Absolutely continuous spectrum of perturbed Stark operators

Kiselev, Alexander

doi:10.1090/S0002-9947-99-02450-2

Absolutely continuous spectrum of perturbed Stark operators
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by Alexander Kiselev

Trans. Amer. Math. Soc. 352 (2000), 243-256

DOI: https://doi.org/10.1090/S0002-9947-99-02450-2

Published electronically: September 21, 1999

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Abstract:

We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate $(1+x) ^{-\frac {1}{3}-\epsilon }$ or if it is continuously differentiable with derivative from the Hölder space $C_{\alpha }(R),$ with any $\alpha >0.$

References

Joachim Asch, Pierre Duclos, and Pavel Exner, Stark-Wannier Hamiltonians with pure point spectrum, Differential equations, asymptotic analysis, and mathematical physics (Potsdam, 1996) Math. Res., vol. 100, Akademie Verlag, Berlin, 1997, pp. 10–25. MR 1456171
Joseph E. Avron and Barry Simon, Transient and recurrent spectrum, J. Functional Analysis 43 (1981), no. 1, 1–31. MR 639794, DOI 10.1016/0022-1236(81)90034-3
J.E. Avron, P. Exner and Y. Last, Periodic Schrödinger operators with large gaps and Wannier-Stark ladders, Phys. Rev. Lett. 72(1994), 896–899.
F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, and R. Weder, Schrödinger operators with an electric field and random or deterministic potentials, Comm. Math. Phys. 88 (1983), no. 3, 387–397. MR 701924
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, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 213–229. MR 1014651, DOI 10.1017/S0308210500018680
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
Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490
A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Comm. Math. Phys. 179 (1996), no. 2, 377–400. MR 1400745
A. Kiselev, Stability of the absolutely continuous spectrum of Schrödinger equation under perturbations by slowly decreasing potentials and a.e. convergence of integral operators, Duke Math. J. 94(1998), 619–646.
A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), no. 2, 345–356. MR 1335385, DOI 10.1006/jfan.1995.1074
Yoram Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142 (1996), no. 2, 406–445. MR 1423040, DOI 10.1006/jfan.1996.0155
S. N. Naboko, On the dense point spectrum of Schrödinger and Dirac operators, Teoret. Mat. Fiz. 68 (1986), no. 1, 18–28 (Russian, with English summary). MR 875178
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
J. von Neumann and E.P. Wigner, Über merkwürdige diskrete eigenwerte, Z. Phys. 30(1929), 465–467.
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Jaouad Sahbani, Propagation theorems for some classes of pseudo-differential operators, J. Math. Anal. Appl. 211 (1997), no. 2, 481–497. MR 1458508, DOI 10.1006/jmaa.1997.5481
B. Simon, Spectral analysis of rank one perturbations and applications, Proc. 1993 Vancouver Summer School in Mathematical Physics.
Barry Simon, Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3361–3369. MR 1350963, DOI 10.1090/S0002-9939-96-03599-X
Günter Stolz, Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl. 169 (1992), no. 1, 210–228. MR 1180682, DOI 10.1016/0022-247X(92)90112-Q
Johann Walter, Absolute continuity of the essential spectrum of $-d^{2}/dt^{2}+q(t)$ without monotony of $q$, Math. Z. 129 (1972), 83–94. MR 326053, DOI 10.1007/BF01229545
A. Zygmund, A remark on Fourier transforms, Proc. Camb. Phil. Soc. 32, 321–327 (1936).