Asymptotic presentation of eigenfunctions of a two-dimensional harmonic oscillator

The asymptotics of eigenfunctions of a two-dimensional harmonic oscillator has been obtained all over the space. The need of such presentation arises when studying the spectral characteristics of non compact perturbation of the two-dimensional harmonic oscillator. The absence of exact asymptotic equalities for fundamental systems of solutions to the differential equation complicates the study, because eigenfunctions of the two-dimensional harmonic oscillator are represented in the form of a product of normalized eigenfunctions of the one-dimensional harmonic oscillator. The usage of standard solutions helps to solve the problem.