On extensions of symmetric operators

TL;DR

This paper provides a detailed description of all minimal self-adjoint extensions of a specific class of symmetric operators in Hilbert spaces with deficiency indices (1, 1).

Contribution

It offers an explicit characterization of all such extensions, advancing understanding of operator theory in Hilbert spaces.

Findings

Explicit description of minimal self-adjoint extensions.

Complete classification for deficiency indices (1, 1).

Enhanced understanding of symmetric operator extensions.

Abstract

We give an explicit description of all minimal self-adjoint extensions of a densely defined, closed symmetric operator in a Hilbert space with deficiency indices $(1, 1)$.