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Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs
Bound states of the Hamiltonian describing a quantum particle living on three
dimensional straight strip of width are investigated. We impose the Neumann
boundary condition on the two concentric windows of the radii and
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 .
When and 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.Comment: 25 pages, 7 figure
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