Geometry of Weyl theory for Jacobi matrices with matrix entries

TL;DR

This paper explores the geometric structure of Weyl theory for Jacobi matrices with matrix entries, analyzing boundary conditions, isotropic subspaces, and explicit resolvent calculations for semi-infinite cases.

Contribution

It introduces a geometric interpretation of the Weyl surface as isotropic subspaces and develops a limit surface theory for semi-infinite matrices with arbitrary deficiency indices.

Findings

Weyl surface as maximally isotropic subspace

Explicit resolvent formulas for extensions

Limit point/limit surface theory for degenerate cases

Abstract

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green's matrix on the boundary conditions is interpreted as the set of maximally isotropic subspace of a quadratic from given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/limit surface theory of maximal symmetric extensions for semi-infinite Jacobi matrices with matrix entries with arbitrary deficiency indices. The resolvent of the extensions is explicitly calculated.