Fritz Gesztesy and Barry Simon
A new approach to inverse spectral theory, II. General real potentials and the
connection to the spectral measure
(124K, LaTeX)
ABSTRACT. We continue the study of the $A$-amplitude associated to
a half-line Schr\"odinger operator, $-\f{d^2}{dx^2}+ q$ in $L^2 ((0,b))$,
$b\leq \infty$. $A$ 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 $\veps >0$. 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.