On the discrete spectrum of a spatial quantum waveguide with a disc window

TL;DR

This paper analyzes the bound states of a quantum particle in a three-dimensional strip with a disc-shaped boundary condition, revealing discrete eigenvalues below the essential spectrum and providing numerical estimates and asymptotic behavior.

Contribution

It introduces a detailed analysis of the discrete spectrum for a quantum waveguide with a disc window, including existence proofs and asymptotic eigenvalue behavior.

Findings

Discrete eigenvalues exist below the essential spectrum for all positive radii a.

Number of discrete eigenvalues depends on the ratio a/d.

Eigenvalues asymptotically approach a limit as a tends to infinity.

Abstract

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$. We impose the Neumann boundary condition on a disc window of radius $a$ and Dirichlet boundary conditions on the remained part of the boundary of the strip. We prove that such system exhibits discrete eigenvalues below the essential spectrum for any $a>0$. We give also a numeric estimation of the number of discrete eigenvalue as a function of $\displaystyle \frac{a}{d}$. When $a$ tends to the infinity, the asymptotic of the eigenvalue is given.