Stark-Wannier ladders and cubic exponential sums

Stark-Wannier Ladders and Cubic Exponential Sums

Alexander Fedotov and Frédéric Klopp St. Petersburg State University, 7/9 Universitetskaya nab., St.Petersburg, 199034, Russia a.fedotov@spbu.ru Sorbonne Universités, UPMC Univ. Paris 06, UMR 7586, IMJ-PRG, F-75005, Paris, France Univ. Paris Diderot, Sorbonne Paris Cité, UMR 7586, IMJ-PRG, F-75205 Paris, France CNRS, UMR 7586, IMJ-PRG, F-75005, Paris, France frederic.klopp@imj-prg.fr

The present work was supported by the Russian foundation of basic research under grant 14-01-00760-a. A.F. acknowledges support by the Fondation Sciences Mathématiques de Paris. F.K. acknowledges support by the Chebyshev Laboratory and the French Embassy in Russia through the Chaire Lamé. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme “Periodic and Ergodic Spectral Problems” supported by EPSRC Grant Number EP/K032208/1. F.K. also acknowledges support from the Simons Foundation during his stay at the INI.

On $L^{2} (R)$ , we consider the Schrödinger operator

(1.1)

H_{ϵ} = - \frac{\partial^{2}}{\partial x^{2}} + v (x) - ϵ x,

where $v$ is a real analytic $1$ -periodic function and $ϵ$ is a positive constant. This operator is a model to study a Bloch electron in a constant electric field ([1]). The parameter $ϵ$ is proportional to the electric field. The operator (1.1) was studied both by physicists (see, e.g., the review [6]) and by mathematicians (see, e.g., [9]). Its spectrum is absolutely continuous and fills the real axis. One of main features of $H_{ϵ}$ is the existence of Stark-Wannier ladders. These are $ϵ$ -periodic sequences of resonances, which are poles of the analytic continuation of the resolvent kernel in the lower half plane through the spectrum (see, e.g., [2]). Most of the mathematical work studied the case of small $ϵ$ (see, e.g., [9, 3] and references therein). When $ϵ$ is small, there are ladders exponentially close to the real axis. Actually, only the case of finite gap potentials $v$ was relatively well understood. For these potentials, there is only a finite number of ladders exponentially close to the real axis. It was further noticed that the ladders non-trivially “interact” as $ϵ$ changes, and conjectured that the behavior of the resonances strongly depends on number theoretical properties of $ϵ$ (see, e.g., [1]).

In the present note, we only consider the periodic potential $v (x) = 2 cos (2 π x)$ and study the reflection coefficient $r (E)$ of the Stark-Wannier operator (1.1) in the lower half of the complex plane of the spectral parameter $E$ . The resonances are the poles of the reflection coefficient. We show that, as $I m E \to - \infty$ , the function $E \mapsto \frac{1}{r (E)}$ can be asymptotically described in terms of a regularized cubic exponential sum that is a close relative of the cubic exponential sums often encountered in analytic number theory. This explains the dependence of the reflection coefficient on the arithmetic nature of $ϵ$ . For $\frac{π^{2}}{3 ϵ} \in Q$ , we describe the asymptotics of the Stark-Wannier ladders situated far from the real axis.

Let us recall the definition of the reflection coefficient for (1.1) following [2]. Consider the equation

(1.2)

- ψ^{''} (x) + (v (x) - ϵ x) ψ (x) = E ψ (x), x \in C,

For the sake of simplicity, assume that the potential $v$ is entire. Assume also $\int_{0}^{1} v (x) d x = 0$ . For any $E \in C$ , there are unique solutions $ψ_{\pm}$ to (1.2) that admit the asymptotic representations

(1.3)		$ψ_{-} (x, E)$	$= \frac{1}{\sqrt[4]{- ϵ x - E}} e^{- \int_{x}^{- E / ϵ} \sqrt{- ϵ t - E} d t + o (1)}, x \to - \infty,$
(1.3)		$ψ_{+} (x, E)$	$= \frac{1}{\sqrt[4]{ϵ x + E}} e^{i \int_{- E / ϵ}^{x} \sqrt{ϵ t + E} d t + o (1)}, x \to + \infty,$

where the determinations of $\sqrt{\cdot}$ and $\sqrt[4]{\cdot}$ are analytic in $C ∖ R_{-}$ and positive along $R_{+}$ . Consider also the solution $ψ_{+}^{*} (x, E) = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ψ_{+} (¯ x, ¯ E)$ . The solutions $ψ_{+}$ and $ψ_{+}^{*}$ being linearly independent, one has

(1.4)

ψ_{-} (x, E) = w (E) ψ_{+}^{*} (x, E) + w^{*} (E) ψ_{+} (x, E), x \in R,

where the coefficient $w (E)$ is independent of $x$ and the function $E \mapsto w (E)$ is entire. The ratio $r (E) = w^{*} (E) / w (E)$ is the reflection coefficient. It is an $ϵ$ -periodic meromorphic function of $E$ . The reflection coefficient is analytic in $C_{+}$ , and, for $E \in R$ , one has $| r (E) | = 1$ . The poles of $r$ are the resonances of $H_{ϵ}$ .

Let us now state the first of our results. Represent $1 / r$ by its Fourier series $1 / r (E) = \sum m \in Z e^{2 π n i E / ϵ} p (m)$ for $Im E \leq 0$ . Let $a (ϵ) = \sqrt{\frac{2}{ϵ}} π e^{i π / 4}$ . One has

Theorem 1.

Let $v (x) = 2 cos (2 π x)$ . Then, as $m \to \infty$ ,

(1.5)

p (m) = a (ϵ) \sqrt{m} e^{- 2 π i ω m^{3} - 2 m log (2 π m / e) + δ (m)}, ω = {\frac{π^{2}}{3 ϵ}},

where, for $x$ real, ${x}$ denotes the fractional part of $x$ , and $δ (m) = O ({log}^{2} m / m)$ . This estimate is locally uniform in $ϵ > 0$ .

Clearly, the asymptotic behavior of $1 / r (E)$ as $Im E \to - \infty$ is determined by the Fourier series terms with large positive $m$ , and so, roughly,

(1.6)

\frac{1}{r (E)} \approx a (ϵ) P (E / ϵ), P (s) = \sum m \geq 1 \sqrt{m} e^{- 2 π i ω m^{3} - 2 m log (2 π m / e) + 2 π i m s} .

It is worth to compare the function $P$ with the cubic exponential sums $N \sum n = 1 e^{- 2 π i ω n^{3}}$ . Such sums were extensively studied in analytic number theory, see, e.g., [4]. They were proved to depend strongly on the arithmetic nature of $ω$ . This appears to be true in our case too. We have

Theorem 2.

Let $v (x) = 2 cos (2 π x)$ . Assume that $ω \in Q$ and represent it in the form $ω = \frac{p}{q}$ , where $0 \leq p < q$ are co-prime integers. If $p = 0$ , we take $q = 1$ . For $ξ \in R$ , we set $I_{q} (ξ) := {m \in Z : | ξ - \frac{m}{q} | \leq 1 / 2}$ . As $Im E \to - \infty$ , one has

(1.7)

r^{- 1} (E) = \frac{b (ϵ) ρ}{q} \sum m \in I_{q} (ξ) S_{q} (p, m) e^{ρ e^{i π (ξ - m / q)} + i π (ξ - m / q) + O ({log}^{2} ρ / ρ)} + e^{O (\frac{ρ}{ln ρ})},

where $b (ϵ) = π^{\frac{3}{2}} e^{i π / 4} / \sqrt{2 ϵ}$ , $ξ = R e E / ϵ$ , $ρ = e^{- π I m E / ϵ}$ , and

S_{q} (p, m) = q - 1 \sum l = 0 e^{- 2 π i \frac{p l^{3} - m l}{q}} .

The error estimates are locally uniform in $ϵ > 0$ .

Let us discuss this result. First, assume that $ω = 0$ . By Theorem 2,

(1.8)

(b (ϵ) r (E))^{- 1} = \sqrt{z} e^{\sqrt{z} + O (\frac{{ln}^{2} z}{\sqrt{z}})} + e^{O (\frac{\sqrt{z}}{ln z})}, z = e^{2 i π E / ϵ},

where the determination of $\sqrt{\cdot}$ is analytic in $C ∖ R_{-}$ and positive along $R_{+}$ . Recall that $1 / r$ is $ϵ$ -periodic. Let $B_{ϵ} = {E \in C : Im E \leq 0, 0 \leq Re E \leq ϵ}$ . Representation (1.8) implies

Corollary 1.

Assume $ω = 0$ . The resonances located in $B_{ϵ}$ have the following properties :

for sufficiently large $y > 0$ , the resonances with $Im E < - ϵ y$ are located in the domain $| Re E - ϵ / 2 | \leq C ϵ^{2} / | Im E |$ , where $C > 0$ is a constant;
let $n (y)$ be the number of resonances in the rectangle $[0, ε] - i [0, ε y]$ ; then, one has

$n (y) = \frac{1}{π} e^{π y + o (1)} as y \to \infty .$

The first statement immediately follows from Theorem 2; to prove the second one has to use Jensen formula and Levin lower bounds for the absolute values of entire functions, see, e.g., [8].
When $ω = 0$ , it is difficult to obtain the asymptotics of the resonances as, in a neighborhood of the line $Re E / ϵ = 1 / 2 mod 1$ , they are determined by the first Fourier coefficients of $1 / r$ , i.e., by $p (m)$ with $m = 1, 2, 3 \dots$ . Hence, the problem is not asymptotic in nature.

If $ω \neq 0$ , then the description of the resonances is determined by the values of $S_{q} (p, m)$ for $m = 1, 2, \dots q - 1$ (the map $m \to S_{q} (p, m)$ is $q$ -periodic). The $S_{q} (p, m)$ are cubic complete rational exponential sums, see, e.g., [7]. One easily checks

Lemma 1.

For any $q \in N$ , $q - 1 \sum m = 0 | S_{q} (p, m) |^{2} = q^{2}$ .

This implies that, for $q \geq 1$ , there is at least one integer $0 \leq m_{0} < q - 1$ such that $S_{q} (p, m_{0}) \neq 0$ .

If $S_{q} (p, m)$ is non zero for only one $0 \leq m_{0} < q$ (this happens, for example, for $q = 2, 3, 6$ ), then one can characterize the resonances as when $ω = 0$ . Now, they live near the lines ${Re E / ϵ = m_{0} / q + 1 / 2 + n}$ , $n \in Z$ .

For large $q$ , there are actually many non-zero values $S_{q} (p, m)$ :

Lemma 2.

There exists a constant $C > 0$ such that, for any co-prime $q > p > 0$ , one has $# {0 \leq m < q : S_{q} (p, m) \neq 0} \geq C q^{\frac{2}{3}}$ .

This statement follows from Lemma 1 and the well-known upper bound for general complete rational exponential sums of Hua ([7]).

In general, the behavior of $m \mapsto S_{q} (p, m)$ is nontrivial; it is known to depend strongly on the prime factorization of $q$ . Computer calculations lead to the following conjecture: if $q$ is prime, $0 < p < q$ , and $0 < m < q$ , then $S_{q} (p, m) \neq 0$ .

If $S_{q} (p, m)$ is non zero for at least two values of $m$ such that $0 \leq m < q$ , then, using (1.5), one can describe asymptotically all the resonances with sufficiently negative imaginary part. One has

Corollary 2.

Assume that, for some integers $m_{1} < m_{2}$ such that $m_{2} - m_{1} < q$ , one has $S_{q} (p, m_{1}) \neq 0$ , $S_{q} (p, m_{2}) \neq 0$ , and $S_{q} (p, m) = 0$ for all $m_{1} < m < m_{2}$ . Then, for sufficiently large $y > 0$ , in the vertical half-strip

{E \in C : - Im E \geq ϵ y, \frac{m_{1}}{q} \leq \frac{Re E}{ϵ} \leq \frac{m_{2}}{q}},

there are resonances, and they are described by the asymptotic formulas:

(1.9)

\frac{E}{ϵ} = - i (\frac{ln (π k)}{π} - ln sin \frac{π (m_{2} - m_{1})}{q}) + \frac{m_{2} + m_{1}}{q} + o (1), k \in N,

where $o(1)\operatornamewithlimits→k→+∞0$ .

This statement easily follows from Theorem 2.

Finally, let us describe very briefly the ideas leading to Theorems 1 and 2. Buslaev’s solutions $ψ_{\pm}$ used to define the reflection coefficient (see (1.3)) are entire functions of $x$ and $E$ ; they satisfy the relations $ψ_{\pm} (x + 1, E) = ψ_{\pm} (x, E + ϵ)$ . It appears that the analytic properties of such solutions can naturally be described in terms of a system of two first order difference equations on the complex plane (see, for example, [5]). To get the asymptotics of the Fourier coefficients of the reflection coefficient, we study the solutions of this system far from the origin. The idea leading from Theorem 1 to Theorem 2 is analogous to one used to study the behavior of the exponential sums $N \sum n = 1 e^{- 2 π i ω n^{3}}$ with $ω \in Q$ for large $N$ , see [4]. However, to use it successfully, one has to carry out a non trivial analysis of properties of the error term in (1.5).

References

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Buslaev V.S., Dmitrieva L.A. A Bloch electron in an external field. Leningrad Mathematical Journal, 1(2):287–320 (1990).
Buslaev V. and Grigis A. Imaginary parts of Stark-Wannier resonances. Journal of mathematical physics, 39:2520-2550 (1998).
Davenport H. Analytic methods for Diophantine equations and Diophantine inequalities, University Press, Cambridge, 2005.
Fedotov A. and Klopp F. Anderson transitions for a family of almost periodic Schrödinger equations in the adiabatic case. Communications in Mathematical Physics, 227(1):1-92, 2002.
Gluck M., Kolovsky A.R. and Korsch H.J. Wannier-Stark resonances in optical and semiconductor superlattices. Physics Reports, 366:103–182 (2002).
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Levin, B. Ja. Distribution of zeros of entire functions, volume 5 of Translations of Mathematical Monographs. American Mathematical Society, Providence,
Sacchetti, A. Existence of the Stark-Wannier quantum resonances. J. Math. Phys., 55(12):122104, 2014.

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