A New Approach to Inverse Spectral Theory, Ii. General Real Potentials and the Connection to the Spectral Measure

We continue the study of the A-amplitude associated to a half-line Schrödinger operator, − d dx + q in L((0, b)), b ≤ ∞. A is related to the Weyl-Titchmarsh m-function via m(−κ2) = −κ− ∫ a 0 A(α)e −2ακ dα+O(e) for all ε > 0. We discuss five issues here. First, we extend the theory to general q in L((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 ρ: A(α) = −2 ∫ ∞ −∞ λ− 1 2 sin(2α √ λ) dρ(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 WeylTitchmarsh m-function. Finally, we discuss some examples where one can compute A exactly.