TY - JOUR

T1 - Hill's operators with the potentials analytically dependent on energy

AU - Badanin, Andrey

AU - Korotyaev, Evgeny L.

N1 - Funding Information: A. Badanin was supported by the RFBR grant number 19-01-00094 . E. Korotyaev was supported by the RSF grant number 18-11-00032 . Publisher Copyright: © 2020 Elsevier Inc. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2021/1/15

Y1 - 2021/1/15

N2 - We consider Schrödinger 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.

AB - We consider Schrödinger 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.

KW - Asymptotics

KW - Eigenvalues

KW - Energy-dependent potential

KW - Hill's equation

UR - http://www.scopus.com/inward/record.url?scp=85091226497&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2020.09.016

DO - 10.1016/j.jde.2020.09.016

M3 - Article

AN - SCOPUS:85091226497

VL - 271

SP - 638

EP - 664

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -