TY - JOUR

T1 - Functional-Difference Equations and Their Link with Perturbations of the Mehler Operator

AU - Лялинов, Михаил Анатольевич

N1 - Publisher Copyright: © 2022, Pleiades Publishing, Ltd.

PY - 2022/9/1

Y1 - 2022/9/1

N2 - This work deals with the spectral properties of the functional-dierence equations, thatarise in a number of applications in the diraction of waves and quantum scattering. Their linkwith some of the spectral properties of perturbations of the Mehler operator is addressed. The latternaturally arise in studies of functional-dierence equations of the second order with a meromorphicpotential which depend on a characteristic parameter. In particular, this kind of equations is frequentlyencountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge-or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of themodied MehlerFock transform. Its resolvent and spectral measure are described. These resultsare obtained by use of some additional analysis applied to the known Mehler formulas. For a class ofcompact perturbations of this operator, sucient conditions of existence and niteness of the discretespectrum are then discussed. Applications to the functional-dierence equations are also addressed.An example of a problem leading to the study of the spectral properties for a functional-dierenceequation is considered. The corresponding eigenfunctions and characteristic values are found explicitlyin this case.

AB - This work deals with the spectral properties of the functional-dierence equations, thatarise in a number of applications in the diraction of waves and quantum scattering. Their linkwith some of the spectral properties of perturbations of the Mehler operator is addressed. The latternaturally arise in studies of functional-dierence equations of the second order with a meromorphicpotential which depend on a characteristic parameter. In particular, this kind of equations is frequentlyencountered with in the asymptotic treatment of eigenfunctions of the Robin Laplacians in wedge-or cone-shaped domains. The unperturbed selfadjoint Mehler operator is studied by means of themodied MehlerFock transform. Its resolvent and spectral measure are described. These resultsare obtained by use of some additional analysis applied to the known Mehler formulas. For a class ofcompact perturbations of this operator, sucient conditions of existence and niteness of the discretespectrum are then discussed. Applications to the functional-dierence equations are also addressed.An example of a problem leading to the study of the spectral properties for a functional-dierenceequation is considered. The corresponding eigenfunctions and characteristic values are found explicitlyin this case.

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UR - https://www.mendeley.com/catalogue/09ae5bb0-b898-3e5c-8979-fc01ce8fe986/

U2 - 10.1134/s1061920822030062

DO - 10.1134/s1061920822030062

M3 - Article

VL - 29

SP - 378

EP - 396

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 3

ER -