ON RANK-ONE PERTURBATIONS OF SELF-ADJOINT OPERATORS

Let A be a selfadjoint operator in a Hilbert space h. Its rank one per turbations A+tau(.,omega)omega, tau is an element of R, are studied wh en omega belongs to the scare space h(-2) associated with h(+2) = dom A and (.,.) is the corresponding duality. If A is nonnegative and omeg a belongs to the scale space h(-1), Gesztesy and Simon [4] prove that the spectral Measures of A(tau), tau is an element of R, converge weak ly to the spectral measure of the limiting perturbation A(infinity). I n fact A(infinity) can be identified as a Friedrichs extension, Furthe r results for nonnegative operators A were obtained by Kiselev and Sim on [14] by allowing omega is an element of h(-2). Our purpose is to sh ow that most results of Gesztesy, Kiselev, and Simon are valid for ran k one perturbations of selfadjoint operators, which are not necessaril y semibounded. We use the fact Chat rank one perturbations constitute selfadjoint extensions oi an associated symmetric operator, The use of so-called Q-functions [6, 8] facilitates the descriptions. In the spe cial case that omega belongs to the scale space h(-1) associated with h(+1) = dom \A\(1/2), the limiting perturbation A(infinity) is shown t o be the generalized Friedrichs extension [5].