Hill's operators with the potentials analytically dependent on energy

TL;DR

This paper investigates Schrödinger operators with energy-dependent potentials, establishing conditions under which the high energy spectrum is real and deriving asymptotics, with applications to the good Boussinesq equation.

Contribution

It introduces new spectral analysis results for Schrödinger operators with analytically energy-dependent potentials, including spectrum reality and asymptotics.

Findings

High energy spectrum is real under certain conditions.

Asymptotic behavior of the spectrum is characterized.

Applications to the good Boussinesq equation are provided.

Abstract

We consider Schr\"odinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.