Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generalized potentials and obstacle scattering


Author: Richard L. Ford
Journal: Trans. Amer. Math. Soc. 329 (1992), 415-431
MSC: Primary 35P25; Secondary 35J10, 35R05
DOI: https://doi.org/10.1090/S0002-9947-1992-1042287-0
MathSciNet review: 1042287
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] S. Agmon, Spectral properties of Schràdinger operators and scattering theory, Ann. Scuola Norm Sup. Pisa 2 (1975), 151. MR 0397194 (53:1053)
  • [2] W. O. Amrein and V. Georgescu, Strong asymptotic completeness of wave operators for highly singular potentials, Helv. Phys. Acta 47 (1974), 517-533. MR 0366278 (51:2526)
  • [3] P. Constantin, Scattering for Schràdinger operators in a class of domains with noncompact boundaries, J. Funct. Anal. 44 (1981), 87-113. MR 638296 (83c:35096)
  • [4] E. B. Davies, Eigenfunction expansions for singular Schràdinger operators, Arch. Rational Mech. Anal. 63 (1976), 261-272. MR 0426690 (54:14629)
  • [5] A. Devinatz, Schràdinger operators with singular potentials, J. Operator Theory 4 (1980), 25-35. MR 587366 (82c:35056)
  • [6] R. Ford, The use of delta functions in scattering past an obstacle, Doctoral Dissertation, University of California, Irvine, 1989.
  • [7] T. Ikebe, Eigenfunction expansions associated with the Schràdinger operator and their applications to scattering theory, Arch. Rational Mech. Anal. 5 (1960), 1-34. MR 0128355 (23:B1398)
  • [8] -, On eigenfunction expansions connected with the exterior problem for the Schràdinger equation, Japan J. Math. 36 (1967), 33-35. MR 0247297 (40:566a)
  • [9] E. M. Il'in, The principle of limiting absorbtion and scattering on non-compact obstacles, Soviet Math. (Iz. Vuz) 28 (1984), 46-55. MR 739763 (87b:35127a)
  • [10] T. Kato, Schràdinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148. MR 0333833 (48:12155)
  • [11] J. Kupsch and W. Sandhas, Moeller operators for scattering on singular potentials, Comm. Math. Phys. 2 (1966), 147-154. MR 0195419 (33:3619)
  • [12] S. T. Kuroda, Scattering theory for differential operators. III: Exterior problems, Lecture Notes in Math., vol. 448, Springer-Verlag, 1975, p. 277. MR 0417595 (54:5645)
  • [13] P. Lax and R. Phillips, Scattering theory, Academic Press, 1967. MR 0217440 (36:530)
  • [14] D. B. Pearson, Time dependent scattering theory for highly singular potentials, Helv. Phys. Acta MR 0420033 (54:8050)
  • [15] A. G. Ramm, Scattering by obstacles, Reidel, Dordrecht, 1986. MR 847716 (87k:35197)
  • [16] M. Schechter, Spectra of partial differential operators, North-Holland, Amsterdam, 1986. MR 869254 (88h:35085)
  • [17] -, Operator methods in quantum mechanics, North-Holland, Amsterdam, 1981. MR 597895 (83b:81004)
  • [18] -, Selfadjoint realizations in another Hilbert space, Amer. J. Math. 106 (1984), 43-65. MR 729754 (85m:47007)
  • [19] N. A. Shenk, Eigenfunction expansions and scattering theory for the wave equation in an exterior domain, Arch. Rational Mech. Anal. 21 (1966), 120-150. MR 0187631 (32:5081)
  • [20] Y. Shizuta, Eigenfunction expansion associated with the operator $ - \Delta $ in the exterior domain, Proc. Japan Acad. 39 (1963), 656-660. MR 0195420 (33:3620)
  • [21] D. W. Thoe, Eigenfunction expansions associated with Schràdinger operators in $ {R_n},n \geq 4$, Arch. Rational Mech. Anal. 26 (1967), 335-356. MR 0218772 (36:1856)
  • [22] R. Weder and J. M. Combes, A new criterion for existence and completeness of wave operators and applications to scattering by unbounded obstacles, Comm. Partial Differential Equations 6 (1981), 1179-1223. MR 640022 (84h:35128)
  • [23] C. Wilcox, Scattering theory for the d'Alembert equation in exterior domains, Lecture Notes in Math., vol. 442, Springer-Verlag, 1975. MR 0460927 (57:918)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35P25, 35J10, 35R05

Retrieve articles in all journals with MSC: 35P25, 35J10, 35R05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1042287-0
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society