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

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Bifurcation of spectra of periodic self-adjoint operators on axis with small PT-symmetric potential. / Borisov, Denis Ivanovich; Taimanov, Iskander Asanovich.

в: Journal of Inverse and Ill-Posed Problems, 25.06.2026.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{e203070fa27448379af4b42e828418bd,

title = "Bifurcation of spectra of periodic self-adjoint operators on axis with small PT-symmetric potential",

abstract = "We consider an one-dimensional Schr{\"o}dinger operator with an even periodic potential perturbed by a small 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.",

keywords = "Dirichlet spectra, One-dimensional periodic Schr{\"o}dinger operator, PT-symmetric perturbation, asymptotic expansions, finite-gap potential, non-real spectrum, Одномерный периодический оператор Шрёдингера, PT-симметричное возмущение, невещественный спектр, асимптотические разложения, конечнозонный потенциал, спектры Дирихле",

author = "Borisov, {Denis Ivanovich} and Taimanov, {Iskander Asanovich}",

note = "Borisov, Denis Ivanovich and Taimanov, Iskander Asanovich. {"}Bifurcation of spectra of periodic self-adjoint operators on axis with small 𝒫𝒯-symmetric potential{"} Journal of Inverse and Ill-posed Problems, 2026. https://doi.org/10.1515/jiip-2026-0056 The research of the second author Iskander A. Taimanov was supported by the Russian Science Foundation, project no. 24-11-00281",

year = "2026",

month = jun,

day = "25",

doi = "10.1515/jiip-2026-0056",

language = "English",

journal = "Journal of Inverse and Ill-Posed Problems",

issn = "0928-0219",

publisher = "Walter de Gruyter GmbH",

}

RIS

TY - JOUR

T1 - Bifurcation of spectra of periodic self-adjoint operators on axis with small PT-symmetric potential

AU - Borisov, Denis Ivanovich

AU - Taimanov, Iskander Asanovich

N1 - Borisov, Denis Ivanovich and Taimanov, Iskander Asanovich. "Bifurcation of spectra of periodic self-adjoint operators on axis with small 𝒫𝒯-symmetric potential" Journal of Inverse and Ill-posed Problems, 2026. https://doi.org/10.1515/jiip-2026-0056 The research of the second author Iskander A. Taimanov was supported by the Russian Science Foundation, project no. 24-11-00281

PY - 2026/6/25

Y1 - 2026/6/25

N2 - We consider an one-dimensional Schrödinger operator with an even periodic potential perturbed by a small 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.

AB - We consider an one-dimensional Schrödinger operator with an even periodic potential perturbed by a small 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.

KW - Dirichlet spectra

KW - One-dimensional periodic Schrödinger operator

KW - PT-symmetric perturbation

KW - asymptotic expansions

KW - finite-gap potential

KW - non-real spectrum

KW - Одномерный периодический оператор Шрёдингера

KW - PT-симметричное возмущение

KW - невещественный спектр

KW - асимптотические разложения

KW - конечнозонный потенциал

KW - спектры Дирихле

UR - https://www.mendeley.com/catalogue/afd74d21-b6bc-3116-a3db-d035ae82ef68/

UR - https://www.scopus.com/pages/publications/105043330985

U2 - 10.1515/jiip-2026-0056

DO - 10.1515/jiip-2026-0056

M3 - Article

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

ER -

ID: 83243833