79 research outputs found
Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces
The abstract theory of self-adjoint extensions of symmetric operators is used
to construct self-adjoint realizations of a second-order elliptic operator on
with linear boundary conditions on (a relatively open part of)
a compact hypersurface. Our approach allows to obtain Krein-like resolvent
formulas where the reference operator coincides with the "free" operator with
domain ; this provides an useful tool for the scattering
problem from a hypersurface. Concrete examples of this construction are
developed in connection with the standard boundary conditions, Dirichlet,
Neumann, Robin, and -type, assigned either on a
dimensional compact boundary or on a relatively open
part . Schatten-von Neumann estimates for the difference
of the powers of resolvents of the free and the perturbed operators are also
proven; these give existence and completeness of the wave operators of the
associated scattering systems.Comment: Final revised version, to appear in Journal of Differential Equation
with point interactions
Let and be the self-adjoint, one-dimensional
Dirac and Schr\"odinger operators in and
respectively. It is well known that, in absence
of an external potential, the two operators are related through the equality
. We show that such a
kind of relation also holds in the case of -point singular perturbations:
given any self-adjoint realization of the formal sum
, we explicitly determine
the self-adjoint realization of such that
. The
found correspondence preserves the subclasses of self-adjoint realizations
corresponding to both the local and the separating boundary conditions. The
case on nonlocal boundary conditions allows the study of the relation for quantum graphs with (at most) two ends; in
particular, the square of the extension corresponding to Kirchhoff-type
boundary conditions for the Dirac operator on the graph gives the direct sum of
two Schr\"odinger operators on the same graph, one with the usual Kirchhoff
boundary conditions and the other with a sort of reversed Kirchhoff ones
Schroedinger operators with singular interactions: a model of tunneling resonances
We discuss a generalized Schr\"odinger operator in , with an attractive singular interaction supported by a
-dimensional hyperplane and a finite family of points. It can be
regarded as a model of a leaky quantum wire and a family of quantum dots if
, or surface waves in presence of a finite number of impurities if .
We analyze the discrete spectrum, and furthermore, we show that the resonance
problem in this setting can be explicitly solved; by Birman-Schwinger method it
is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 page
Point interactions in acoustics: one dimensional models
A one dimensional system made up of a compressible fluid and several
mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed
for different settings of the oscillators array. The dynamical models are
formulated in terms of singular perturbations of the decoupled dynamics of the
acoustic field and the mechanical oscillators. Detailed spectral properties of
the generators of the dynamics are given for each model we consider. In the
case of a periodic array of mechanical oscillators it is shown that the energy
spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure
Scattering by local deformations of a straight leaky wire
We consider a model of a leaky quantum wire with the Hamiltonian in , where is a compact
deformation of a straight line. The existence of wave operators is proven and
the S-matrix is found for the negative part of the spectrum. Moreover, we
conjecture that the scattering at negative energies becomes asymptotically
purely one-dimensional, being determined by the local geometry in the leading
order, if is a smooth curve and .Comment: Latex2e, 15 page
Jointly Optimal Channel Pairing and Power Allocation for Multichannel Multihop Relaying
We study the problem of channel pairing and power allocation in a
multichannel multihop relay network to enhance the end-to-end data rate. Both
amplify-and-forward (AF) and decode-and-forward (DF) relaying strategies are
considered. Given fixed power allocation to the channels, we show that channel
pairing over multiple hops can be decomposed into independent pairing problems
at each relay, and a sorted-SNR channel pairing strategy is sum-rate optimal,
where each relay pairs its incoming and outgoing channels by their SNR order.
For the joint optimization of channel pairing and power allocation under both
total and individual power constraints, we show that the problem can be
decoupled into two subproblems solved separately. This separation principle is
established by observing the equivalence between sorting SNRs and sorting
channel gains in the jointly optimal solution. It significantly reduces the
computational complexity in finding the jointly optimal solution. It follows
that the channel pairing problem in joint optimization can be again decomposed
into independent pairing problems at each relay based on sorted channel gains.
The solution for optimizing power allocation for DF relaying is also provided,
as well as an asymptotically optimal solution for AF relaying. Numerical
results are provided to demonstrate substantial performance gain of the jointly
optimal solution over some suboptimal alternatives. It is also observed that
more gain is obtained from optimal channel pairing than optimal power
allocation through judiciously exploiting the variation among multiple
channels. Impact of the variation of channel gain, the number of channels, and
the number of hops on the performance gain is also studied through numerical
examples.Comment: 15 pages. IEEE Transactions on Signal Processin
Rigorous Dynamics and Radiation Theory for a Pauli-Fierz Model in the Ultraviolet Limit
The present paper is devoted to the detailed study of quantization and
evolution of the point limit of the Pauli-Fierz model for a charged oscillator
interacting with the electromagnetic field in dipole approximation. In
particular, a well defined dynamics is constructed for the classical model,
which is subsequently quantized according to the Segal scheme. To this end, the
classical model in the point limit is reformulated as a second order abstract
wave equation, and a consistent quantum evolution is given. This allows a study
of the behaviour of the survival and transition amplitudes for the process of
decay of the excited states of the charged particle, and the emission of
photons in the decay process. In particular, for the survival amplitude the
exact time behaviour is found. This is completely determined by the resonances
of the systems plus a tail term prevailing in the asymptotic, long time regime.
Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift
correction to the unperturbed frequencies of the oscillator.Comment: Shortened version. To appear in J. Math. Phy
Scattering into Cones and Flux across Surfaces in Quantum Mechanics: a Pathwise Probabilistic Approach
We show how the scattering-into-cones and flux-across-surfaces theorems in
Quantum Mechanics have very intuitive pathwise probabilistic versions based on
some results by Carlen about large time behaviour of paths of Nelson
diffusions. The quantum mechanical results can be then recovered by taking
expectations in our pathwise statements.Comment: To appear in Journal of Mathematical Physic
Dynamics and Lax-Phillips scattering for generalized Lamb models
This paper treats the dynamics and scattering of a model of coupled
oscillating systems, a finite dimensional one and a wave field on the half
line. The coupling is realized producing the family of selfadjoint extensions
of the suitably restricted self-adjoint operator describing the uncoupled
dynamics. The spectral theory of the family is studied and the associated
quadratic forms constructed. The dynamics turns out to be Hamiltonian and the
Hamiltonian is described, including the case in which the finite dimensional
systems comprises nonlinear oscillators; in this case the dynamics is shown to
exist as well. In the linear case the system is equivalent, on a dense
subspace, to a wave equation on the half line with higher order boundary
conditions, described by a differential polynomial explicitely
related to the model parameters. In terms of such structure the Lax-Phillips
scattering of the system is studied. In particular we determine the incoming
and outgoing translation representations, the scattering operator, which turns
out to be unitarily equivalent to the multiplication operator given by the
rational function , and the Lax-Phillips semigroup,
which describes the evolution of the states which are neither incoming in the
past nor outgoing in the future
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
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