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 ≤ ∞ A is related to the Weyl-Titchmarsh m-function via m(-k^2) = -k- ʃ^a_0 A(α)e^(-2αk) dα+O(e^(-(2α-Є)k)) for all Є > 0. We discuss five issues here. First, we extend the theory to general q in L^1((0,α)) 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 p: A(α) = -2 ^ʃ∞_(-∞)λ^(-1/2) sin (2α√λ) dp(λ) (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 < ∞. 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.