15,308 research outputs found
Quantum-classical correspondence on compact phase space
We propose to study the -norm distance between classical and quantum
phase space distributions, where for the latter we choose the Wigner function,
as a global phase space indicator of quantum-classical correspondence. For
example, this quantity should provide a key to understand the correspondence
between quantum and classical Loschmidt echoes. We concentrate on fully chaotic
systems with compact (finite) classical phase space. By means of numerical
simulations and heuristic arguments we find that the quantum-classical fidelity
stays at one up to Ehrenfest-type time scale, which is proportional to the
logarithm of effective Planck constant, and decays exponentially with a maximal
classical Lyapunov exponent, after that time.Comment: 26 pages. 9 figures (31 .epz files), submitted to Nonlinearit
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
Slow time behavior of the semidiscrete Perona-Malik scheme in dimension one
We consider the long time behavior of the semidiscrete scheme for the
Perona-Malik equation in dimension one. We prove that approximated solutions
converge, in a slow time scale, to solutions of a limit problem. This limit
problem evolves piecewise constant functions by moving their plateaus in the
vertical direction according to a system of ordinary differential equations.
Our convergence result is global-in-time, and this forces us to face the
collision of plateaus when the system singularizes.
The proof is based on energy estimates and gradient-flow techniques,
according to the general idea that "the limit of the gradient-flows is the
gradient-flow of the limit functional". Our main innovations are a uniform
H\"{o}lder estimate up to the first collision time included, a well preparation
result with a careful analysis of what happens at discrete level during
collisions, and renormalizing the functionals after each collision in order to
have a nontrivial Gamma-limit for all times.Comment: 42 page
The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends
A simple sufficient condition on curved end of a straight cylinder is found
that provides a localization of the principal eigenfunction of the mixed
boundary value for the Laplace operator with the Dirichlet conditions on the
lateral side. Namely, the eigenfunction concentrates in the vicinity of the
ends and decays exponentially in the interior. Similar effects are observed in
the Dirichlet and Neumann problems, too.Comment: 25 pages, 10 figure
A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface
It is proved that small periodic singular perturbation of a cylindrical
waveguide surface may open a gap in the continuous spectrum of the Dirichlet
problem for the Laplace operator. If the perturbation period is long and the
caverns in the cylinder are small, the gap certainly opens.Comment: 24 pages, 9 figure
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