THE INVERSE PROBLEM FOR A SPECTRAL ASYMMETRY FUNCTION OF THE SCHRÖDINGER OPERATOR ON A FINITE INTERVAL

For the Schrödinger equation − d 2 u / d x 2 + q ( x ) u = λ u 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 a ( λ ) . For any given a ( λ ) , there is one potential for each Dirichlet spectral sequence.