1,413 research outputs found
Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics
This paper is devoted to estimates of the exponential decay of eigenfunctions
of difference operators on the lattice Z^n which are discrete analogs of the
Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our
investigation of the essential spectra and the exponential decay of
eigenfunctions of the discrete spectra is based on the calculus of so-called
pseudodifference operators (i.e., pseudodifferential operators on the group
Z^n) with analytic symbols and on the limit operators method. We obtain a
description of the location of the essential spectra and estimates of the
eigenfunctions of the discrete spectra of the main lattice operators of quantum
mechanics, namely: matrix Schr\"{o}dinger operators on Z^n, Dirac operators on
Z^3, and square root Klein-Gordon operators on Z^n
Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory
We isolate a large class of self-adjoint operators H whose essential spectrum
is determined by their behavior at large x and we give a canonical
representation of their essential spectrum in terms of spectra of limits at
infinity of translations of H. The configuration space is an arbitrary abelian
locally compact not compact group.Comment: 63 pages. This is the published version with several correction
Essential spectra of difference operators on \sZ^n-periodic graphs
Let (\cX, \rho) be a discrete metric space. We suppose that the group
\sZ^n acts freely on and that the number of orbits of with respect to
this action is finite. Then we call a \sZ^n-periodic discrete metric
space. We examine the Fredholm property and essential spectra of band-dominated
operators on where is a \sZ^n-periodic discrete metric space.
Our approach is based on the theory of band-dominated operators on \sZ^n and
their limit operators.
In case is the set of vertices of a combinatorial graph, the graph
structure defines a Schr\"{o}dinger operator on in a natural way. We
illustrate our approach by determining the essential spectra of Schr\"{o}dinger
operators with slowly oscillating potential both on zig-zag and on hexagonal
graphs, the latter being related to nano-structures
Nonlinear properties of left-handed metamaterials
We analyze nonlinear properties of microstructured materials with negative
refraction, the so-called left-handed metamaterials. We consider a
two-dimensional periodic structure created by arrays of wires and split-ring
resonators embedded into a nonlinear dielectric, and calculate the effective
nonlinear electric permittivity and magnetic permeability. We demonstrate that
the hysteresis-type dependence of the magnetic permeability on the field
intensity allows changing the material from left- to right-handed and back.
These effects can be treated as the second-order phase transitions in the
transmission properties induced by the variation of an external field.Comment: 4 pages, 3 figure
Demonstration of the difference Casimir force for samples with different charge carrier densities
A measurement of the Casimir force between a gold coated sphere and two Si
plates of different carrier densities is performed using a high vacuum based
atomic force microscope. The results are compared with the Lifshitz theory and
good agreement is found. Our experiment demonstrates that by changing the
carrier density of the semiconductor plate by several orders of magnitude it is
possible to modify the Casimir interaction. This result may find applications
in nanotechnology.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Let
- …