The problem of deficiency indices for discrete Schr\"odinger operators on locally finite graphs

TL;DR

This paper investigates the deficiency indices of discrete Schr"odinger operators on locally finite graphs, establishing conditions for essential self-adjointness and conjecturing a dichotomy for deficiency indices in such graphs.

Contribution

It proves that on trees, deficiency indices are either zero or infinite, and provides criteria for essential self-adjointness, advancing understanding of operator extensions on graphs.

Findings

Deficiency indices on trees are either zero or infinite.

Almost surely, certain trees have all Schr"odinger operators essentially self-adjoint.

Criteria for essential self-adjointness are established.

Abstract

The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schr\"odinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schr\"odinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite unoriented simple graph, its the deficiency indices are either null or infinite. Besides that, we consider some generalizations of trees and weighted graphs.