# Shnol's theorem and the spectrum of long range operators

## TL;DR

This paper extends classical spectral results to long-range operators with decay, establishing a version of Shnol's theorem and linking random operator spectra to deterministic periodic spectra.

## Contribution

It introduces an extension of Shnol's theorem for long-range operators and connects the spectra of random and periodic long-range operators.

## Key findings

- Proved an analogue of Shnol's theorem for long-range operators.
- Established the relationship between spectra of random and periodic long-range operators.

## Abstract

We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogy of Sch'nol's theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.