Singular Schrödinger operators as self-adjoint extensions of N-entire operators

Abstract

We investigate the connections between Weyl–Titchmarsh– Kodaira theory for one-dimensional Schrödinger operators and the theory of n-entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be n-entire in terms of square integrability of derivatives (w.r.t. the spectral parameter) of the Weyl solution. We also show that this is equivalent to the Weyl function being in a generalized Herglotz–Nevanlinna class. As an application we show that perturbed Bessel operators are n-entire, improving the previously known conditions on the perturbation.