The spectrum of the Schr\"{o}dinger Hamiltonian for trapped particles in a cylinder with a topological defect perturbed by two attractive delta interactions

TL;DR

This paper analyzes the discrete energy spectrum of a quantum particle confined in a cylindrical region with a topological defect and two attractive delta interactions, revealing how the bound states depend on the cylinder's radius and the defect.

Contribution

The study introduces a rigorous method to analyze delta interactions in curved spacetime with topological defects, deriving an approximate energy formula and exploring bound state behavior.

Findings

Number of bound states increases with cylinder radius

Existence of states with zero total energy (quasi free states)

Mixture of positive, zero, and negative energy states depending on parameters

Abstract

In this paper we exploit the technique used in \cite{A}-\cite{5b} to deal with delta interactions in a rigorous way in a curved spacetime represented by a cosmic string along the $z$ axis. This mathematical machinery is applied in order to study the discrete spectrum of a point-mass particle confined in an infinitely long cylinder with a conical defect on the $z$ axis and perturbed by two identical attractive delta interactions symmetrically situated around the origin. We derive a suitable approximate formula for the total energy. As a consequence, we found the existence of a mixing of states with positive or zero energy with the ones with negative energy (bound states). This mixture depends on the radius $R$ of the trapping cylinder. The number of quantum bound states is an increasing function of the radius $R$. It is also interesting to note the presence of states with zero total energy (quasi free states). Apart from the gravitational background, the model presented in this paper is of interest in the context of nanophysics and graphene modeling. In particular, the graphene with double layer in this framework, with the double layer given by the aforementioned delta interactions and the string on the $z-$axis modeling topological defects connecting the two layers. As a consequence of these setups, we obtain the usual mixture of positive and negative bound states present in the graphene literature.