A new approach to inverse spectral theory , 11 . General real potentials and the connection to the spectral measure

We continue the study of the A-amplitude associated to a half-line d2 Schrodinger operator, -=t 4 in L2((0, b ) ) , b 5 oo.A is related to the iieylTitchmarsh m-function via m(-fi2) = A(a)e-2ff" d c x + ~ ( e ( ~ ~ & ) " ) -6-J: for all E > 0. We discuss five issues here. First, we extend the theory to general q in L1((O, 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 p: A(a) = -2 JFw x-+ sin(2cuA) dp(X) (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 < GO. 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.