Analytic Differential Operators on the Unit Disk

TL;DR

This paper studies symmetric differential operators on weighted Hardy-Hilbert spaces, characterizing their self-adjointness, eigenvalue asymptotics, and extensions, with applications to classical equations like Riemann and Heun.

Contribution

It provides a comprehensive analysis of symmetric differential operators on the unit disk, including self-adjointness criteria and eigenvalue asymptotics, extending to non-self-adjoint cases.

Findings

Characterization of symmetric minimal operators.

Essential self-adjointness for operators with non-vanishing leading coefficients on the unit circle.

Eigenvalue asymptotics for these operators.

Abstract

Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical Riemann and Heun equations. Symmetric minimal operators are characterized. A regular class whose leading coefficients have no zeros on the unit circle are shown to be essentially self-adjoint. Eigenvalue asymptotics are established. Some extensions to non-self-adjoint operators are also considered.