# Spectral analysis of the Laplacian acting on discrete cusps and funnels

## TL;DR

This paper investigates the spectral properties of the discrete Laplacian on structures resembling cusps and funnels, using perturbations and positive commutator techniques to establish propagation estimates and the Limiting Absorption Principle.

## Contribution

It introduces a novel analysis of the discrete Laplacian on perturbed cusp and funnel geometries, extending spectral theory methods to these structures.

## Key findings

- Established propagation estimates for the perturbed Laplacian.
- Proved the Limiting Absorption Principle away from embedded eigenvalues.
- Applied positive commutator techniques to discrete geometric settings.

## Abstract

We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique.