The inverse problem for a spectral asymmetry function of the Schrödinger operator on a finite interval

Abstract

For the Schrödinger equation $-d^{2}u/d{x}^2+q(x)u=\lambda u$ on a finite $x$-interval, there is defined an "asymmetry function" $a(\lambda ;q)$, which is entire of order $1/2$ and type $1$ in $\lambda$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function. For any given $a(\lambda )$, there is one potential for each Dirichlet spectral sequence.