# Asymptotics and estimates of the discrete spectrum of the Schrodinger operator on a discrete periodic graph

## TL;DR

This paper investigates the asymptotic behavior and estimates of the discrete spectrum for a perturbed Schrödinger operator on a discrete periodic graph, focusing on large coupling constants and potentials with power asymptotics.

## Contribution

It provides new asymptotic estimates for the discrete spectrum of the Schrödinger operator under specific potential decay conditions on a discrete periodic graph.

## Key findings

- Derived asymptotics of the discrete spectrum for large coupling constants.
- Established estimates for the discrete spectrum based on potential asymptotics.
- Analyzed the impact of decreasing potentials on the spectral properties.

## Abstract

The periodic Schrodinger operator $ H $ on a discrete periodic graph is considered. We estimate the discrete spectrum of the perturbed operator $ H _ {-} (t) = H-tV $, $ t> 0 $, where the potential $ V \ ge 0 $ is decreasing and $t>0$ is the coupling constant. If the potential has a power asymptotics at infinity, then we obtain asymptotics of the discrete spectrum of operator $ H_ {-} (t) $ with a large coupling constant.