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

Abstract

Bound states of the Hamiltonian describing a quantum particle living on a 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 asymptotics 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 wavefunction 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.