The elements of scattering theory in the harmonic oscillator representation

The problem is discussed to solve the Schroedinger equation for a particle in a potential well V in terms of the harmonic oscillator representation. The matrix of the free-motion Hamiltonian in the oscillator basis has a simple three-diagonal form. An explicit formula has been obtained for the matrix eigenvectors corresponding to both the regular and irregular solutions for free motion at an arbitrary energy E. The expressions obtained make it possible to calculate the energies of the bound and resonant states and the scattering phase shifts. In the asymptotic case of a high radial quantum number n, the Schroedinger difference equation in harmonic oscillator representation is shown to turn into the conventional Schroedinger differential equation in n-space with a nonlocal potential V(n,n') approximately (n/V/n'). (author)