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On the spectrum of the Wannier-Stark operator


Author: A. A. Pozharskii
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 16 (2004), nomer 3.
Journal: St. Petersburg Math. J. 16 (2005), 561-581
MSC (2000): Primary 34L40
Published electronically: May 2, 2005
MathSciNet review: 2083569
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  • [AGZ] J. Avron, L. Gunter, and J. Zak, Energy uncertainty in ``Stark ladder'', Solid State Comm. 16 (1975), no. 2, 189-191.
  • [AZ] J. E. Avron and J. Zak, Instability of the continuous spectrum: the 𝑁-band Stark ladder, J. Mathematical Phys. 18 (1977), no. 5, 918–921. MR 0438948
  • [B1] 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
  • [DK] 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, 10.1007/s002200050615
  • [DSS] François Delyon, Barry Simon, and Bernard Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 3, 283–309 (English, with French summary). MR 797277
  • [E] P. Exner, The absence of the absolutely continuous spectrum for 𝛿’ Wannier-Stark ladders, J. Math. Phys. 36 (1995), no. 9, 4561–4570. MR 1347099, 10.1063/1.530908
  • [GP] 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, 10.1016/0022-247X(87)90212-5
  • [K] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
  • [LS] 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, 10.1007/s002220050288
  • [NN] A. Nenciu and G. Nenciu, Dynamics of Bloch electrons in external electric fields. I. Bounds for interband transitions and effective Wannier Hamiltonians, J. Phys. A 14 (1981), no. 10, 2817–2827. MR 629327
  • [BS] M. Š. Birman and M. Z. Solomjak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
    M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782
  • [B2] V. S. Buslaev, Adiabatic perturbation of a periodic potential, Teoret. Mat. Fiz. 58 (1984), no. 2, 233–243 (Russian, with English summary). MR 743409
  • [Ba] M. V. Buslaeva, The one-dimensional Schrödinger operator with accelerating potential, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 65–66 (Russian). MR 739094
  • [BD1] V. S. Buslaev and L. A. Dmitrieva, Adiabatic perturbation of a periodic potential. II, Teoret. Mat. Fiz. 73 (1987), no. 3, 430–442 (Russian, with English summary). MR 939788
  • [BD2] V. S. Buslaev and L. A. Dmitrieva, A Bloch electron in an external field, Algebra i Analiz 1 (1989), no. 2, 1–29 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 2, 287–320. MR 1025153
  • [BF] 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
  • [Z] A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. MR 933759
  • [L] B. M. Levitan, Obratnye zadachi Shturma-Liuvillya, “Nauka”, Moscow, 1984 (Russian). MR 771843
    B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
  • [P] 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

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Additional Information

A. A. Pozharskii
Affiliation: Department of Physics, St. Petersburg State University, Ulyanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia
Email: lehman@sbor.net

DOI: https://doi.org/10.1090/S1061-0022-05-00865-4
Keywords: Schr\"odinger operator, essentially selfadjoint operator, continuous spectrum
Received by editor(s): August 10, 2003
Published electronically: May 2, 2005
Article copyright: © Copyright 2005 American Mathematical Society