# Stark–Wannier ladders and cubic exponential sums

```@article{Klopp2016StarkWannierLA,
title={Stark–Wannier ladders and cubic exponential sums},
author={Fr{\'e}d{\'e}ric Klopp and Alexander Fedotov},
journal={Functional Analysis and Its Applications},
year={2016},
volume={50},
pages={233-236}
}```
• Published 22 April 2016
• Mathematics
• Functional Analysis and Its Applications
Given a one-dimensional Stark–Wannier operator, we study the reflection coefficient and its poles in the lower half of the complex plane far from the real axis. In particular, the reflection coefficient is described asymptotically in terms of regularized infinite cubic exponential sums.
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The exponential sums Sqam=∑l=1qexp2πial3+mlq−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}

## References

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In this paper, we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field.
• Mathematics, Physics
• 2002
Abstract: This work is devoted to the study of a family of almost periodic one-dimensional Schrödinger equations. Using results on the asymptotic behavior of a corresponding monodromy matrix in the
Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use of
Let f(z) be any transcendental entire function. Let rk denote the absolute value of the zero z k of f(k)(z) which is nearest to the origin. Alander, Erdos and Renyi, and Po'lya have investigated the
Complete Exponential Sums.- Weyl's Sums.- Fractional Parts Distribution, Normal Numbers, and Quadrature Formulas.
• Physics
• 1998
We consider a one-dimensional Stark–Wannier Hamiltonian, H=−d2/dx2+p(x)−ex, x∈R, where p is a smooth periodic, finite-gap potential, and e>0 is small enough. We compute rigorously the imaginary parts