Absolutely continuous spectrum of Stark type operators
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A. A. Pozharskiĭ
Translated by: B. M. Bekker - St. Petersburg Math. J. 20 (2009), 473-492
- DOI: https://doi.org/10.1090/S1061-0022-09-01057-7
- Published electronically: April 8, 2009
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Abstract:
Some new conditions are obtained for the absolutely continuous spectrum of a Stark operator to fill the entire real line.References
- V. S. Buslaev and L. D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator, Soviet Math. Dokl. 1 (1960), 451–454. MR 0120417
- S. N. Naboko and A. B. Pushnitskiĭ, A point spectrum, lying on a continuous spectrum, for weakly perturbed operators of Stark type, Funktsional. Anal. i Prilozhen. 29 (1995), no. 4, 31–44, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 29 (1995), no. 4, 248–257 (1996). MR 1375539, DOI 10.1007/BF01077472
- A. Zygmund, Trigonometric series. Vol. I, II, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR 1963498
- A. A. Pozharskiĭ, On operators of Wannier-Stark type with singular potentials, Algebra i Analiz 14 (2002), no. 1, 158–193 (Russian); English transl., St. Petersburg Math. J. 14 (2003), no. 1, 119–145. MR 1893324
- A. A. Pozharskiĭ, On the nature of the spectrum of the Stark-Wannier operator, Algebra i Analiz 16 (2004), no. 3, 171–200 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 3, 561–581. MR 2083569, DOI 10.1090/S1061-0022-05-00865-4
- 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
- V. S. Buslaev, Kronig-Penney electron in a homogeneous electric field, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 45–57. MR 1730502, DOI 10.1090/trans2/189/04
- Michael Christ and Alexander Kiselev, Absolutely continuous spectrum of Stark operators, Ark. Mat. 41 (2003), no. 1, 1–33. MR 1971938, DOI 10.1007/BF02384565
- P. Deift and R. Killip, On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), no. 2, 341–347. MR 1697600, DOI 10.1007/s002200050615
- D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30–56. MR 915965, DOI 10.1016/0022-247X(87)90212-5
- Yoram Last and Barry Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329–367. MR 1666767, DOI 10.1007/s002220050288
- Galina Perelman, On the absolutely continuous spectrum of Stark operators, Comm. Math. Phys. 234 (2003), no. 2, 359–381. MR 1962465, DOI 10.1007/s00220-002-0776-2
Bibliographic Information
- A. A. Pozharskiĭ
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Prospekt 28, Petrodvorets, St. Petersburg, 198504, Russia
- Email: pozharsky@math.nw.ru
- Received by editor(s): October 31, 2006
- Published electronically: April 8, 2009
- Additional Notes: Supported by RFBR (grant nos. 05-01-01076 and 05-01-02944)
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 473-492
- MSC (2000): Primary 34L40
- DOI: https://doi.org/10.1090/S1061-0022-09-01057-7
- MathSciNet review: 2454457