891 research outputs found
Emergence of Periodic Structure from Maximizing the Lifetime of a Bound State Coupled to Radiation
Consider a system governed by the time-dependent Schr\"odinger equation in
its ground state. When subjected to weak (size ) parametric forcing
by an "ionizing field" (time-varying), the state decays with advancing time due
to coupling of the bound state to radiation modes. The decay-rate of this
metastable state is governed by {\it Fermi's Golden Rule}, , which
depends on the potential and the details of the forcing. We pose the
potential design problem: find which minimizes (maximizes
the lifetime of the state) over an admissible class of potentials with fixed
spatial support. We formulate this problem as a constrained optimization
problem and prove that an admissible optimal solution exists. Then, using
quasi-Newton methods, we compute locally optimal potentials. These have the
structure of a truncated periodic potential with a localized defect. In
contrast to optimal structures for other spectral optimization problems, our
optimizing potentials appear to be interior points of the constraint set and to
be smooth. The multi-scale structures that emerge incorporate the physical
mechanisms of energy confinement via material contrast and interference
effects.
An analysis of locally optimal potentials reveals local optimality is
attained via two mechanisms: (i) decreasing the density of states near a
resonant frequency in the continuum and (ii) tuning the oscillations of
extended states to make , an oscillatory integral, small. Our
approach achieves lifetimes, , for locally
optimal potentials with as compared with
for a typical potential. Finally, we
explore the performance of optimal potentials via simulations of the
time-evolution.Comment: 33 pages, 6 figure
Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
We prove sharp stability estimates for the variation of the eigenvalues of
non-negative self-adjoint elliptic operators of arbitrary even order upon
variation of the open sets on which they are defined. These estimates are
expressed in terms of the Lebesgue measure of the symmetric difference of the
open sets. Both Dirichlet and Neumann boundary conditions are considered
Rendezvous of Two Robots with Constant Memory
We study the impact that persistent memory has on the classical rendezvous
problem of two mobile computational entities, called robots, in the plane. It
is well known that, without additional assumptions, rendezvous is impossible if
the entities are oblivious (i.e., have no persistent memory) even if the system
is semi-synchronous (SSynch). It has been recently shown that rendezvous is
possible even if the system is asynchronous (ASynch) if each robot is endowed
with O(1) bits of persistent memory, can transmit O(1) bits in each cycle, and
can remember (i.e., can persistently store) the last received transmission.
This setting is overly powerful.
In this paper we weaken that setting in two different ways: (1) by
maintaining the O(1) bits of persistent memory but removing the communication
capabilities; and (2) by maintaining the O(1) transmission capability and the
ability to remember the last received transmission, but removing the ability of
an agent to remember its previous activities. We call the former setting
finite-state (FState) and the latter finite-communication (FComm). Note that,
even though its use is very different, in both settings, the amount of
persistent memory of a robot is constant.
We investigate the rendezvous problem in these two weaker settings. We model
both settings as a system of robots endowed with visible lights: in FState, a
robot can only see its own light, while in FComm a robot can only see the other
robot's light. We prove, among other things, that finite-state robots can
rendezvous in SSynch, and that finite-communication robots are able to
rendezvous even in ASynch. All proofs are constructive: in each setting, we
present a protocol that allows the two robots to rendezvous in finite time.Comment: 18 pages, 3 figure
Wave operator bounds for 1-dimensional Schr\"odinger operators with singular potentials and applications
Boundedness of wave operators for Schr\"odinger operators in one space
dimension for a class of singular potentials, admitting finitely many Dirac
delta distributions, is proved. Applications are presented to, for example,
dispersive estimates and commutator bounds.Comment: 16 pages, 0 figure
Green functions and dimensional reduction of quantum fields on product manifolds
We discuss Euclidean Green functions on product manifolds P=NxM. We show that
if M is compact then the Euclidean field on P can be approximated by its zero
mode which is a Euclidean field on N. We estimate the remainder of this
approximation. We show that for large distances on N the remainder is small. If
P=R^{D-1}xS^{beta}, where S^{beta} is a circle of radius beta, then the result
reduces to the well-known approximation of the D dimensional finite temperature
quantum field theory to D-1 dimensional one in the high temperature limit.
Analytic continuation of Euclidean fields is discussed briefly.Comment: 17 page
Residence time and collision statistics for exponential flights: the rod problem revisited
Many random transport phenomena, such as radiation propagation,
chemical/biological species migration, or electron motion, can be described in
terms of particles performing {\em exponential flights}. For such processes, we
sketch a general approach (based on the Feynman-Kac formalism) that is amenable
to explicit expressions for the moments of the number of collisions and the
residence time that the walker spends in a given volume as a function of the
particle equilibrium distribution. We then illustrate the proposed method in
the case of the so-called {\em rod problem} (a 1d system), and discuss the
relevance of the obtained results in the context of Monte Carlo estimators.Comment: 9 pages, 8 figure
Spectral Analysis for Matrix Hamiltonian Operators
In this work, we study the spectral properties of matrix Hamiltonians
generated by linearizing the nonlinear Schr\"odinger equation about soliton
solutions. By a numerically assisted proof, we show that there are no embedded
eigenvalues for the three dimensional cubic equation. Though we focus on a
proof of the 3d cubic problem, this work presents a new algorithm for verifying
certain spectral properties needed to study soliton stability. Source code for
verification of our comptuations, and for further experimentation, are
available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.Comment: 57 pages, 22 figures, typos fixe
L^p boundedness of the wave operator for the one dimensional Schroedinger operator
Given a one dimensional perturbed Schroedinger operator H=-(d/dx)^2+V(x) we
consider the associated wave operators W_+, W_- defined as the strong L^2
limits as s-> \pm\infty of the operators e^{isH} e^{-isH_0} We prove that the
wave operators are bounded operators on L^p for all 1<p<\infty, provided
(1+|x|)^2 V(x) is integrable, or else (1+|x|)V(x) is integrable and 0 is not a
resonance. For p=\infty we obtain an estimate in terms of the Hilbert
transform. Some applications to dispersive estimates for equations with
variable rough coefficients are given.Comment: 26 page
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
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