On the spectrum of the Wannier–Stark operator

Pozharskiĭ, A.

doi:10.1090/S1061-0022-05-00865-4

On the spectrum of the Wannier–Stark operator
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by A. A. Pozharskiĭ
Translated by: B. M. Bekker

St. Petersburg Math. J. 16 (2005), 561-581

DOI: https://doi.org/10.1090/S1061-0022-05-00865-4

Published electronically: May 2, 2005

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References

J. Avron, L. Gunter, and J. Zak, Energy uncertainty in “Stark ladder”, Solid State Comm. 16 (1975), no. 2, 189–191.
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 438948, DOI 10.1063/1.523360
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
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
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
P. Exner, The absence of the absolutely continuous spectrum for $\delta ’$ Wannier-Stark ladders, J. Math. Phys. 36 (1995), no. 9, 4561–4570. MR 1347099, DOI 10.1063/1.530908
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
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
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
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
M. Š. Birman and M. Z. Solomjak, Spektral′naya teoriya samosopryazhennykh operatorov v gil′bertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
V. S. Buslaev, Adiabatic perturbation of a periodic potential, Teoret. Mat. Fiz. 58 (1984), no. 2, 233–243 (Russian, with English summary). MR 743409
M. V. Buslaeva, The one-dimensional Schrödinger operator with accelerating potential, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 65–66 (Russian). MR 739094
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
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
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
A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. MR 933759
B. M. Levitan, Obratnye zadachi Shturma-Liuvillya, “Nauka”, Moscow, 1984 (Russian). MR 771843
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