# Eigenvalues of periodic difference operators on lattice octant

## TL;DR

This paper constructs specific periodic difference operators on lattice octants with prescribed eigenvalues within a bounded interval, revealing detailed spectral properties and extending results to higher dimensions.

## Contribution

It introduces a method to realize operators with any given finite spectrum on an interval, advancing inverse spectral theory for periodic difference operators.

## Key findings

- Existence of operators with prescribed eigenvalues on a bounded interval.
- Spectral gaps and infinite-dimensional subspaces adjacent to the eigenvalues.
- Extension of results to various domains and higher dimensions.

## Abstract

Consider a difference operator $H$ with periodic coefficients on the octant of the lattice. We show that for any integer $N$ and any bounded interval $I$, there exists an operator $H$ having $N$ eigenvalues, counted with multiplicity on this interval, and does not exist other spectra on the interval. Also right and to the left of it are spectra and the corresponding subspaces have an infinite dimension. Moreover, we prove similar results for other domains and any dimension. The proof is based on the inverse spectral theory for periodic Jacobi operators.