Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.

Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Michael Demuth and M. Krishna
Title: Determining spectra in quantum theory
Additional book information: Progress in Mathematical Physics, vol. 44, Birkhäuser, Boston, 2005, x+219 pp., ISBN 978-0-8176-4366-9, US$99.00

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

  • 1. Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
  • 2. N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equations, Amer. J. Math. 79 (1957), 597–610. MR 0088623,
  • 3. Jiří Blank, Pavel Exner, and Miloslav Havlíček, Hilbert space operators in quantum physics, AIP Series in Computational and Applied Mathematical Physics, American Institute of Physics, New York, 1994. MR 1275370
  • 4. René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675
  • 5. H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
  • 6. E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825
  • 7. Michael Demuth and Jan A. van Casteren, Stochastic spectral theory for selfadjoint Feller operators, Probability and its Applications, Birkhäuser Verlag, Basel, 2000. A functional integration approach. MR 1772266
  • 8. Jan Dereziński and Christian Gérard, Scattering theory of classical and quantum 𝑁-particle systems, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997. MR 1459161
  • 9. William F. Donoghue Jr., On the perturbation of spectra, Comm. Pure Appl. Math. 18 (1965), 559–579. MR 0190761,
  • 10. Kurt O. Friedrichs, Perturbation of spectra in Hilbert space, Marc Kac, editor. Lectures in Applied Mathematics (Proceedings of the S ummer Seminar, Boulder, Colorado, (1960), Vol. III, American Mathematical Society, Providence, R.I., 1965. MR 0182883
  • 11. Stephen J. Gustafson and Israel Michael Sigal, Mathematical concepts of quantum mechanics, Universitext, Springer-Verlag, Berlin, 2003. MR 2002159
  • 12. W. Heisenberg: Z. Physik 33, 897 (1925); 43, 172 (1927); see also The Physical Principles of Quantum Theory, Chicago: University of Chicago Press, 1930.
  • 13. P. D. Hislop and I. M. Sigal, Introduction to spectral theory, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. With applications to Schrödinger operators. MR 1361167
  • 14. Vojkan Jakšić and Yoram Last, Spectral structure of Anderson type Hamiltonians, Invent. Math. 141 (2000), no. 3, 561–577. MR 1779620,
  • 15. A. Jensen and M. Krishna, New criteria to identify spectrum, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 2, 217–226. MR 2142467,
  • 16. Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
  • 17. Shinichi Kotani, One-dimensional random Schrödinger operators and Herglotz functions, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985) Academic Press, Boston, MA, 1987, pp. 219–250. MR 933826
  • 18. John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, 1955. Translated by Robert T. Beyer. MR 0066944
  • 19. Leonid Pastur and Alexander Figotin, Spectra of random and almost-periodic operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR 1223779
  • 20. M. Reed, B. Simon: Methods of Modern Mathematical Physics, volumes 1-4, Academic Press, New York, 1980 (vol. 1, second ed.), 1975, 1979, 1978.
  • 21. Franz Rellich, Perturbation theory of eigenvalue problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0240668
  • 22. E. Schrödinger: Quantisation as a problem of proper values, Ann. Physik (1926), Part I 79, 361; Part II 79, 489; Part III 80, 437; Part IV 81, 109; reprinted in Collected Papers on Wave Mechanics, London, Glasgow, Blackie & Son Limited, 1928.
  • 23. E. Schrödinger: On the relation between the quantum mechanics of Heisenberg, Born, and Jordan, and that of Schrödinger, Ann. Physik 79 (1926).
  • 24. Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153
  • 25. Barry Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995
  • 26. Barry Simon and Tom Wolff, Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), no. 1, 75–90. MR 820340,
  • 27. Peter Stollmann, Caught by disorder, Progress in Mathematical Physics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 2001. Bound states in random media. MR 1935594
  • 28. M. Stone: Linear Transformation on a Hilbert Space and Applications to Analysis, AMS Colloquium Publications vol. XV, Amer. Math. Soc., Providence RI, 1966.
  • 29. H. Weyl: The Theory of Groups and Quantum Mechanics, Princeton University Press, Princeton NY, 1931.
  • 30. D. Yafaev: Mathematical Scattering Theory, Amer. Math. Soc., Providence RI, 1992.

Review Information:

Reviewer: Peter D. Hislop
Affiliation: University of Kentucky
Journal: Bull. Amer. Math. Soc. 45 (2008), 469-477
MSC (2000): Primary 81C10; Secondary 35P05, 47A10
Published electronically: April 21, 2008
Review copyright: © Copyright 2008 American Mathematical Society
American Mathematical Society