Bound states for non-symmetric evolution Schrödinger potentials

Abstract

We consider the spectral problem associated with the evolution Schrodinger equation, (D2 + k2) = u, where u is a matrix-square-valued function, with entries in the Schwartz class defined on the real line. The solution , called the wavefunction, consists of a function of one real variable, matrix-square-valued with entries in the Schwartz class. This problem has been dealt for symmetric potentials u. We found for the present case that the bound states are localized similarly to the scalar and symmetric cases, but by the zeroes of an analytic matrix-valued function. If we add an extra condition to the potential u, we can determine these states by an analytic scalar function. We do this by generalizing the scalar and symmetric cases but without using the fact that the Wronskian of a pair of wavefunction is constant.