Scattering by magnetic fields
Author:
D. R. Yafaev
Translated by:
the author
Original publication:
Algebra i Analiz, tom 17 (2005), nomer 5.
Journal:
St. Petersburg Math. J. 17 (2006), 875-895
MSC (2000):
Primary 47A40, 81U05
DOI:
https://doi.org/10.1090/S1061-0022-06-00934-4
Published electronically:
July 27, 2006
MathSciNet review:
2241430
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Abstract | References | Similar Articles | Additional Information
Abstract: Consider the scattering amplitude ,
,
, corresponding to an arbitrary short-range magnetic field
,
. This is a smooth function of
and
away from the diagonal
, but it may be singular on the diagonal. If
, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac
-function and a singular denominator. Such a structure is typical of the long-range magnetic scattering. This phenomenon is referred to as the long-range Aharonov-Bohm effect. On the contrary, for
scattering is essentially of a short-range nature, although, for example, the magnetic potential
such that
and
, decays at infinity as
only. More precisely, it is shown that, up to the diagonal Dirac function (times an explicit function of
), the scattering amplitude has only a weak singularity in the forward direction
. The approach is based on the construction (in the dimension
) of a short-range magnetic potential
corresponding to a given short-range magnetic field
.
- 1. G. N. Afanasiev, Topological effects in quantum mechanics, Fundamental Theories of Physics, vol. 107, Kluwer, Acad. Publ. Group, Dordrecht, 1999. MR 1730630 (2001c:81098)
- 2. Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in the quantum theory, Phys. Rev. (2) 115 (1959), 485-491. MR 0110458 (22:1336)
- 3. R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 169 (1995), 237-259. MR 1329195 (96a:81026)
- 4. W. C. Henneberger, Some aspects of the Aharonov-Bohm effect, Phys. Rev. A (3) 22 (1980), 1383-1388. MR 0587271 (81j:81004)
- 5.
I. Herbst, On the connectedness structure of the Coulomb
-matrix, Comm. Math. Phys. 35 (1974), 181-191. MR 0351325 (50:3814)
- 6. M. Loss and B. Thaller, Scattering of particles by long-range magnetic fields, Ann. Physics 176 (1987), 159-180. MR 0893482 (88f:81133)
- 7. M. Peshkin and A. Tonomura, The Aharonov-Bohm effect, Lecture Notes in Phys., vol. 340, Springer-Verlag, Berlin, 1989. MR 1016149 (91c:81004)
- 8. P. Roux, La matrice de diffusion pour l'opérateur de Schrödinger avec un potentiel électro- magnétique à longue portée, Ph. D. thesis, Univ. de Rennes-1, 2001.
- 9. P. Roux and D. R. Yafaev, On the mathematical theory of the Aharonov-Bohm effect, J. Phys. A 35 (2002), 7481-7492. MR 1945524 (2003m:81085)
- 10. -, The scattering matrix for the Schrödinger operator with a long-range electromagnetic potential, J. Math. Phys. 44 (2003), 2762-2786. MR 1982788 (2004i:81260)
- 11. S. N. M. Ruijsenaars, The Aharonov-Bohm effect and scattering theory, Ann. Physics 146 (1983), 1-34. MR 0701261 (85e:81117)
- 12. M. A. Shubin, Pseudodifferential operators and spectral theory, ``Nauka'', Moscow, 1978; English transl., Springer-Verlag, Berlin, 1987. MR 0509034 (80h:47057); MR 0883081 (88c:47105)
- 13. D. R. Yafaev, On the classical and quantum Coulomb scattering, J. Phys. A 30 (1997), 6981-6992. MR 1481360 (99b:81259)
- 14. -, Scattering theory: Some old and new problems, Lecture Notes in Math., vol. 1735, Springer-Verlag, Berlin, 2000. MR 1774673 (2001j:81248)
- 15. -, High-energy and smoothness asymptotic expansion of the scattering amplitude, J. Funct. Anal. 202 (2003), 526-570. MR 1990537 (2004f:35134)
- 16. -, Scattering matrix for magnetic potentials with Coulomb decay at infinity, Integral Equations Operator Theory 47 (2003), 217-249. MR 2002666 (2004f:47018)
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Additional Information
D. R. Yafaev
Affiliation:
IRMAR, Université Rennes-I, Campus Beaulieu, 35042 Rennes Cedex, France
Email:
yafaev@univ-rennes1.fr
DOI:
https://doi.org/10.1090/S1061-0022-06-00934-4
Keywords:
Magnetic fields,
scattering matrix,
gauge transformations,
long-range Aharonov--Bohm effect
Received by editor(s):
January 20, 2005
Published electronically:
July 27, 2006
Dedicated:
Dedicated to the memory of Ol$^{′}$ga Aleksandrovna Ladyzhenskaya
Article copyright:
© Copyright 2006
American Mathematical Society