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

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.