Finite lifetime eigenfunctions of coupled systems of harmonic oscillators

Abstract

We find a Hermite-type basis for which the eigenvalue problem associated to the operator $H_{A,B}:=B(-{\partial }_x^2)+A{x}^2$ acting on $L^2(\mathbf{R};{\mathbf{C}}^2)$ becomes a three-terms recurrence. Here $A$ and $B$ are two constant positive definite matrices with no other restriction. Our main result provides an explicit characterization of the eigenvectors of $H_{A,B}$ that lie in the span of the first four elements of this basis when $AB\ne BA$.

11 pages, 1 figure. Some typos where corrected in this new version