# Restricted three particle quantum walk on ${\bf Z_{\bf +}}$: explicit solution

## TL;DR

This paper analyzes a quantum walk involving three particles on the positive integers, with one particle fixed at zero and interactions only at that point, providing a complete spectral characterization of the system.

## Contribution

It offers an explicit solution and full spectral description for a restricted three-particle quantum walk with a fixed particle and localized interactions.

## Key findings

- Explicit spectral decomposition of the Hamiltonian.
- Complete characterization of essential, point, and discrete spectra.
- Insight into localized interactions in quantum walks.

## Abstract

We consider 3 particles on ${\bf Z}{}_{+}$. One of them has infinite mass and stands still at $0$. The particles interact only if all of them are at the point $0$. We give a full description of essential, point and discrete spectra of the corresponding hamiltonian.