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Journal of Mathematical Physics

Nondiffractive scattering: Scattering from kaleidoscopes

Bill Sutherland

Published Jul 1, 1980• 23 citations
Publisher

Abstract

We introduce and exactly solve a series of multidimensional scattering problems. The potential is a generalization of the delta function potential and has the symmetry of a kaleidoscope. The solution demonstrates the property of nondiffractive scattering. The bound state energies are calculated exactly for all cases. We prove a remarkable identity for certain finite groups.

DOI
10.1063/1.524628

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We introduce and exactly solve a series of multidimensional scattering problems. The potential is a generalization of the delta function potential and has the symmetry of a kaleidoscope. The solution demonstrates the property of nondiffractive scattering. The bound state energies are calculated exactly for all cases. We prove a remarkable identity for certain finite groups.
The exact solution to the one-dimensional Schr\"odinger equation which describes the motion of a particle scattering off a finite set of $\ensuremath{\delta}$-function potentials of arbitrary strengths and positions is found. The behavior of the TTT matrix both on and off the energy shell is also precisely given. Localization of states is discussed.
In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.