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On the spectrum of the Wannier-Stark operator. (English. Russian original) Zbl 1071.81041

St. Petersbg. Math. J. 16, No. 3, 561-581 (2005); translation from Algebra Anal. 16, No. 3, 171-200 (2004).
In the paper it is considered the one-dimensional Schrödinger operator \[ H=-\frac{d^2}{dx^2}-Fx+p(x) \] in \(L_2(\mathbb{R}_+)\) with domain dom\((H)=\{ \varphi: \varphi'\) is absolutely continuous, \(\varphi(0)=0\), supp\((\varphi)\) is bounded, \(-\varphi''+p\varphi\in L_2(\mathbb{R}_+) \}. \) Here the potential \(p(x)\) is a periodic real valued function such that \(p(x)\in L_1[0,1]\). It is proved that \(H\) is essentially selfadjoint. By \(H_d\) it is denoted the closure of \(H\).
The result of the author’s previous paper [St. Petersbg. Math. J. 14, 119–145 (2003; Zbl 1044.81041)] is generalized. This can be seen putting \[ r(l)=e^{i\frac{3\pi}{4}}\frac{1}{\sqrt{2F}}\int_0^1p(t)e^{2i\pi lt}\,dt, \qquad w(l)=\int_0^1\int_0^tp(y)p(t)e^{2i\pi lt}\,dy\,dt. \] The main result of the paper is the following
Theorem. If \(p(x)\) satisfies the conditions \[ \sum_{l=1}^{\infty}| r(l)| l^{-1}<\infty, \qquad \sum_{l=1}^{\infty}| w(l)| l^{-1}<\infty, \] then the absolute continuous spectrum of \(H_d\) fills the real axis.
It should be noted that the proof of the theorem is based on the results of the mentioned paper. In the proof also the methods of the following papers are used: [V. S. Buslaev, Differential operators and Spectral Theory, 45–57 (1999; Zbl 0935.34072)], [P. Deift and R. Killip, Commun. Math. Phys. 189, 341–347 (1999; Zbl 0934.34075), D. J. Gilbert and D. B. Pearson, J. Math. Anal. Appl. 128, 30–56 (1987; Zbl 0666.34023), Y. Last and B. Simon, Invent. Math. 135, 329–367 (1999; Zbl 0931.34066)].

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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[1] J. Avron, L. Gunter, and J. Zak, Energy uncertainty in “Stark ladder”, Solid State Comm. 16 (1975), no. 2, 189-191.
[2] J. E. Avron and J. Zak, Instability of the continuous spectrum: the \?-band Stark ladder, J. Mathematical Phys. 18 (1977), no. 5, 918 – 921.
[3] 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. · Zbl 0935.34072
[4] 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. · Zbl 0934.34075
[5] 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). · Zbl 0579.60056
[6] P. Exner, The absence of the absolutely continuous spectrum for \?’ Wannier-Stark ladders, J. Math. Phys. 36 (1995), no. 9, 4561 – 4570. · Zbl 0884.47049
[7] 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. · Zbl 0666.34023
[8] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009
[9] 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. · Zbl 0931.34066
[10] 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.
[11] Спектрал\(^{\приме}\)ная теория самосопряженных операторов в гил\(^{\приме}\)бертовом пространстве, Ленинград. Унив., Ленинград, 1980 (Руссиан). М. Ш. Бирман анд М. З. Соломјак, Спецтрал тхеоры оф селфађоинт операторс ин Хилберт спаце, Матхематицс анд иц Апплицатионс (Совиет Сериес), Д. Реидел Публишинг Цо., Дордречт, 1987. Транслатед фром тхе 1980 Руссиан оригинал бы С. Хрущёв анд В. Пеллер.
[12] V. S. Buslaev, Adiabatic perturbation of a periodic potential, Teoret. Mat. Fiz. 58 (1984), no. 2, 233 – 243 (Russian, with English summary). · Zbl 0534.34064
[13] M. V. Buslaeva, The one-dimensional Schrödinger operator with accelerating potential, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 65 – 66 (Russian). · Zbl 0549.34027
[14] 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). · Zbl 0643.34068
[15] 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. · Zbl 0714.34128
[16] V. S. Buslaev and L. D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator, Soviet Math. Dokl. 1 (1960), 451 – 454. · Zbl 0129.06501
[17] A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. · Zbl 0628.42001
[18] Обратные задачи Штурма-Лиувилля, ”Наука”, Мосцощ, 1984 (Руссиан). Б. М. Левитан, Инверсе Стурм-Лиоувилле проблемс, ВСП, Зеист, 1987. Транслатед фром тхе Руссиан бы О. Ефимов.
[19] 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.
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