Jacobi matrices on trees

TL;DR

This paper investigates symmetric Jacobi matrices on homogeneous trees, revealing how the tree's structure influences their essential selfadjointness and describing defect spaces for nonselfadjoint cases.

Contribution

It characterizes essential selfadjointness of Jacobi matrices on trees based on their structure and relates defect spaces to the Poisson kernel for nonselfadjoint cases.

Findings

Matrices on trees with one end and infinite origins are always selfadjoint.

Selfadjointness depends on the tree's structure, not just coefficients.

Defect spaces are described via the Poisson kernel.

Abstract

Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by the restriction to the so called radial functions. For nonselfadjoint matrices the defect spaces are described in terms of the Poisson kernel associated with the boundary of the tree.