Schrödinger operator in a half-plane with the Neumann condition on the boundary and a singular δ-potential supported by two half-lines, and systems of functional-difference equations

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Schrödinger operator in a half-plane with the Neumann condition on the boundary and a singular δ-potential supported by two half-lines, and systems of functional-difference equations. / Лялинов, Михаил Анатольевич.

In: Theoretical and Mathematical Physics, Vol. 213, No. 2, 11.2022, p. 1560-1588.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{e125a380eaaf4d38bf2ded94d712be50,

title = "Schr{\"o}dinger operator in a half-plane with the Neumann condition on the boundary and a singular δ-potential supported by two half-lines, and systems of functional-difference equations",

abstract = "Abstract: We study the asymptotics with respect to distance for the eigenfunction of the Schr{\"o}dinger operator in a half-plane with a singular δ-potential supported by two half-lines. Such an operator occurs in problems of scattering of three one-dimensional quantum particles with point-like pair interaction under some additional restrictions, as well as in problems of wave diffraction in wedge-shaped and cone-shaped domains. Using the Kontorovich–Lebedev representation, the problem of constructing an eigenfunction of an operator reduces to studying a system of homogeneous functional-difference equations with a characteristic (spectral) parameter. We study the properties of solutions of such a system of second-order homogeneous functional-difference equations with a potential from a special class. Depending on the values of the characteristic parameter in the equations, we describe their nontrivial solutions, the eigenfunctions of the equation. The study of these solutions is based on reducing the system to integral equations with a bounded self-adjoint operator, which is a completely continuous perturbation of the matrix Mehler operator. For a perturbed Mehler operator, sufficient conditions are proposed for the existence of a discrete spectrum to the right of the essential spectrum. Conditions for the finiteness of the discrete spectrum are studied. These results are used in the considered problem in the half-plane. The transformation from the Kontorovich–Lebedev representation to the Sommerfeld integral representation is used to construct the asymptotics with respect to the distance for the eigenfunction of the Schr{\"o}dinger operator under consideration.",

keywords = "asymptotics of eigenfunctions, functional-difference equations, perturbed Mehler operator, spectrum",

author = "Лялинов, {Михаил Анатольевич}",

note = "Publisher Copyright: {\textcopyright} 2022, Pleiades Publishing, Ltd.",

year = "2022",

month = nov,

doi = "10.1134/s0040577922110058",

language = "English",

volume = " 213",

pages = "1560--1588",

journal = "Theoretical and Mathematical Physics (Russian Federation)",

issn = "0040-5779",

publisher = "Springer Nature",

number = "2",

}

RIS

TY - JOUR

T1 - Schrödinger operator in a half-plane with the Neumann condition on the boundary and a singular δ-potential supported by two half-lines, and systems of functional-difference equations

AU - Лялинов, Михаил Анатольевич

PY - 2022/11

Y1 - 2022/11

N2 - Abstract: We study the asymptotics with respect to distance for the eigenfunction of the Schrödinger operator in a half-plane with a singular δ-potential supported by two half-lines. Such an operator occurs in problems of scattering of three one-dimensional quantum particles with point-like pair interaction under some additional restrictions, as well as in problems of wave diffraction in wedge-shaped and cone-shaped domains. Using the Kontorovich–Lebedev representation, the problem of constructing an eigenfunction of an operator reduces to studying a system of homogeneous functional-difference equations with a characteristic (spectral) parameter. We study the properties of solutions of such a system of second-order homogeneous functional-difference equations with a potential from a special class. Depending on the values of the characteristic parameter in the equations, we describe their nontrivial solutions, the eigenfunctions of the equation. The study of these solutions is based on reducing the system to integral equations with a bounded self-adjoint operator, which is a completely continuous perturbation of the matrix Mehler operator. For a perturbed Mehler operator, sufficient conditions are proposed for the existence of a discrete spectrum to the right of the essential spectrum. Conditions for the finiteness of the discrete spectrum are studied. These results are used in the considered problem in the half-plane. The transformation from the Kontorovich–Lebedev representation to the Sommerfeld integral representation is used to construct the asymptotics with respect to the distance for the eigenfunction of the Schrödinger operator under consideration.

AB - Abstract: We study the asymptotics with respect to distance for the eigenfunction of the Schrödinger operator in a half-plane with a singular δ-potential supported by two half-lines. Such an operator occurs in problems of scattering of three one-dimensional quantum particles with point-like pair interaction under some additional restrictions, as well as in problems of wave diffraction in wedge-shaped and cone-shaped domains. Using the Kontorovich–Lebedev representation, the problem of constructing an eigenfunction of an operator reduces to studying a system of homogeneous functional-difference equations with a characteristic (spectral) parameter. We study the properties of solutions of such a system of second-order homogeneous functional-difference equations with a potential from a special class. Depending on the values of the characteristic parameter in the equations, we describe their nontrivial solutions, the eigenfunctions of the equation. The study of these solutions is based on reducing the system to integral equations with a bounded self-adjoint operator, which is a completely continuous perturbation of the matrix Mehler operator. For a perturbed Mehler operator, sufficient conditions are proposed for the existence of a discrete spectrum to the right of the essential spectrum. Conditions for the finiteness of the discrete spectrum are studied. These results are used in the considered problem in the half-plane. The transformation from the Kontorovich–Lebedev representation to the Sommerfeld integral representation is used to construct the asymptotics with respect to the distance for the eigenfunction of the Schrödinger operator under consideration.

KW - asymptotics of eigenfunctions

KW - functional-difference equations

KW - perturbed Mehler operator

KW - spectrum

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U2 - 10.1134/s0040577922110058

DO - 10.1134/s0040577922110058

M3 - Article

VL - 213

SP - 1560

EP - 1588

JO - Theoretical and Mathematical Physics (Russian Federation)

JF - Theoretical and Mathematical Physics (Russian Federation)

SN - 0040-5779

IS - 2

ER -

ID: 100852654