86 research outputs found
Time-Frequency Integrals and the Stationary Phase Method in Problems of Waves Propagation from Moving Sources
The time-frequency integrals and the two-dimensional stationary phase method
are applied to study the electromagnetic waves radiated by moving modulated
sources in dispersive media. We show that such unified approach leads to
explicit expressions for the field amplitudes and simple relations for the
field eigenfrequencies and the retardation time that become the coupled
variables. The main features of the technique are illustrated by examples of
the moving source fields in the plasma and the Cherenkov radiation. It is
emphasized that the deeper insight to the wave effects in dispersive case
already requires the explicit formulation of the dispersive material model. As
the advanced application we have considered the Doppler frequency shift in a
complex single-resonant dispersive metamaterial (Lorenz) model where in some
frequency ranges the negativity of the real part of the refraction index can be
reached. We have demonstrated that in dispersive case the Doppler frequency
shift acquires a nonlinear dependence on the modulating frequency of the
radiated particle. The detailed frequency dependence of such a shift and
spectral behavior of phase and group velocities (that have the opposite
directions) are studied numerically
The Fredholm index of locally compact band-dominated operators on
We establish a necessary and sufficient criterion for the Fredholmness of a
general locally compact band-dominated operator on and solve the
long-standing problem of computing its Fredholm index in terms of the limit
operators of . The results are applied to operators of convolution type with
almost periodic symbol
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 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
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
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