Section: New Results

Analysis and numerical simulation of the Schrödinger equation

The linear or nonlinear Schrödinger equation with potential is one of the basic equations of quantum mechanics and it arises in many areas of physical and technological interest, e.g. in quantum semiconductors, in electromagnetic wave propagation, and in seismic migration. The Schrödinger equation is the lowest order one-way approximation (paraxial wave equation) to the Helmholtz equation and is called Fresnel equation in optics, or standard parabolic equation in underwater acoustics. The solution of the equation is defined on an unbounded domain. If one wants to solve such a whole space evolution problem numerically, one has to restrict the computational domain by introducing artificial boundary conditions. So, the objective is to approximate the exact solution of the whole-space problem, restricted to a finite computational domain. A review article [39] was written this year to describe and compare the different current approaches of constructing and discretizing the transparent boundary conditions in one and two dimensions. However, these approaches are limited to the linear case (or nonlinear with the classical cubic nonlinearity: an article written was dedicated to this case this year [41] ) and constant potentials. Therefore, in collaboration with X. Antoine (IECN Nancy and Inria Lorraine), we proposed to P. Klein to study, in her PhD thesis, the case of the Schrödinger equation with variable potentials. The study of the non-stationary one-dimensional case has already led to one publication [40] and some preliminary results in the stationary case are really promising. These cases are relevant since for example the equations appear in the Bose Einstein condensate with a quadratic potential.

This problem is obviously not limited to the Schrödinger equation and new developments are in progress on the Korteweg de Vries equation with M. Ehrhardt. This equation is more difficult to study due to its third order derivative in space.

Dispersive equations, such as the Schrödinger equation are also considered as boundary-value problems. For example, in [57] , G.D. studies the long time asymptotics of the solutions of linear Schrodinger equations considered as initial-boundary value problems on the half-line and on bounded intervals when the boundary data are periodic functions of time. G.D. obtains theoretical results using a transformation method introduced by T. Fokas and provides several numerical experiments to support them.

The Diffusion Monte-Carlo method is a powerful strategy used by chemists to estimate the groundstate energy of a N-body Schrödinger Hamiltonian with high accuracy.However, the method suffers from two major limitations :

• The quantity of physical interest is more the energy variation with respect to a parameter than the energy itself.

• The case of Fermions (as electrons of atoms) relies on constraining the random walkers in some nodal pocket (which imposes skew-symmetry on the distribution of walkers) which is only approximately known (the Fixed Node Approximation).

In the article [63] , Mathias Rousset proposed a new strategy for the case of Fermions to compute the energy variation with respect to a variation of the nodal pocket. The ultimate goal is to design a Monte-Carlo strategy able to optimize the Fixed Node Approximation.