Absence of absolutely continuous spectrum for some one dimensional random but deterministic Schrödinger operators
W. Kirsch; S. Kotani; B. Simon
Annales de l'I.H.P. Physique théorique (1985)
- Volume: 42, Issue: 4, page 383-406
- ISSN: 0246-0211
Access Full Article
topHow to cite
topReferences
top- [1] J. Avron and B. Simon, Almost periodic Schrödinger operators, II. The integrated density of states, Duke Math. J., t. 50, 1983, p. 369-391. Zbl0544.35030MR700145
- [2] J. Bellissard, R. Lima and D. Testard, A metal-insulator transition for the almost Mathieu model, Commun. Math. Phys., t. 88, 1983, p. 207-234. Zbl0542.35059MR696805
- [3] W. Craig and B. Simon, Subharmonicity of the Lyaponov index, Duke Math. J., t. 50, 1983, p. 551-560. Zbl0518.35027MR705040
- [4] V. De Alfaro and T. Regge, Potential Scattering, North Holland, Amsterdam, 1965. Zbl0141.23202MR191316
- [5] P. Deift and E. Trubowitz, Inverse Scattering on the Line, Comm. Pure Appl. Math., t. 32, 1979, p. 121-251. Zbl0388.34005MR512420
- [6] E. Dinaburg and Y. Sinai, On the one dimensional Schrödinger equation with quasi–periodic potential, Funk Anal. i Pril., t. 9, 1975, p. 8-21. Zbl0333.34014MR470318
- [7] I. Goldsheid, S. Molchanov and L. Pastur, A pure point spectrum of the stochastic and one dimensional Schrödinger equation, Funct. Anal. Appl., t. 11, 1977, p. 1-10. Zbl0368.34015MR470515
- [8] I. Herbst and J. Howland, The Stark ladder and other one-dimensional external field problems, Commun. Math. Phys., t. 80, 1981, p. 23. Zbl0473.47037MR623150
- [9] J. Keller, Discriminant, transmission coefficients and stability bands of Hill's equation, J. Math. Phys., to appear. Zbl0558.34026
- [10] W. Kirsch, On a class of random Schrödinger operators, to appear inAdv. Appl. Math. Zbl0578.60059MR789852
- [11] W. Kirsch, F. Martinelli, On the spectrum of Schrödinger operators with a random potential: Commun. Math. Phys., t. 85, 1982, p. 329-350. Zbl0506.60058MR678150
- [12] W. Kirsch, F. Martinelli, On the ergodic properties of the spectrum of general random operators: J. Reine Angew. Math., t. 334, 1982, p. 141-156. Zbl0476.60058MR667454
- [13] S. Kotani, Lyaponov Indices Determine Absolutely Continuous Spectra of Stationary Random One-Dimensional Schrödinger Operators, Proc. Stoch. Anal., Kyoto, 1982. Zbl0549.60058
- [14] S. Kotani, Support Theorems for Random Schrödinger Operators, Commun. Math. Phys., to appear. Zbl0573.60054MR778625
- [15] H. Kunz and B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Commun. Math. Phys., t. 78, 1980, p. 201-246. Zbl0449.60048MR597748
- [16] W. Magnus and S. Winkler, Hill's Equation, Interscience, 1966; Dover Edition available. Zbl0158.09604
- [17] N. Mott and W. Twose, The theory of impurity conduction, Adv. in Physics, t. 10, 1961, p. 107-155.
- [18] R. Newton, Inverse scattering by a local impurity in a periodic potential in one dimension. J. Math. Phys., t. 24, 1983, p. 2152. Zbl0524.34026MR713548
- [19] M. Reed and B. Simon, Methods of modern mathematical physics. III. Scattering theory, Academic Press, New York, 1979. Zbl0405.47007MR529429
- [20] M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978. Zbl0401.47001MR493421
- [21] B. Simon, Kotani theory for one dimensional stochastic Jacobi matrices. Comm. Math. Phys., t. 89, 1983, p. 227. Zbl0534.60057MR709464
- [22] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw–Hill, New York, 1955. Zbl0064.33002MR69338
- [23] N. Dunford and J. Schwartz, Linear Operators, Vol. II, Wiley, New York, 1963. Zbl0128.34803MR188745
- [24] T. Kato, Schrödinger operators with singular potentials. Israel J. Math., t. 13, 1973, p. 135-148. Zbl0246.35025MR333833
- [25] W. Kirsch and F. Martinelli, On the essential selfadjointness of stochastic Schrödinger operators, Duke Math. J., t. 50, 1983, p. 1255-1260. Zbl0543.60069MR726328