Scattering with penetrable wall potentials

Abstract

In this paper we shall consider the Schrodinger operator with a penetrable wall potential formally of the form Hformal = -A + q(x)x x6(Ix I a), where q(x) is real and continuous on S a = {x ~ R 3 ; Ixl = a}, a > 0, and 6 denotes the delta function. (This operator is said to provide a simple model for the s-decay [i].) The form counterpart of Hformal is h[u,v] = (HformalU,V) = (Vu,Vv) + (qu,v) a with domain Dom[h] = HI(R3), where ( , ) means the L2(R3) inner product, ( , )a the L2(S a) inner product, and Hm(G) the Sobolev space of order m over G. h is shown to be a lower semibounded closed form, and thus determines a lower semibounded selfadjoint operator H. We should note here that while h is a small perturbation of h o (ho[U,V] = (Vu,Vv), Dom[h o] = HI(R3)) via an infinitesimally ho-bounded form, H H o is not Ho-bounded, where H o = -A, Dom(H o) = H2(R3), is the selfadjoint operator associated with h o. We shall show, however, that the difference of the resolvents R(z) and Ro(Z) of H and H o lies in the trace class. This implies by a well-known theorem that the wave operators intertwining H and H O exist and are complete.