New Results on Hard Disk and Hard Ball Scattering

1 Generalities In the following I will describe new data (see gures 1-16) about the following systems: 1. A1 resonances in the two-dimensional 2-disk scattering system, 2. B1 resonances in the two-dimensional 2-disk scattering system, 3. A1 resonances in the two-dimensional 3-disk scattering system, 4. the shape resonances in the two-dimensional 1-disk scattering system, 5. m = 0 and jmj = 1 resonances in the three-dimensional 2-ball system, 6. A1 and A2 00 resonances in the three-dimensional 3-ball system. The two-dimensional results I have obtained in collaboration with GG abor Vattay (Bu-dapest) and Per E. Rosenqvist (Copenhagen) (see also refs..1, 2]), on the three-dimensional ones I have worked with Michael Henseler (Darmstadt) and in the initial stages with Thomas Guhr (Copenhagen). The scattering problems are always the ones of a point particle scattered from up to three hard, circular, equally sized (and for multi-disk or multi-ball systems) equally spaced disks or balls in two or three dimensions, respectively. Thus the corresponding wave equation is the two-or three-dimensional scalar Helmholtz equation with Dirichlet boundary conditions on the disks or balls. In all these cases I plot in the complex wave-number k plane the resonance positions as function of the real part, Re k, and imaginary part, Imk, of the wave number. Re k and Imk are measured in terms of 1=a where a is the radius of the disks in the two-dimensional cases and of the balls in the three-dimensional situation. The distance R between the centers of the disks or balls in the multi-disk or multi-ball scattering systems is always: R = 6a. In the multi-disk or multi-ball problems only the genuine resonances are plotted, that means the 1-disk or 1-ball resonances have been taken out, respectively. 2 The two-dimensional 2-disk scattering system The 2-disk scattering system is classiied according to the group C 2v by just four 1-dimensional representations: A1, A2, B1, B2. The rst two transform symmetrically with respect to reeection at the system axis joining the two centers, the latter two transform anti-symmetrically. Furthermore the A1 and B1 representations are symmetric with respect to the mirror line perpendicular to the system axis passing thru its center, the other two transform anti-symmetrically. Therefore the latter two do not involve new physics, but describe more or less the same spectrum as the former two shifted by half a unit.