An Addendum to Krein's Formula

TL;DR

This paper extends Krein's formula by explicitly deriving the linear fractional transformation that relates the Weyl-Titchmarsh M-functions of two self-adjoint extensions of a symmetric operator, enhancing understanding of their resolvent differences.

Contribution

It provides explicit derivation of the linear fractional transformation connecting M-functions for two self-adjoint extensions, adding detailed analytical tools to Krein's formula.

Findings

Explicit formula relating M_1(z) and M_2(z)

Enhanced understanding of resolvent differences

Analytical framework for self-adjoint extensions

Abstract

We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A_1 and A_2 of a densely defined closed symmetric linear operator A with (possibly infinite) equal deficiency indices. In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M_1(z) and M_2(z) corresponding to A_1 and A_2.