# On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KDV potentials

@article{Batchenko2003OnTS, title={On the spectrum of Schr{\"o}dinger operators with quasi-periodic algebro-geometric KDV potentials}, author={Volodymyr Batchenko and Fritz Gesztesy}, journal={Journal d’Analyse Math{\'e}matique}, year={2003}, volume={95}, pages={333-387} }

We characterize the spectrum of one-dimensional Schrödinger operatorsH=−d2/dx2+V inL2(ℝdx) with quasi-periodic complex-valued algebro-geometric potentialsV, i.e., potentialsV which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy, associated with nonsingular hyperelliptic curves. The spectrum ofH coincides with the conditional stability set ofH and can be described explicitly in terms of the mean value of the inverse of the diagonal Green’s…

## 21 Citations

### On the spectrum of Jacobi operators with quasi-periodic algebro-geometric coefficients

- Mathematics
- 2005

We characterize the spectrum of one-dimensional Jacobi operators H=aS^{+}+a^{-}S^{-}+b in l^{2}(\Z) with quasi-periodic complex-valued algebro-geometric coefficients (which satisfy one (and hence…

### On the shape of spectra for non-self-adjoint periodic Schrödinger operators

- Mathematics, Physics
- 2004

The spectra of the Schrodinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of, at most, countably many line segments (energy bands)…

### On the spectra of real and complex lamé operators

- Mathematics
- 2016

We study Lame operators of the form L = − dx/dx 2 2 + m(m + 1)ω 2 ℘(ωx + z 0 ), with m ∈ N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z 0 such that the…

### Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

- Mathematics
- 2021

The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the…

### The Algebro-Geometric Initial Value Problem for the Relativistic Lotka-Volterra Hierarchy and Quasi-Periodic Solutions

- Mathematics
- 2012

We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric…

### A criterion for Hill operators to be spectral operators of scalar type

- Mathematics
- 2006

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = −d2/dx2 + V to be a spectral operator of scalar type. The conditions…

### Finite gap integration of the derivative nonlinear Schrödinger equation: A Riemann–Hilbert method

- Mathematics
- 2020

### The algebro-geometric Toda hierarchy initial value problem for complex-valued initial data

- Mathematics
- 2006

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvabil- ity globally in time for a set of initial (Dirichlet divisor)…

## References

SHOWING 1-10 OF 71 REFERENCES

### Ljapunov Indices Determine Absolutely Continuous Spectra of Stationary Random One-dimensional Schrödinger Operators

- Mathematics
- 1984

### NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES

- Mathematics
- 1976

The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical…

### NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES

- Mathematics
- 1976

The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical…

### Soliton Equations and their Algebro-Geometric Solutions

- Mathematics
- 2003

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial…

### Picard potentials and Hill's equation on a torus

- Mathematics
- 1996

Hill's equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness. Of particular importance in the last 20 years is its connection…

### Spectra of non-selfadjoint Hill's operators and a class of Riemann surfaces

- Mathematics
- 1996

*I am grateful to A. Boutet de Monvel for the invitation to Laboratory of Mathematical Physics and Geometry, University Paris-7, where the initial version of this paper was written. It gives me a…

### On Hill's Equation with a Singular Complex‐Valued Potential

- Mathematics
- 1998

In this paper Hill's equation y″ + qy = Ey, where q is a complex‐valued function with inverse square singularities, is studied. Results on the dependence of solutions to initial value problems on the…

### The spectrum of Hill's equation

- Mathematics
- 1975

AbstractLetq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=−d2/dx2+q(x) in the class of functions of period 2 is a discrete series -…