A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure

Abstract

We continue the study of the A-amplitude associated to a half-line Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related to the Weyl-Titchmarsh m-function via m(-\kappa^2) =-\kappa - \int_0^a A(\alpha) e^{-2\alpha\kappa} d\alpha +O(e^{-(2a -\epsilon)\kappa}) for all \epsilon > 0. We discuss five issues here. First, we extend the theory to general q in L^1 ((0,a)) for all a, including q's which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure \rho: A(\alpha) = -2\int_{-\infty}^\infty \lambda^{-\frac12} \sin (2\alpha \sqrt{\lambda})\, d\rho(\lambda) (since the integral is divergent, this formula has to be properly interpreted). Third, we provide a Laplace transform representation for m without error term in the case b<\infty. Fourth, we discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.