Resonance and virtual bound state solutions of the radial Schr\"odinger equation

Abstract

The formulation of the eigenvalue problem for the Schr\"odinger equation is studied, for the numerical solution a new approach is applied. With the usual exponentially rising free-state asymptotical behavior, and also with a first order correction to it, the lower half of the $k$ plane is systematically explored. Note that no other method, including the complex rotation one, is suitable for calculating virtual bound states far from the origin. Various phenomena are studied about how the bound and the virtual bound states are organized into a system. Even for short range potentials, the free state asymptotical condition proves to be inadequate at some distance from the real $k$ axis; the first order correction doesn't solve this problem. Therefore, the structure of the space spanned by the mathematically possible solutions is studied and the notion of minimal solutions is introduced. Such solutions provide the natural boundary condition for bound states, their analytical continuation to the lower half plane does the same for resonances and virtual bound states. The eigenvalues coincide with the poles of the S-matrix. The possibility of continuation based on numerical data is also demonstrated. The proposed scheme is suitable for the definition of the in- and outgoing solutions for positive energies; therefore, also for the calculation of the so called distorted S-matrix in case of long range interactions.