On the integrated density of states for the Schrödinger operators with certain random electromagnetic potentials

Hiroyuki MATSUMOTO

doi:10.2969/jmsj/04520197

Hiroyuki MATSUMOTO

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JOURNAL FREE ACCESS

1993 Volume 45 Issue 2 Pages 197-214

DOI https://doi.org/10.2969/jmsj/04520197

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Published: 1993 Received: December 17, 1991 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323 (1981), 53-67.
2) J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, I. General interactions, Duke Math. J., 45 (1978), 847-883.
3) R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990.
4) Y. Colin de Verdière, L'asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., 105 (1986), 327-335.
5) B. Helffer and J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators, (eds. H. Holden and A. Jensen), Lecture Notes in Phys., 345, Springer, 1989, pp. 118-197.
6) W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, J. Reine. Angew. Math., 334 (1982), 141-156.
7) W. Kirsch and F. Martinelli, On the density of states of Schrödinger operators with a random potential, J. Phys. A, 15 (1982), 2139-2156.
8) W. Kirsch and F. Martinelli, On the essential selfadjointness of stochastic Schrödinger operators, Duke Math. J., 50 (1983), 1255-1260.
9) H. Kunz, The quantum hall effect for electrons in a random potential, Comm. Math. Phys., 112 (1987), 121-145.
10) E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, (eds. E. H. Lieb, et al), Princeton Series in Physics, 1976, pp. 269-303.
11) I. M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Soviet Phys. Uspekhi, 7 (1965), 549-573.
12) S. Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math., 3 (1977), 111-139.
13) L. A. Pastur, Spectra of random selfadjoint operators, Russian Math. Surveys, 28 (1973), 1-67.
14) M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-adjointness, Academic Press, London, 1975.
15) M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, London, 1978.
16) N. Ueki, Spectral asymptotics for random Schrödinger operators with magnetic fields, preprint.
17) F. Wegner, Exact density of states for lowest Landau level in white noise potential, superfield representation for interacting systems, Z. Phys. B, 51 (1983), 279-285.

Right : [1] M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math., 323 (1981), 53-67.
[2] J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields, I. General interactions, Duke Math. J., 45 (1978), 847-883.
[3] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston, 1990.
[4] Y. Colin de Verdière, L'asymptotique de Weyl pour les bouteilles magnétiques, Comm. Math. Phys., 105 (1986), 327-335.
[5] B. Helffer and J. Sjöstrand, Equation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger operators, (eds. H. Holden and A. Jensen), Lecture Notes in Phys., 345, Springer, 1989, pp. 118-197.
[6] W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, J. Reine. Angew. Math., 334 (1982), 141-156.
[7] W. Kirsch and F. Martinelli, On the density of states of Schrödinger operators with a random potential, J. Phys. A, 15 (1982), 2139-2156.
[8] W. Kirsch and F. Martinelli, On the essential selfadjointness of stochastic Schrödinger operators, Duke Math. J., 50 (1983), 1255-1260.
[9] H. Kunz, The quantum hall effect for electrons in a random potential, Comm. Math. Phys., 112 (1987), 121-145.
[10] E. H. Lieb and W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, (eds. E. H. Lieb, et al), Princeton Series in Physics, 1976, pp. 269-303.
[11] I. M. Lifshitz, Energy spectrum structure and quantum states of disordered condensed systems, Soviet Phys. Uspekhi, 7 (1965), 549-573.
[12] S. Nakao, On the spectral distribution of the Schrödinger operator with random potential, Japan. J. Math., 3 (1977), 111-139.
[13] L. A. Pastur, Spectra of random selfadjoint operators, Russian Math. Surveys, 28 (1973), 1-67.
[14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II. Fourier Analysis, Self-adjointness, Academic Press, London, 1975.
[15] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, London, 1978.
[16] N. Ueki, Spectral asymptotics for random Schrödinger operators with magnetic fields, preprint.
[17] F. Wegner, Exact density of states for lowest Landau level in white noise potential, superfield representation for interacting systems, Z. Phys. B, 51 (1983), 279-285.

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