STARK-WANNIER LADDERS AND CUBIC EXPONENTIAL SUMS
Résumé
On L 2 (R), we consider the Schrödinger operator (1.1) H ǫ = − ∂ 2 ∂x 2 + v(x) − ǫx, where v is a real analytic 1-periodic function and ǫ is a positive constant. This operator is a model to study a Bloch electron in a constant electric field ([1]). The parameter ǫ 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 ǫ is the existence of Stark-Wannier ladders. These are ǫ-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 ǫ (see, e.g., [9, 3] and references therein). When ǫ 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 ǫ changes, and conjectured that the behavior of the resonances strongly depends on number theoretical properties of ǫ (see, e.g., [1]). In the present note, we only consider the periodic potential v(x) = 2 cos(2πx) 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 Im E → −∞, the function E → 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 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..
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