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 n -entire operators. As our main result we find a necessary and sufficient condition for a one-dimensional Schrödinger operator to be n 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 n -entire, improving the previously known conditions on the perturbation.