Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs

TL;DR

This paper investigates the spectral and localization properties of a quantum waveguide with two concentric Neumann discs, revealing discrete eigenvalues, asymptotic behaviors, and wave function localization changes, with implications for quantum confinement and bound states.

Contribution

It introduces a detailed analysis of the spectral properties of a three-dimensional waveguide with two Neumann discs, including eigenvalue asymptotics and localization effects, extending understanding beyond single-window systems.

Findings

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

Eigenvalue asymptotics are derived as radii tend to infinity.

Wave function localization undergoes drastic changes at anticrossings, with true bound states at critical Neumann radii.

Abstract

Bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$ are investigated. We impose the Neumann boundary condition on the two concentric windows of the radii $a$ and $ b$ located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip. We prove that such a system exhibits discrete eigenvalues below the essential spectrum for any $a,b>0$. When $a$ and $b$ tend to the infinity, the asymptotic of the eigenvalue is derived. A comparative analysis with the one-window case reveals that due to the additional possibility of the regulating energy spectrum the anticrossing structure builds up as a function of the inner radius with its sharpness increasing for the larger outer radius. Mathematical and physical interpretation of the obtained results is presented; namely, it is derived that the anticrossings are accompanied by the drastic changes of the wave function localization. Parallels are drawn to the other structures exhibiting similar phenomena; in particular, it is proved that, contrary to the two-dimensional geometry, at the critical Neumann radii true bound states exist.