5,965 research outputs found
Complete asymptotic expansions for eigenvalues of Dirichlet Laplacian in thin three-dimensional rods
We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The
rod is finite. Its cross-section is constant and small, and rotates along the
reference curve in an arbitrary way. We find a two-parametric set of the
eigenvalues of such operator and construct their complete asymptotic
expansions. We show that this two-parametric set contains any prescribed number
of the first eigenvalues of the considered operator. We obtain the complete
asymptotic expansions for the eigenfunctions associated with these first
eigenvalues
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
Homogenization of the planar waveguide with frequently alternating boundary conditions
We consider Laplacian in a planar strip with Dirichlet boundary condition on
the upper boundary and with frequent alternation boundary condition on the
lower boundary. The alternation is introduced by the periodic partition of the
boundary into small segments on which Dirichlet and Neumann conditions are
imposed in turns. We show that under the certain condition the homogenized
operator is the Dirichlet Laplacian and prove the uniform resolvent
convergence. The spectrum of the perturbed operator consists of its essential
part only and has a band structure. We construct the leading terms of the
asymptotic expansions for the first band functions. We also construct the
complete asymptotic expansion for the bottom of the spectrum
Quantum star-graph analogues of PT-symmetric square wells
We pick up a solvable symmetric quantum square well on an
interval of (with an dependent
non-Hermiticity given by Robin boundary conditions) and generalize it. In
essence, we just replace the support interval (reinterpreted
as an equilateral two-pointed star graph with the Kirchhoff matching at the
vertex ) by a pointed equilateral star graph
endowed with the simplest complex-rotation-symmetric external
dependent Robin boundary conditions. The remarkably compact form of
the secular determinant is then deduced. Its analysis reveals that (1) at any
integer , there exists the same, independent and infinite
subfamily of the real energies, and (2) at any special , there
exists another, additional and dependent infinite subfamily of the real
energies. In the spirit of the recently proposed dynamical construction of the
Hilbert space of a quantum system, the physical bound-state interpretation of
these eigenvalues is finally proposed.Comment: 20 pp, 1 figur
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