Representation of the Green function of a two-dimensional harmonic oscillator

Abstract

In 1933, Courant R. and Hilbert D. considered a formal decomposition of the source function by eigenfunctions of the Dirichlet problem of the Laplace operator on a rectangle. It turned out that the specified series cannot converge absolutely for any pair of internal points of the rectangle. Therefore, the convergence of a series can only be conditional. Then the summation order is important for conditional convergence. Systemically similar decompositions are studied in the works of V. A. Ilyin. In this paper, we investigate the convergence of the source function decomposition with respect to the eigenfunctions of a two-dimensional harmonic oscillator. A representation of the green function of a two-dimensional harmonic oscillator is obtained. The features of the green function are highlighted. As a result, it follows that the green function of a two-dimensional harmonic oscillator has two singular points. The features are located symmetrically relative to the origin. This effect was not observed in the studies of V. A. Ilyin. Fractional order kernels studied By V. A. Ilyin had only one singular point. Another circumstance distinguishes the green function of a two-dimensional harmonic oscillator from the green function of boundary-value problems in a bounded domain. The green function of a boundary value problem on a flat bounded domain has a logarithmic singularity. At the same time, the green function of a two-dimensional harmonic oscillator has power-law features. However, the degree of this singularity is much less than the power-law singularity of the green function of a three-dimensional boundary value problem in a bounded domain.