Transactions of the American Mathematical Society

Author(s): Alexander Kiselev
Journal: Trans. Amer. Math. Soc. 352 (2000), 243-256.
MSC (1991): Primary 34L40, 81Q10
Posted: 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 $\alpha>0.$

1.: J. Asch, P. Duclos and P. Exner, Stark-Wannier Hamiltonians with pure point spectrum, in Differential Equations, Asymptotic Analysis, and Mathematical Physics (Potsdam, 1996), Akademie Verlag, Berlin, 1997. MR 98f:81066
2.: J. Avron and B. Simon, Transient and recurrent spectrum, J. Funct. Anal. 43(1981), 1-31. MR 83c:47008
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.
4.: F. Bentosela et al., Schrödinger operators with an electric field and random or deterministic potentials, Commun. Math. Phys. 88(1983), 387-397. MR 85f:34036
5.: P. Exner, The absence of the absolutely continuous spectrum for $\delta '$ Wannier-Stark ladders, J. Math. Phys. 36(9) (1995), 4561-4570. MR 96g:81050
6.: 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), 213-229. MR 90j:47061
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), 30-56. MR 89a:34033
8.: P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston 1982. MR 83e:34002
9.: A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decaying potentials, Commun. Math. Phys.179(1996), 377-400. MR 97e:81027
10.: 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. CMP 98:17
11.: A. Kiselev and B. Simon, Rank one perturbations in infinitesmal coupling, J. Funct. Anal. 130(1)(1995), 345-356. MR 96e:47012
12.: Y. Last, Quantum dynamics and decompositions of singular continuous spectra, J. Funct. Anal. 142(2)(1996), 406-445. MR 97k:81044
13.: S.N. Naboko, Dense point spectra of Dirac and Schrödinger operators, Theor. Math. 68(1986), 18-28. MR 88h:81029
14.: S.N. Naboko and A.B. Pushnitskii, Point spectrum on a continuous spectrum for weakly perturbed Stark type operators, Funct. Anal. Appl. 29(4)(1995), 248-257. MR 96k:34187
15.: J. von Neumann and E.P. Wigner, Über merkwürdige diskrete eigenwerte, Z. Phys. 30(1929), 465-467.
16.: W. Rudin, Real and Complex Analysis, New York, McGraw-Hill, 1987. MR 88k:00002
17.: J. Sahbani, Propagation theorems for some classes of pseudodifferential operators, J. Math. Anal. Appl. 211 (1997), 481-497. MR 98e:47078
18.: B. Simon, Spectral analysis of rank one perturbations and applications, Proc. 1993 Vancouver Summer School in Mathematical Physics.
19.: B. Simon, Bounded eigenfunctions and absolutely continuous specta for one-dimensional Schrödinger operators, Proc. Amer. Math. Soc. 124(11), 3361-3369. MR 97a:34223
20.: G. Stolz, Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl. 169(1992), 210-228. MR 93f:34141
21.: J. Walter, Absolute continuity of the essential spectrum of $-d^{2}/dx^{2} +q(x)$ without monotony of Math. Z. 129(1)(1972), 83-94. MR 48:4399
22.: A. Zygmund, A remark on Fourier transforms, Proc. Camb. Phil. Soc. 32, 321-327 (1936).

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34L40, 81Q10

Alexander Kiselev
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637
Email: kiselev@math.uchicago.edu

DOI: 10.1090/S0002-9947-99-02450-2
PII: S 0002-9947(99)02450-2
Received by editor(s): April 14, 1997
Posted: September 21, 1999
Copyright of article: Copyright 1999, American Mathematical Society

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