RANK-ONE PERTURBATIONS AT INFINITE COUPLING

We discuss rank one perturbations A(alpha) = A + alpha(phi,.)phi, alph a is an element of R, A greater than or equal to 0 self-adjoint. Let d mu(alpha)(x) be the spectral measure defined by (phi, (A(alpha) - z)( -1) phi) = integral d mu(alpha)(x)/(x - z). We prove there is a measur e d rho(infinity) which is the weak limit of (1 + alpha(2)) d mu(alpha )(x) as alpha --> infinity. If phi is cyclic for A, then A(infinity), the strong resolvent limit of A(alpha), is unitarily equivalent to mul tiplication by x on L(2)(R, d rho(infinity)). This generalizes results known for boundary condition dependence of Sturm-Liouville operators on half-lines to the abstract rank one case. (C) 1995 Academic Press, Inc.