On the phase-shift formula for the scattering operator

Abstract

Let us denote by S\ k \ the scattering operator attached to a fixed value | k | 2 of the kinetic energy. We shall show by a method different from that of Buslaev [2] that S\ k \ is the identity plus a Hilbert-Schmidt integral operator under a weaker assumption on q(x), and give a way of unique determination of the phase shifts in terms of which S [k \ can be represented as the orthogonal direct sum of multiplication operators by a number with absolute value equal to unity.