Work

Bifurcation of spectra of periodic self-adjoint operators on axis with small 𝒫𝒯-symmetric potential

Year: 2026

Type: article

Abstract: We consider an one-dimensional Schrödinger operator with an even periodic potential perturbed by a small 𝒫 ⁢ 𝒯 {\mathcal{PT}} -symmetric potential. The unperturbed spectrum can have points associated with two periodic or antiperiodic eigenfunctions. We study how such points bifurcate under the perturbation. We obtain a three-terms asymptotic expansion for the corresponding perturbed band functions with rigorous estimates for the error terms. These asymptotics allow us to establish sufficient conditions for the emergence and absence of a non-real spectrum. The emerging non-real spectrum is a complex curve, the shape of which is described by our asymptotic expansions. We also describe how the Dirichlet eigenavules, which satisfy the Dubrovin equations for finite-gap potentials, depend on the perturbation.

Source: Journal of Inverse and Ill-Posed Problems

DOI: 10.1515/jiip-2026-0056

Authors: Denis Ivanovich Borisov, Iskander Asanovich Taimanov

Institutions: Peoples' Friendship University of Russia, M. Akmullah Bashkir State Pedagogical University, Novosibirsk State University, Lomonosov Moscow State University

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,712

Funder: Russian Science Foundation