Generalized potentials and obstacle scattering

Ford, Richard L.

doi:10.1090/S0002-9947-1992-1042287-0

Generalized potentials and obstacle scattering
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by Richard L. Ford PDF

Trans. Amer. Math. Soc. 329 (1992), 415-431 Request permission

Abstract:

Potential scattering theory is a very well-developed and understood subject. Scattering for Schràdinger operators represented formally by $- \Delta + V$, where $V$ is a generalized function such as a $\delta$-function, is less understood and requires form perturbation techniques. A general scattering theory for a large class of such singular perturbations of the Laplacian is developed. The theory has application to obstacle scattering. One considers an alternative mathematical model of an obstacle in ${{\mathbf {R}}^n}$. Instead of representing the obstacle by deleting the region inhabited by the obstacle from ${{\mathbf {R}}^n}$, the surface of the obstacle is treated as impenetrable. The impenetrable surface is understood to be the limiting case of a sequence of highly spiked potentials whose support converges to the surface of the obstacle and whose peaks grow without bound. The limiting case is identified as a $\delta$-function acting on the surface of the obstacle. Hamiltonians for the limiting case are constructed and the conditions governing the existence and completeness of the associated wave operators are determined through application of the general theory.

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