# 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} }

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.

## 5 Citations

### Spacing gain and absorption in a simple PT-symmetric model: spectral singularities and ladders of eigenvalues and resonances

- PhysicsJournal of Physics A: Mathematical and Theoretical
- 2019

We consider a parity-time () symmetric waveguide consisting of a localized gain and loss element separated by a variable distance. The situation is modeled by a Schrödinger operator with localized…

### On the spectrum of the Kronig–Penney model in a constant electric field

- MathematicsProbability and Mathematical Physics
- 2022

. We are interested in the nature of the spectrum of the one-dimensional Schr¨odinger operator with F > 0 and two diﬀerent choices of the coupling constants { g n } n ∈ Z . In the ﬁrst model g n ≡ λ…

### Number of Nonzero Cubic Sums

- Materials ScienceJournal of Mathematical Sciences
- 2019

The exponential sums Sqam=∑l=1qexp2πial3+mlq−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

## References

SHOWING 1-10 OF 11 REFERENCES

### Existence of the Stark-Wannier quantum resonances

- Mathematics
- 2014

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.…

### Anderson Transitions for a Family of Almost Periodic Schrödinger Equations in the Adiabatic Case

- 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…

### Analytic methods for Diophantine equations and Diophantine inequalities, by Harold Davenport

- Mathematics
- 2005

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…

### On the distribution of zeros of entire functions

- Mathematics
- 1974

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…

### Analytic Methods for Diophantine Equations and Diophantine Inequalities: Cubic forms: bilinear equations

- Mathematics
- 2005

### Exponential Sums and their Applications

- Mathematics
- 1992

Complete Exponential Sums.- Weyl's Sums.- Fractional Parts Distribution, Normal Numbers, and Quadrature Formulas.

### Imaginary parts of Stark–Wannier resonances

- 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…