On a comparison between absolute and relative self-adjoint extension schemes

TL;DR

This paper compares two schemes for classifying self-adjoint extensions of symmetric operators, establishing quantitative links between their parameters, especially as deficiency spaces converge at spectral points.

Contribution

It provides a detailed analysis connecting the absolute von Neumann and relative boundary-triplet extension schemes for self-adjoint operators, with explicit parameter relations.

Findings

Quantitative connections between extension parameters are established.

Limit behavior of deficiency spaces at spectral points is analyzed.

The comparison clarifies the relationship between two extension schemes.

Abstract

The problem of connecting the operator parameters that label the same self-adjoint extension of a given symmetric operator, respectively, within the 'absolute' von Neumann extension scheme and the 'relative' boundary-triplet-induced extension scheme (i.e., a la Kre\u{i}n-Vi\v{s}ik-Birman) is discussed, and quantitative connections between the two parameters are established in the limit of deficiency spaces at complex spectral points converging to the deficiency space at a real spectral point.