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Inverse scattering by a local impurity in a periodic potential in one dimension

J. Math. Phys. 24, 2152 (1983); doi:10.1063/1.525968

Issue Date: August 1983

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Roger G. Newton
Physics Department, Indiana University, Bloomington, Indiana 47405
Hill's equation, modified by a potential that vanishes as x --> ±[infinity], is considered. The direct scattering problem is studied; analytic and asymptotic properties of solutions of Hill's equation as well as of solutions of the modified equation are established. A new version of Levinson's theorem is proved. The inverse scattering problem is solved by means of a Marchenko-like equation. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.
History: Received 12 October 1982; accepted 25 March 1983
Permalink: http://link.aip.org/link/?JMAPAQ/24/2152/1
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KEYWORDS and PACS

Keywords
PACS
  • 03.65.Nk
    Classical and quantum physics: mechanics and fields Quantum theory quantum mechanics Nonrelativistic scattering theory
  • 72.10.Fk
    Electronic transport in condensed matter Theory of electronic transport; scattering mechanisms Scattering by point defects, dislocations, surfaces, and other imperfections (including Kondo effect)
  • YEAR: 1983

PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (20)
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  1. T. H. Dupree, Ann. Phys. (N.Y.) 15, 63 (1961);
  2. G. J. Morgan, Proc. Phys. Soc. 89, 365 (1966);
  3. N. A. W. Holzwarth, Phys. Rev. B 11, 3718 (1975).
  4. P. Ungar, Comm. Pure Appl. Math. 14, 707 (1961);
  5. H. Hochstadt, Arch. Rat. Mech. Anal. 19, 353 (1965);
  6. W. Magnus and S. Winkler, Hill's Equation (Interscience, New York, 1966);
  7. H. P. McKean and P. van Moerbeke, Invenciones Math. 30, 217 (1975);
  8. E. Trubowitz, Comm. Pure Appl. Math. 30, 321 (1977).
  9. L. D. Faddeev, T. Mat. Inst. Steklov 73, 314 (1964)
  10. [Am. Math. Soc. Transl. 2, 139 (1964)];
  11. P. Deift and E. Trubowitz, Comm. Pure Appl. Math. 32, 121 (1979).
  12. R. G. Newton, J. Math. Phys. 21, 493 (1980).
  13. N. Levinson, K. Dan. Vidensk. Selsk. Mat.-Phys. Medd. 25(9) (1949).
  14. R. G. Newton, J. Math. Phys. 21, 1698 (1980);
    15. 22, 631 (1981);
    23, 693 (1982).
  15. R. G. Newton, J. Math. Phys. 22, 2191 (1981);
    16. 23, 693 (1982).
  16. R. G. Newton, J. Math. Phys. 23, 594 (1982).
  17. R. G. Newton, J. Math. Phys. 23, 2257 (1982).
  18. A tilde denotes the transpose.
  19. An asterisk denotes the complex conjugate.
  20. A dagger denotes the adjoint.
  21. This is known as the reciprocity theorem.
  22. C± mean the upper and lower half-planes, not including the real axis.
  23. See, for example, Magnus and Winkler, Ref. 2.
  24. E. Trubowitz, Ref. 2.
  25. B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory (Am. Math. Soc., Providence, RI, 1975), p. 21.
  26. H. Hochstadt, Ref. 2.
  27. For results on the distribution of bound states among the band gaps, see F. S. Rofe-Beketov, Dokl. Akad. Nauk SSSR 156, 1029 (1964)
    28. [Sov. Math. 5, 772 (1964)];
    V. A. Zheludev, in Topics in Mathematical Physics, edited by M. Sh. Birman (Consultants Bureau, New York, 1968, 1971), Vol. 2, p. 87 and Vol. 4, p. 55;
    M. Klaus, Helv. Phys. Acta 55, 49 (1982).
  28. The statement of the Levinson theorem in Ref. 4, Sec. 3, is in error. When T(0) = 0, which is the generic case, then delta(0)−delta([infinity]) = pi(n−(1/2)); when there is a half-bound state, i.e., T(0) = 1, then delta(0)−delta([infinity]) = pin.

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