Mathematical Aspects of the Schroedinger Equation
Oslo, June 14, 2004

ABSTRACTS (chronologically ordered):

Morten Hjorth-Jensen
I will present several quantum mechanical systems of current interest with large overlaps into the realm of mathematical physics. Systems of interest are for example vortices in superfluids and superconductors. Within the framework of Landau-Ginzburg theory such systems can be modelled through a time-dependent Schroedinger equation with non linear terms. I will present the physics behind these approximations, its limitations and point to interesting areas of overlap between physics and mathematics. Landau Ginzburg theory is based on weakly interacting particles or very dilute systems. For strongly interacting and/or dense many-body systems, Monte-Carlo methods are often the preferred starting point. An exposition of these methods will also be given, with an emphasis on systems from solid state and atomic physics.

Giuseppe Maria Coclite
In this lecture, we consider a quantistic non - relativistic charged particle (say an electron), that is moving under the action of an external force field. It generates a wave function (solution of the Schrodinger Equation) and an electromagnetic field (solution of the Maxwell Equations). We study the steady states generated by the interaction of these two fields with the external one. In particular, we consider the case of the Hydrogen atom, where the external force field is the electromagnetic one generated by the nucleus. The equations that describes this phenomenon are the coupled Maxwell Schrodinger ones. We prove the existence of infinitely many standing waves that are rapidly decaying at infinity. Mathematically, this consists in the analysis of an eigenvalue problem for a nonlinear system of two second order elliptic equations. The main property of this system is its variational structure, namely its equations are the Euler-Lagrange ones of an Energy Functional. These results were obtained in collaboration with V. Georgiev.

Per Christian Moan

Simen Kvaal
Global propagation schemes represent a class of methods for solving the time dependent Schroedinger equation that for some reason have been neglected in the physics community, despite their simplicity, accuracy and ease of implementation. They are based on polynomial expansions of the propagator (i.e., the exponential of a skew-Hermitian operator), and different methods utilize different classes of (orthogonal) polynomials. We consider methods based on Chebyshev and Hermite polynomials in addition to the Taylor expansion. Numerical properties of the methods will presented in a rather informal way in addition to some open questions and ideas for applications and extentions, in particular approaches to time dependent problems. We will also give a brief overview of Krylov subspace methods for exponentiation and relate these to the polynomial expansions.