B. Giraud (CEA, Saclay)

NEW FAMILY OF ORTHOGONAL POLYNOMIALS:

The Hohenberg-Kohn theorem (existence of a density functional) states that the energy of a system depends directly on the density rather than on the potentialdriving the particles. To study the functional it is necessary to make variations of the density which are particle number conserving and variations of the potential which are orthogonal to a flat potential. To expand the variations we construct a basis of orthogonal states. These states involve special polynomials which are compatible with the constraints.

TAMING OF THE SHREW RESONANCES FOR COMPLETENESS:
Annals of Physics 308 (2003) 115-142

Long lived states are narrow resonances and quantum mechanics describes them by non square-integrable wave functions (Gamow states), which are a nuisance. This lecture briefly describes a system of two coupled-channel equations and its Jost and regular solutions, in order to generate the Green's function Gand a specific contour integral of G. The point is, a transformation $r -> r exp(i\theta),$ $p -> p exp(-i\theta),$ $H -> H exp(2i\theta)$ (complex scaling) was performed, and that this system of equations does not corrrespond to a Hermitian problem. But the system now describes narrow resonances by square integrable wave functions, and we prove that these can be included in a suitable resolution of the identity.