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

TL;DR

This paper disproves a previous conjecture by showing that for any non-negative integer, there exists a locally finite graph whose adjacency matrix has deficiency indices equal to that integer, highlighting the diversity of spectral properties.

Contribution

The authors demonstrate that deficiency indices of adjacency matrices on locally finite graphs can be any non-negative integer, countering earlier conjectures.

Findings

Existence of graphs with arbitrary deficiency indices

Negative answer to the previous conjecture

Diversity in spectral properties of graph adjacency matrices

Abstract

In this note we answer negatively to our conjecture concerning the deficiency indices. More precisely, given any non-negative integer $n$, there is locally finite graph on which the adjacency matrix has deficiency indices $(n,n)$.