The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

Details The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs

2010-05-02

arxiv.org/abs/1005.0165v4

Mathematics

Functional Analysis

Typos corrected. References and ToC added. Paper slightly reorganized. Section 3.2, about the diagonalization has been much im

Scientific paper

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.

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