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 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.

25 pages, 7 figures

References in corpus (4)

• On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition
• Homogenization of the planar waveguide with frequently alternating boundary conditions
• Tuning Fano resonances by magnetic forces for electron transport through a quantum wire side-coupled to a quantum ring
• Magnetic forces and localized resonances in electron transfer through quantum rings

Cited by in corpus (2)

• Electric-Field Control of Bound States and Optical Spectrum in Window-Coupled Quantum Waveguides
• Scattering through a straight quantum waveguide with combined boundary conditions