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

Sylvain Golénia; Christoph Schumacher

doi:10.1063/1.3596179

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arXiv · 1005.0165

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

Sylvain Golénia·Christoph Schumacher

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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 Schrödinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schrödinger 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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BibTeX RIS

Sylvain Golénia, Christoph Schumacher. 2011-05-13. The problem of deficiency indices for discrete Schrödinger operators on locally finite graphs. https://doi.org/10.1063/1.3596179

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