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

We study the asymptotics with respect to distance for the eigenfunction of the Schrödinger operator in a half-plane with a singular $\delta$-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.