43 research outputs found
The magnetic formalism; new results
We review recent results on the magnetic pseudo-differential calculus both in
symbolic and in -algebraic form. We also indicate some applications to
spectral analysis of pseudo-differential operators with variable magnetic
fields
Magnetic Pseudodifferential Operators
In previous papers, a generalization of the Weyl calculus was introduced in
connection with the quantization of a particle moving in under
the influence of a variable magnetic field . It incorporates phase factors
defined by and reproduces the usual Weyl calculus for B=0. In the present
article we develop the classical pseudodifferential theory of this formalism
for the standard symbol classes . Among others, we obtain
properties and asymptotic developments for the magnetic symbol multiplication,
existence of parametrices, boundedness and positivity results, properties of
the magnetic Sobolev spaces. In the case when the vector potential has all
the derivatives of order bounded, we show that the resolvent and the
fractional powers of an elliptic magnetic pseudodifferential operator are also
pseudodifferential. As an application, we get a limiting absorption principle
and detailed spectral results for self-adjoint operators of the form
, where is an elliptic symbol, and is the
vector potential corresponding to a short-range magnetic field
An efficient Monte Carlo method for calculating ab initio transition state theory reaction rates in solution
In this article, we propose an efficient method for sampling the relevant
state space in condensed phase reactions. In the present method, the reaction
is described by solving the electronic Schr\"{o}dinger equation for the solute
atoms in the presence of explicit solvent molecules. The sampling algorithm
uses a molecular mechanics guiding potential in combination with simulated
tempering ideas and allows thorough exploration of the solvent state space in
the context of an ab initio calculation even when the dielectric relaxation
time of the solvent is long. The method is applied to the study of the double
proton transfer reaction that takes place between a molecule of acetic acid and
a molecule of methanol in tetrahydrofuran. It is demonstrated that calculations
of rates of chemical transformations occurring in solvents of medium polarity
can be performed with an increase in the cpu time of factors ranging from 4 to
15 with respect to gas-phase calculations.Comment: 15 pages, 9 figures. To appear in J. Chem. Phy
Eigenfunctions decay for magnetic pseudodifferential operators
We prove rapid decay (even exponential decay under some stronger assumptions)
of the eigenfunctions associated to discrete eigenvalues, for a class of
self-adjoint operators in defined by ``magnetic''
pseudodifferential operators (studied in \cite{IMP1}). This class contains the
relativistic Schr\"{o}dinger operator with magnetic field