Book Review
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MathSciNet review:
3077139
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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$99.00$
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
N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equations, Amer. J. Math. 79 (1957), 597–610. MR 88623, DOI 10.2307/2372564
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
René Carmona and Jean Lacroix, Spectral theory of random Schrödinger operators, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1102675, DOI 10.1007/978-1-4612-4488-2
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
E. B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR 1349825, DOI 10.1017/CBO9780511623721
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, DOI 10.1007/978-3-0348-8460-0
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, DOI 10.1007/978-3-662-03403-3
William F. Donoghue Jr., On the perturbation of spectra, Comm. Pure Appl. Math. 18 (1965), 559–579. MR 190761, DOI 10.1002/cpa.3160180402
Kurt O. Friedrichs, Perturbation of spectra in Hilbert space, Lectures in Applied Mathematics, American Mathematical Society, Providence, R.I., 1965. Marc Kac, editor. MR 0182883
Stephen J. Gustafson and Israel Michael Sigal, Mathematical concepts of quantum mechanics, Universitext, Springer-Verlag, Berlin, 2003. MR 2002159, DOI 10.1007/978-3-642-55729-3
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.
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, DOI 10.1007/978-1-4612-0741-2
Vojkan Jakšić and Yoram Last, Spectral structure of Anderson type Hamiltonians, Invent. Math. 141 (2000), no. 3, 561–577. MR 1779620, DOI 10.1007/s002220000076
A. Jensen and M. Krishna, New criteria to identify spectrum, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 2, 217–226. MR 2142467, DOI 10.1007/BF02829628
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
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
John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, N.J., 1955. Translated by Robert T. Beyer. MR 0066944
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, DOI 10.1007/978-3-642-74346-7
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.
Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. 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).
Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
Barry Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005. MR 2105995, DOI 10.1090/chel/351
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, DOI 10.1002/cpa.3160390105
Peter Stollmann, Caught by disorder, Progress in Mathematical Physics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 2001. Bound states in random media. MR 1935594, DOI 10.1007/978-1-4612-0169-4
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.
- 1.
- M. Aizenman, S. Molchanov: Localization at large disorder and at extreme energies: an elementary derivation, Commun. Math. Phys. 157 (1993), 245-278. MR 1244867
- 2.
- N. Aronszajn: On a problem of Weyl, Amer. J. Math. 79 (1957), 596-620. MR 0088623
- 3.
- J. Blank, P. Exner, M. Havlíček: Hilbert Space Operators in Quantum Physics, American Institute of Physics, New York, 1994. MR 1275370
- 4.
- R. Carmona, J. Lacroix: Spectral Theory of Random Schrödinger Operators, Birkhaüser Boston, 1990. MR 1102675
- 5.
- H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon: Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 0883643
- 6.
- E. B. Davies: Spectral Theory and Differential Operators, Cambridge University Press, Cambridge, 1995. MR 1349825
- 7.
- M. Demuth, J. van Casteren: Stochastic Spectral Theory for Selfadjoint Feller Operators, Birkhäuser, Basel, 2000. MR 1772266
- 8.
- J. Dereziński, C. Gérard: Scattering Theory of Classical and Quantum
-Particle Systems, Texts and Monographs in Physics, Springer, Berlin, 1997. MR 1459161
- 9.
- W. F. Donoghue, Jr.: On the perturbation of spectra, Commun. Pure Appl. Math. 18 (1965), 559-579. MR 0190761
- 10.
- K. Friedrichs: Perturbation of Spectra in Hilbert Space, Amer. Math. Soc., Providence, RI, 1965. MR 0182883
- 11.
- S. Gustafson, I. M. 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, I. M. Sigal: Introduction to Spectral Theory with Applications to Schrödinger Operators, Applied Mathematics Series vol. 113, Springer, New York, 1996. MR 1361167
- 14.
- V. Jaksić, Y. Last: Spectral properties of Anderson type operators, Inventiones Math. 141 (2000), 561-577. MR 1779620
- 15.
- A. Jensen, M. Krishna: New criteria to identify spectra, Proc. Ind. Acad. Sci. 115 (2005), 1-10. MR 2142467
- 16.
- T. Kato: Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976. MR 0407617
- 17.
- S. Kotani: One-dimensional random Schrödinger operators and Herglotz functions, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985), 219-250, Academic Press, Boston, MA, 1987. MR 0933826
- 18.
- J. von Neumann: The Mathematical Foundations of Quantum Mechanics, Investigations in Physics, vol. 2, Princeton: Princeton University Press, 1955. MR 0066944
- 19.
- L. Pastur, A. Figotin: Spectra of Random and Almost-Periodic Operators, 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.
- F. Rellich: Perturbation Theory of Eigenvalue Problems, assisted by J. Berkowitz, 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.
- B. Simon: Trace Ideals and Their Applications, second edition, Mathematical Surveys and Monographs vol. 120, Amer. Math. Soc., Providence, RI, 2005. MR 2154153
- 25.
- B. Simon: Functional Integration and Quantum Physics, second edition, AMS Chelsea Publishing, Providence, RI, 2000. MR 2105995
- 26.
- B. Simon, T. Wolff: Singular continuous spectrum under rank one perturbations and localization for random Hamiltonians, Comm. Pure Appl. Math. 39 (1986), 75-90. MR 0820340
- 27.
- P. Stollmann: Caught by Disorder, Bound States in Random Media, Progress in Mathematical Physics Vol. 20, Birkhäuser Boston, 2001. 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
Email:
hislop@ms.uky.edu
Journal:
Bull. Amer. Math. Soc.
45 (2008), 469-477
DOI:
https://doi.org/10.1090/S0273-0979-08-01193-2
Published electronically:
April 21, 2008
Review copyright:
© Copyright 2008
American Mathematical Society