For more information about this meeting, contact Hope Shaffer, Chun Liu.
|Title:||Discretization of time-dependent quantum systems: propagation of the evolution operator|
|Seminar:||Computational and Applied Mathematics Colloquium|
|Speaker:||Joseph Jerome, Northwestern University|
|The talk represents joint work with Eric Polizzi and is based on a paper of similar title recently published online in Applicable Analysis. We discuss time-dependent quantum systems on bounded domains; these represent closed systems and are relevant for application to Carbon Nanotubes and molecules. Included in our framework are linear iterations involved in time-dependent density functional theory as well as the global nonlinear model which includes the Hartree potential. A key aspect of the analysis of the algorithms is the use of time-ordered evolution operators, which allow for both a well-posed problem and its approximation. The approximation theorems we obtain are operator extensions of classical quadrature theorems. The global existence theorem uses the Leray-Schauder fixed point theorem, coupled to a modified conservation of energy principle. The simulations were performed by Eric Polizzi using his algorithm FEAST. The evolution operators used in the talk are due to T. Kato and their properties will be summarized. Application areas make significant use of these operators, particularly chemical physics. In the mathematical literature, the Euclidean space problem has been studied by T. Cazenave and others, employing the Strichartz inequalities. These are ultimately based on semi-groups. Our results appear to be complementary to results of this type. The solutions we discuss are strong solutions. We are currently studying more general potentials via weak solutions. This work is in-progress.|
Room Reservation Information
|Date:||10 / 27 / 2014|
|Time:||02:30pm - 03:30pm|