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


We continue the study of the $A$-amplitude associated to a half-line Schrödinger operator, $-\frac{d^2}{dx^2} + q$ in $L^2((0,b))$, $b\le \infty$. $A$ is related to the Weyl-Titchmarsh $m$-function via $m(-\kappa^2) = -\kappa – \int_0^a A(\alpha)e^{-2\alpha\kappa}d\alpha+O(e^{-(2a-\varepsilon)\kappa})$ for all $\varepsilon >0$. 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 $\rho\colon A(\alpha) = -2\int_{-\infty}^{\infty} \lambda^{-\frac{1}{2}} \sin (2\alpha \sqrt{\lambda})d\rho(\lambda)$ (since the integral is divergent, this formula has to be property interpreted). Third, we provide a Laplace transform representation for $m$ without error term in the case $b\lt \infty$. Fourth, we discuss $m$-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titschmarsh $m$-function. Finally, we discuss some examples where one can compute $A$ exactly.




Fritz Gesztesy

Barry Simon