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

Abstract

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