Discrete spectrum and eigenfunctions of the Schrödinger operator on the plane with a singular $\delta$-potential of special form

M. A. Lyalinov, E. E. Matveenko

Russian Center of Laser Physics, Research Institute of St. Petersburg State University

Abstract: This paper investigates the discrete spectrum and the eigenfunctions of the Schrödinger operator on the plane with a singular $\delta$-potential supported on three rays originating from a single point. Such kind of operator arises in problems of quantum scattering of three one-dimensional particles with a pairwise point interaction and in diffraction problems in wedge-shaped and conical regions. The problem of calculating the eigenfunction of the discrete spectrum of the operator is posed in the classical framework. Using the Kontorovich–Lebedev integral representation, the problem is reduced to the study of a system of homogeneous functional difference equations of the second order with a spectral parameter. The properties of the solutions to this system are analyzed. Depending on the values of the spectral parameter, nontrivial solutions to the system of functional difference equations are described. The eigenfunctions are studied by reducing the problem to integral equations with an integral operator, which is a compact perturbation of the Mehler operator. A sufficient condition for the non-emptiness of the discrete spectrum is derived.

Key words and phrases: eigenfunctions, discrete spectrum, singular potential, functional difference equations.

UDC: 517.9

Received: 30.09.2025