \cdot \, -y_2)\Vert _2^2\le C \varepsilon . \end{aligned}$$

(58)

Moreover, since

$$\begin{aligned} e_{\mathrm {P}}+\varepsilon \ge {\mathcal {G}}(\psi ,\varphi )={\mathcal {E}}(\psi )+\Vert \varphi +\sigma _{\psi }\Vert _2^2\ge e_{\mathrm {P}}+\Vert \varphi +\sigma _{\psi }\Vert _2^2, \end{aligned}$$

(59)

we also have

$$\begin{aligned} \Vert \varphi +\sigma _{\psi }\Vert _2^2\le \varepsilon . \end{aligned}$$

(60)

In combination, the second bound in (58) and (60) imply

$$\begin{aligned} \Vert \varphi _{\mathrm {P}}(\,\cdot \, - y_2)+\sigma _{\psi }\Vert _2^2\le C\varepsilon . \end{aligned}$$

(61)

Moreover, with the aid of (19) and the first bound in (58), we obtain

$$\begin{aligned}&\Vert \varphi _{\mathrm {P}}(\, \cdot \, -y_1)+\sigma _{\psi }\Vert _2^2=\Vert \sigma _{\psi _{\mathrm {P}}(\,\cdot \, -y_1)}-\sigma _{\psi }\Vert _2^2\nonumber \\&\quad \le C\min _{\theta \in [0,2\pi )}\Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\, \cdot \, -y_1)\Vert _{H^1}^2\le C \varepsilon . \end{aligned}$$

(62)

By putting the second equation in (58), (61) and (62) together, we can hence conclude that

$$\begin{aligned} \Vert \varphi -\varphi _{\mathrm {P}}(\, \cdot \, - y_1)\Vert _2\le & {} \Vert \varphi -\varphi _{\mathrm {P}}(\,\cdot \, -y_2)\Vert _2 +\Vert \varphi _{\mathrm {P}}(\, \cdot \, -y_2)+\sigma _{\psi }\Vert _2+\Vert \sigma _{\psi }+\varphi _{\mathrm {P}}(\, \cdot \, -y_1)\Vert _2\nonumber \\\le & {} C \varepsilon ^{1/2}. \end{aligned}$$

(63)

Therefore, using Lemma 5, we obtain

$$\begin{aligned} \Vert \psi -e^{i\theta }\psi _{\varphi }\Vert _{H^1}&\le \Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\,\cdot \, -y_1)\Vert _{H^1}+\Vert \psi _{\mathrm {P}}(\, \cdot \, -y_1)-\psi _{\varphi }\Vert _{H^1} \nonumber \\&= \Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\, \cdot \, -y_1)\Vert _{H^1}+\Vert \psi _{\varphi _{\mathrm {P}}(\, \cdot \, -y_1)}-\psi _{\varphi }\Vert _{H^1}\nonumber \\&\le \Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\, \cdot \, -y_1)\Vert _{H^1}+C \Vert \varphi _{\mathrm {P}}(\, \cdot \, -y_1)-\varphi \Vert _2 \,. \end{aligned}$$

(64)

This yields (56) after taking the infimum over $\theta \in [0,2\pi )$ and using (63) and the first bound in (58). To prove (57), we use (60), (19), the normalization of $\psi$ and $\psi _{\varphi }$ and (56) to obtain

$$\begin{aligned} \Vert \varphi +\sigma _{\psi _{\varphi }}\Vert _2\le & {} \Vert \varphi +\sigma _{\psi }\Vert _2+\Vert \sigma _{\psi }-\sigma _{\psi _{\varphi }}\Vert _2\nonumber \\\le & {} \varepsilon ^{1/2}+C\min _{\theta \in [0,2\pi )}\Vert \psi -e^{i\theta }\psi _{\varphi }\Vert _{H^1}\le C\varepsilon ^{1/2}. \end{aligned}$$

(65)

$\square$

3 Proof of the main results

The conservation of ${\mathcal {G}}$ 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 $\varOmega ( \varphi _{\mathrm {P}})$ contains only real-valued functions this yields bounds on the $L^2$-norm of the imaginary part of $\varphi _t$. That is, there exists a $C>0$ such that if $( \psi _t, \varphi _t)$ solves the Landau–Pekar equations (1) with initial data $(\psi _0,\varphi _0)$, then

$$\begin{aligned}&\min _{\begin{array}{c} y \in {\mathbb {R}}^3 \\ \theta \in [0, 2 \pi ) \end{array}}\Vert \psi _t - e^{i \theta } \psi _{\mathrm {P}}(\,\cdot \, -y) \Vert _{H^1( {\mathbb {R}}^3)}^2 \le C ({\mathcal {G}}(\psi _0,\varphi _0)-e_{\mathrm {P}}),\nonumber \\&\quad \Vert \mathrm{Im}\, \varphi _t \Vert _2^2 \le C ({\mathcal {G}}(\psi _0,\varphi _0)-e_{\mathrm {P}}), \nonumber \\&\min _{y \in {\mathbb {R}}^3}\Vert \mathrm{Re}\,\varphi _t - \varphi _{\mathrm {P}}(\,\cdot \,-y) \Vert _2^2 \le C ({\mathcal {G}}(\psi _0,\varphi _0)-e_{\mathrm {P}}) \end{aligned}$$

(66)

for all $t\in {\mathbb {R}}$ and $\alpha >0$. It is then straightforward to obtain a proof of Theorem 1.

Proof of Theorem 1

Let $0<\varLambda <\varLambda (\varphi _{\mathrm {P}})$ and let $( \psi _t, \varphi _t)$ denote the solution to the Landau–Pekar equations with initial data $( \psi _0, \varphi _0 )$ satisfying ${\mathcal {G}}(\psi _0,\varphi _0)\le e_{\mathrm {P}}+\varepsilon _{\varLambda }$. From (66) we deduce that for any $t\in {\mathbb {R}}$ there exists $y_t \in {\mathbb {R}}^3$ such that

$$\begin{aligned} \Vert \varphi _t-\varphi _{\mathrm {P}}(\,\cdot \, -y_t)\Vert _2^2\le C\varepsilon _{\varLambda } \end{aligned}$$

(67)

for some $C>0$. Since the spectrum of $h_{\varphi _{\mathrm {P}}(\,\cdot \, -y)}$ and $\Vert \varphi _{\mathrm {P}}(\,\cdot \, -y)\Vert _2$ are independent of $y\in {\mathbb {R}}^3$, Theorem 1 now follows immediately from Lemma 4 by taking $\varepsilon _{\varLambda }= C^{-1}\delta _{\varLambda }^2$, where $\delta _{\varLambda }$ is the same as in Lemma 4. $\square$

Conservation of energy also allows to extend the validity of Lemma 8 for all times. If $(\psi _t,\varphi _t)$ solves (1) with initial data $(\psi _0,\varphi _0)$ satisfying ${\mathcal {G}}(\psi _0,\varphi _0)\le e_{\mathrm {P}}+\varepsilon$ for a sufficiently small $\varepsilon$, then $\psi _{\varphi _t}$ is well defined for all times and

$$\begin{aligned} \min _{\theta \in [0,2\pi )}\Vert \psi _t-e^{i\theta }\psi _{\varphi _t}\Vert _{H^1({\mathbb {R}}^3)}^2\le C\varepsilon , \quad \Vert \varphi _t+\sigma _{\psi _{\varphi _t}}\Vert ^2_2\le C \varepsilon . \end{aligned}$$

(68)

Moreover, Theorem 1 implies that for all times $\varLambda (\varphi _t)\ge \varLambda$ for a suitable $\varLambda >0$. It thus follows from Lemmas 1 and 3 that for some $C>0$

$$\begin{aligned} \Vert R_{\varphi _t} \Vert \le C \quad \mathrm {and} \quad \Vert ( - \varDelta + 1)^{1/2} R_{\varphi _t}^{1/2} \Vert \le C \quad \text {for all} \quad t \in {\mathbb {R}}, \end{aligned}$$

(69)

whereas above $R_{\varphi _t} = q_t \left( h_{\varphi _t} - e( \varphi _t ) \right) ^{-1} q_t$ and $q_t = 1- p_t = 1- \vert \psi _{\varphi _t} \rangle \langle \psi _{\varphi _t} \vert$.

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

$$\begin{aligned} \alpha ^2 \partial _t V_{\varphi _t} = V_{\mathrm{Im}\, \varphi _t}, \quad \alpha ^2 \partial _t V_{\mathrm{Im}\, \varphi _t} = - V_{\mathrm{Re}\, \varphi _t + \sigma _{\psi _t}}. \end{aligned}$$

(70)

Lemmas 1–3 imply, together with (66), that there exists $C>0$ such that

$$\begin{aligned} \Vert R_{\varphi _t} V_{\mathrm{Im}\, \varphi _t} \Vert ^2 \le C \varepsilon \quad \mathrm {for \, all } \quad t \in {\mathbb {R}}. \end{aligned}$$

(71)

In the same way, by the triangle inequality, Lemma 2 and (68), there exists $C>0$ such that

$$\begin{aligned} \Vert R_{\varphi _t} V_{\mathrm{Re}\, \varphi _t + \sigma _{\psi _t}} \Vert ^2\le & {} C \min _{\theta \in (0, 2 \pi ]}\Vert \psi _t - e^{i \theta } \psi _{\varphi _t} \Vert _{H^1( {\mathbb {R}}^3)}^2+ C \Vert \mathrm{Re}\, \varphi _t + \sigma _{\psi _{\varphi _t}} \Vert _2^2 \nonumber \\\le & {} C \varepsilon \quad \mathrm {for \, all } \quad t \in {\mathbb {R}}. \end{aligned}$$

(72)

Moreover, it follows from

$$\begin{aligned} \alpha ^2 \partial _t \psi _{\varphi _t} = - R_{\varphi _t} V_{\mathrm{Im} \, \varphi _t} \psi _{\varphi _t} \end{aligned}$$

(73)

that

$$\begin{aligned} \alpha ^2 \partial _t R_{\varphi _t} = p_t V_{\mathrm{Im}\, \varphi _t} R_{\varphi _t}^2 + R_{\varphi _t}^2 V_{\mathrm{Im}\, \varphi _t} p_t - R_{\varphi _t} \left( V_{\mathrm{Im}\, \varphi _t} - \langle \psi _{\varphi _t}, V_{\mathrm{Im}\, \varphi _t} \psi _{\varphi _t} \rangle \right) R_{\varphi _t}\nonumber \\ \end{aligned}$$

(74)

(see [8, Lemma IV.2]) and by the same arguments as above that

$$\begin{aligned} \Vert \left( - \varDelta + 1\right) ^{1/2} \partial _t R_{\varphi _t} \left( - \varDelta + 1 \right) ^{1/2} \Vert \le C\varepsilon ^{1/2} \alpha ^{-2} \quad \text {for all} \quad t \in {\mathbb {R}}. \end{aligned}$$

(75)

Recall the definitions of ${\widetilde{\psi }}_t$ and $\nu$ in (11). The same computations as in [8, Eqs. (58)–(65)], using

$$\begin{aligned} q_t \, e^{i \int _0^t ds \, e( \varphi _s )} \psi _t = i\, R_{\varphi _t} \, \partial _t \,e^{i \int _0^t ds \, e( \varphi _s )} \psi _{t} \end{aligned}$$

(76)

and integration by parts, lead to

$$\begin{aligned} \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2&= 2 \alpha ^{-2} \mathrm{Im}\, \langle {\widetilde{\psi }}_t, \, R_{\varphi _t}^2 V_{\mathrm{Im}\, \varphi _t} \psi _{\varphi _t} \rangle \end{aligned}$$

(77a)

$$\begin{aligned}&\quad + 2 \alpha ^{-2} \int _0^t ds \, \nu (s) \, \mathrm{Re}\langle {\widetilde{\psi }}_s, \, R_{\varphi _s}^2 V_{\mathrm{Im}\, \varphi _s} \psi _{\varphi _s} \rangle \end{aligned}$$

(77b)

$$\begin{aligned}&\quad + 2 \alpha ^{-4} \int _0^t ds \, \mathrm{Im} \langle {\widetilde{\psi }}_s, \, R_{\varphi _s} \left( R_{\varphi _s} V_{\mathrm{Im}\, \varphi _s} \right) ^2 \psi _{\varphi _s} \rangle \end{aligned}$$

(77c)

$$\begin{aligned}&\quad + 2 \alpha ^{-4} \int _0^t ds \, \mathrm{Im} \langle {\widetilde{\psi }}_s, \, R_{\varphi _s}^2 V_{\mathrm{Re}\, \varphi _s + \sigma _{\psi _s}} \psi _{\varphi _s} \rangle \end{aligned}$$

(77d)

$$\begin{aligned}&\quad - 2 \alpha ^{-2} \int _0^t ds \,\left( \mathrm{Im} \langle {\widetilde{\psi }}_s, \, \left( \partial _s R_{\varphi _s}^2 \right) V_{\mathrm{Im}\, \varphi _s} \psi _{\varphi _s}\rangle + \alpha ^2 \nu (s) \, \mathrm{Im} \langle {\widetilde{\psi }}_s,\, \psi _{\varphi _s} \rangle \right) . \end{aligned}$$

(77e)

The difference to the calculations in [8] are the additional terms (77b) and the second term in (77e) resulting from the phase $\nu$. 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 $\Vert q_t {\widetilde{\psi }}_t \Vert _2 \le \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t}\Vert _2$, we find for the first term using (69) and (71)

$$\begin{aligned} \vert (\hbox {77a}) \vert \le C \alpha ^{-2} \varepsilon ^{1/2} \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2 \le \delta \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2 + C \delta ^{-1} \alpha ^{-4} \varepsilon \end{aligned}$$

(78)

for arbitrary $\delta >0$. Moreover, we have $\vert \nu (s) \vert \le C \alpha ^{-4} \varepsilon$ for all $s\in {\mathbb {R}}$, and find for the second term

$$\begin{aligned} \vert (\hbox {77b}) \vert \le C \alpha ^{-6} \varepsilon ^{3/2} \int _0^t ds\, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2 \,. \end{aligned}$$

(79)

For the third term, we integrate by parts using (76) once more, with the result that

$$\begin{aligned} (\hbox {77c})&= -2 \alpha ^{-4}\, \mathrm{Re}\, \langle {\widetilde{\psi }}_t, \, R_{\varphi _t}^2 \left( R_{\varphi _t} V_{\mathrm{Im}\, \varphi _t} \right) ^2 \psi _{\varphi _t} \rangle \nonumber \\&\quad + 2 \alpha ^{-4} \int _0^t ds \, \nu (s) \, \mathrm{Im}\, \langle {\widetilde{\psi }}_s, R_{\varphi _s}^2 \left( R_{\varphi _s} V_{\mathrm{Im}\, \varphi _s} \right) ^2 \psi _{\varphi _s } \rangle \nonumber \\&\quad + 2 \alpha ^{-4} \int _0^t ds \, \mathrm{Re}\, \langle {\widetilde{\psi }}_s, \, \partial _s \left( R_{\varphi _s}^2 \left( R_{\varphi _s} V_{\mathrm{Im}\, \varphi _s} \right) ^2 \psi _{\varphi _s} \right) \rangle . \end{aligned}$$

(80)

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 $\varphi _s$ only, hence its time derivative leads to another factor of $\alpha ^{-2}$. 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 ${\widetilde{\psi }}_s$ on the l.h.s. of the inner product, in which case we can only bound $\Vert p_s {\widetilde{\psi }}_s \Vert _2 \le 1$. For the remaining terms, we use $\Vert q_s {\widetilde{\psi }}_s \Vert _2 \le \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2$ instead. With the same arguments as above and (72), we obtain

$$\begin{aligned}&\vert (\hbox {77c}) \vert {\le } \delta \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2 {+} C \delta ^{-1} \alpha ^{-8} \varepsilon ^2\nonumber \\&\qquad \quad \quad \; + C \alpha ^{-6} \varepsilon \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2 {+} C \alpha ^{-6} \varepsilon ^{3/2} |t| \end{aligned}$$

(81)

for any $\delta >0$. For the fourth term (77d), we first split

$$\begin{aligned} (\hbox {77d}) = 2 \alpha ^{-4} \int _0^t ds \, \left( \text{ Im }\, \langle {\widetilde{\psi }}_s, \, R_{\varphi _s}^2 V_{\sigma _{\psi _s}- \sigma _{\psi _{\varphi _s}} } \psi _{\varphi _s} \rangle + \text{ Im }\, \langle {\widetilde{\psi }}_s, \, R_{\varphi _s}^2 V_{ \text {Re}\,\varphi _s + \sigma _{\psi _{\varphi _s}} } \psi _{\varphi _s} \rangle \right) .\nonumber \\ \end{aligned}$$

(82)

Lemmas 1–3 and (69) imply that we can bound $\Vert R_{\varphi _s}^2 V_{\sigma _{\psi _s}- \sigma _{\psi _{\varphi _s}} } \Vert \le C \Vert {{\widetilde{\psi }}}_s - \psi _{\varphi _s} \Vert _2$ in the first term. For the second term, we observe that the r.h.s. of the inner product depends on s again only through $\varphi _s$, whose time derivative is of order $\alpha ^{-2}$. We thus again use (76) and integration by parts, and proceed as above. For the calculation, we need to bound the time derivative of $\sigma _{\psi _{\varphi _s}}$, which can be done with the aid [7, Lemma II.4], with the result that $\Vert \partial _s \sigma _{\psi _{\varphi _s}} \Vert _2\le C \varepsilon ^{1/2} \alpha ^{-2}$. Altogether, this shows that

$$\begin{aligned} \vert (\hbox {77d}) \vert&\le C \alpha ^{-4} \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2^2 + \delta \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2 + C \delta ^{-1} \alpha ^{-8} \varepsilon \nonumber \\&\quad +C \alpha ^{-6} \varepsilon ^{1/2} \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2 + C \alpha ^{-6} \varepsilon |t| \end{aligned}$$

(83)

for any $\delta >0$. For the last term, we compute using (74)

$$\begin{aligned} (\hbox {77e})&= -6 \alpha ^{-4} \int _0^t ds \, \mathrm{Im}\langle {\widetilde{\psi }}_s, \, R_{\varphi _s}^3 V_{\mathrm{Im}\, \varphi _s} p_s V_{\mathrm{Im}\, \varphi _s} \psi _{\varphi _s} \rangle \nonumber \\&\quad + 2 \alpha ^{-4} \int _0^t ds \, \mathrm{Im}\langle {\widetilde{\psi }}_s, \, \left( R_{\varphi _s}^2 V_{\mathrm{Im}\, \varphi _s} R_{\varphi _s} + R_{\varphi _s} V_{\mathrm{Im}\, \varphi _s} R_{\varphi _s}^2 \right) V_{\mathrm{Im}\, \varphi _s} \psi _{\varphi _s} \rangle . \end{aligned}$$

(84)

Note that the phase $\nu (s)$ cancels the contribution of $\partial _s R_{\varphi _s}$ projecting onto $\psi _{\varphi _s}$ (the first term of (74)). This cancellation is important, since the integration by parts argument using (76) would not be applicable to this term. It can be applied to all the terms in (84), however, proceeding as above, with the result that

$$\begin{aligned} | (\hbox {77e})|\le & {} \delta \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2 + C \delta ^{-1} \alpha ^{-8} \varepsilon ^2\nonumber \\&\quad +C \alpha ^{-6}\varepsilon \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2 + C \alpha ^{-6} \varepsilon ^{3/2} |t| \end{aligned}$$

(85)

for any $\delta >0$.

Collecting the bounds in (78), (79), (81), (83) and (85), Eq. (77) shows that

$$\begin{aligned} \Vert {\widetilde{\psi }}_t - \psi _{\varphi _t} \Vert _2^2&\le C \alpha ^{-4} \varepsilon + C \alpha ^{-6} \varepsilon ^{1/2} \int _0^t ds\, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2\nonumber \\&\quad + C \alpha ^{-4} \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2^2+ C \alpha ^{-6} \varepsilon |t| \nonumber \\&\le C \alpha ^{-4} \varepsilon + C \alpha ^{-4} \int _0^t ds \, \Vert {\widetilde{\psi }}_s - \psi _{\varphi _s} \Vert _2^2 + C \alpha ^{-6} \varepsilon |t| \end{aligned}$$

(86)

for $\alpha > rsim 1$ and $\varepsilon \lesssim 1$. A Gronwall type argument finally yields the desired bound (12). $\square$

Appendix: Proof of Lemma 6

In this appendix we give the proof of Lemma 6. As already mentioned, the result follows from the work in [6] by standard arguments. We follow closely the proof given in [3] of a corresponding result in the slightly different setting of a confined polaron.

Proof of Lemma 6

Step 1: For any $L^2$-normalized $\psi \in H^1({\mathbb {R}}^3)$, there exists ${\bar{\theta }}\in [0,2\pi )$ and ${\bar{y}}\in {\mathbb {R}}^3$ such that

$$\begin{aligned} \Vert e^{i{\bar{\theta }}}\psi (\, \cdot \, -\bar{y})-\psi _{\mathrm {P}}\Vert _2=\min _{y, \theta } \Vert e^{i\theta }\psi (\, \cdot \, - y)-\psi _{\mathrm {P}}\Vert _2. \end{aligned}$$

(87)

By invariance of ${\mathcal {E}}$ under translations and changes of phase, it is then sufficient to show that for any $L^2$-normalized $\psi$ such that

$$\begin{aligned} \Vert \psi -\psi _{\mathrm {P}}\Vert _2=\min _{y, \theta } \Vert \psi -e^{i\theta }\psi _{\mathrm {P}}(\, \cdot \, -y)\Vert _2, \end{aligned}$$

(88)

the inequality

$$\begin{aligned} {\mathcal {E}}(\psi )-e_{\mathrm {P}}\ge \kappa \Vert \psi -\psi _{\mathrm {P}}\Vert _{H^1({\mathbb {R}}^3)}^2 \end{aligned}$$

(89)

holds (for some $\kappa >0$ independent of $\psi$). In fact, this is stronger than the desired bound (48). We henceforth only work with $L^2$-normalized $\psi$ satisfying (88), and denote $\delta =\psi -\psi _{\mathrm {P}}$. Observe that any $\psi$ satisfying (88) also satisfies

$$\begin{aligned} \langle \psi |\,\psi _{\mathrm {P}}\rangle \ge 0, \quad \langle \psi |\,\partial _i\psi _{\mathrm {P}}\rangle =0 \ \text {for} \ i=1,2,3. \end{aligned}$$

(90)

Step 2: We first prove the quadratic lower bound (89) locally around $\psi _{\mathrm {P}}$ for any $L^2$-normalized $\psi$ satisfying (88). By straightforward computations, using that

$$\begin{aligned} \Vert \delta \Vert _2^2=2-2 \langle \psi _{\mathrm {P}}|\,\psi \rangle =-2\langle \psi _{\mathrm {P}}|\,\delta \rangle \end{aligned}$$

(91)

since both $\psi _{\mathrm {P}}$ and $\psi$ are $L^2$-normalized, we obtain

$$\begin{aligned} {\mathcal {E}}(\psi )-e_{\mathrm {P}}=\text {Hess}_{\psi _{\mathrm {P}}}(\delta )+O(\Vert \delta \Vert ^3_{H^1({\mathbb {R}}^3)}), \end{aligned}$$

(92)

with

$$\begin{aligned} \text {Hess}_{\psi _{\mathrm {P}}}(\delta )&=\langle \mathrm{Im}\,\delta |{QL_{-}Q}|{\mathrm{Im}\, \delta }\rangle +\langle \mathrm{Re}\,\delta |{QL_{+}Q}|{\mathrm{Re}\, \delta }\rangle ,\nonumber \\ Q&=1-|\,\psi _{\mathrm {P}}\rangle \langle \psi _{\mathrm {P}}|,\nonumber \\ L_{-}&=h_{\varphi _{\mathrm {P}}} -e(\varphi _{\mathrm {P}}),\nonumber \\ L_{+}&=L_{-}-4X, \nonumber \\ X&= (2\pi )^3 \psi _{\mathrm {P}} (-\varDelta )^{-1} \psi _{\mathrm {P}} \,, \end{aligned}$$

(93)

where in the last formula for X, $\psi _{\mathrm {P}}$ has to be understood as a multiplication operator.

The Euler–Lagrange equation for the minimization of ${\mathcal {E}}$ reads $L_{-} \psi _{\mathrm {P}}=0$, and since $L_{-}$ is a Schrödinger operator and $\psi _{\mathrm {P}}$ is strictly positive, $L_{-}$ has 0 as its lowest eigenvalue, and a gap above. Therefore, we have

$$\begin{aligned} QL_{-}Q\ge \kappa _1 Q \end{aligned}$$

(94)

for some $\kappa _1>0$. Moreover, it was shown in [6] that the kernel of $L_{+}$ coincides with $\text {span}_{i=1,2,3}\{\partial _i \psi _{\mathrm {P}}\}$ and from this we can infer the existence of a $\kappa _2 > 0$ such that

$$\begin{aligned} QL_{+}Q\ge \kappa _{2} Q' \quad \text {with} \ Q'=Q-\sum _{i=1}^3 \Vert \partial _i \psi _{\mathrm {P}}\Vert _2^{-2} |\,\partial _i \psi _{\mathrm {P}}\rangle \langle \partial _i \psi _{\mathrm {P}}|. \end{aligned}$$

(95)

Recall that $Q' \delta = Q \delta$ by assumption on $\psi$ and orthogonality of $\psi _{\mathrm {P}}$ to its partial derivatives. With $\kappa ' =\min \{\kappa _1,\kappa _2\}$ we thus have

$$\begin{aligned} \text {Hess}_{\psi _{\mathrm {P}}}(\delta )\ge \kappa _1\Vert Q\mathrm{Im}\, \delta \Vert _2^2+\kappa _2\Vert Q' \mathrm{Re}\, \delta \Vert _2^2\ge \kappa ' \Vert Q\delta \Vert ^2_2. \end{aligned}$$

(96)

Using again (91) we see that

$$\begin{aligned} \Vert Q\delta \Vert _2^2=\Vert \delta \Vert _2^2-\langle \psi _{\mathrm {P}}|\,\delta \rangle ^2=\Vert \delta \Vert _2^2\left( 1-\frac{1}{4} \Vert \delta \Vert _2^2\right) \ge \frac{1}{2} \Vert \delta \Vert _2^2, \end{aligned}$$

(97)

which finally implies that

$$\begin{aligned} \text {Hess}_{\psi _{\mathrm {P}}}(\delta )\ge \frac{\kappa '}{2} \Vert \delta \Vert _2^2. \end{aligned}$$

(98)

We now want to improve this bound to include the full $H^1$-norm of $\delta$. Using the regularity of $\psi _{\mathrm {P}}$ it is rather straightforward to show that

$$\begin{aligned} L_{-} =QL_{-}Q&\ge -\varDelta -C \,, \nonumber \\ QL_{+}Q&\ge -\varDelta -C \end{aligned}$$

(99)

which implies that

$$\begin{aligned} \text {Hess}_{\psi _{\mathrm {P}}}(\delta )\ge \Vert \delta \Vert _{H^1}^2-C\Vert \delta \Vert _2^2. \end{aligned}$$

(100)

By interpolating between (98) and (100), we finally obtain

$$\begin{aligned} \text {Hess}_{\psi _{\mathrm {P}}}(\delta )\ge \frac{\kappa '}{\kappa '+2C}\Vert \delta \Vert _{H^1}^2=\kappa ''\Vert \delta \Vert _{H^1}^2. \end{aligned}$$

(101)

In combination with (92), we conclude that

$$\begin{aligned} {\mathcal {E}}(\psi )-e_{\mathrm {P}}\ge \kappa ''\Vert \delta \Vert _{H^1}^2-C\Vert \delta \Vert _{H^1}^3 \end{aligned}$$

(102)

for any $L^2$-normalized $\psi$ satisfying (88), which shows that (89) holds for $\Vert \delta \Vert _{H^1}$ sufficiently small.

Step 3: We now extend the previous local bound to show that (89) holds globally. Suppose by contradiction that there does not exist a universal $\kappa$ such that (89) holds. Then, there exists a sequence $\psi _n$ of $L^2$-normalized functions satisfying (88) such that

$$\begin{aligned} {\mathcal {E}}(\psi _n)\le e_{\mathrm {P}}+\frac{1}{n} \Vert \psi _n-\psi _{\mathrm {P}}\Vert _{H^1}^2\le \frac{2}{n} \Vert \psi _n\Vert _{H^1}^2 + C\,. \end{aligned}$$

(103)

One readily checks that

$$\begin{aligned} {\mathcal {E}}(\psi _n)\ge \frac{1}{2} \Vert \psi _n\Vert _{H^1}^2-C \,, \end{aligned}$$

(104)

hence $\psi _n$ must be bounded in $H^1({\mathbb {R}}^3)$. Again using (103), we conclude that $\psi _n$ must be a minimizing sequence for ${\mathcal {E}}$. It was proven in [9] that any minimizing sequence converges in $H^1({\mathbb {R}}^3)$ to a minimizer of ${\mathcal {E}}$, i.e., an element of $\varTheta (\psi _{\mathrm {P}})$ in (46), and since $\psi _n$ satisfies (88) this implies that $\psi _n\xrightarrow {H^1} \psi _{\mathrm {P}}$. This yields a contradiction, since we already know by (102) that locally the bound (89) holds. $\square$