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Absolutely continuous spectrum of a one-parameter family of Schrödinger operators


Author: O. Safronov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 6.
Journal: St. Petersburg Math. J. 24 (2013), 977-989
MSC (2010): Primary 35J10
DOI: https://doi.org/10.1090/S1061-0022-2013-01275-1
Published electronically: September 23, 2013
MathSciNet review: 3097557
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Abstract | References | Similar Articles | Additional Information

Abstract: Under certain conditions on the potential $ V$, it is shown that the absolutely continuous spectrum of the Schrödinger operator $ -\Delta +\alpha V$ is essentially supported on $ [0,+\infty )$ for almost every $ \alpha \in \mathbb{R}$.


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  • 1. J. Bourgain, On random Schrödinger operators on $ {\mathbb{Z}}^2$, Discrete Contin. Dyn. Syst. 8 (2002), no. 1, 1-15. MR 1877824 (2003f:47063)
  • 2. -, Random lattice Schrödinger operators with decaying potential: some higher dimensional phenomena, Geometric Aspects of Functional Analysis (Israel Seminar 2001-2002) (V. D. Milman and G. Schechtman, eds.), Lecture Notes in Math., vol. 1807, Springer, Berlin, 2003, pp. 70-98. MR 2083389 (2006a:47058)
  • 3. 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 (2000c:34223)
  • 4. S. Denisov, On the absolutely continuous spectrum of Dirac operator, Comm. Partial Differential Equations 29 (2004), no. 9-10, 1403-1428. MR 2103841 (2006c:35242)
  • 5. -, Absolutely continuous spectrum of multidimensional Schrödinger operator, Int. Math. Res. Notices 2004, no. 74, 3963-3982. MR 2103798 (2005h:35047)
  • 6. -, On the preservation of absolutely continuous spectrum for Schrödinger operators, J. Funct. Anal. 231 (2006), no. 1, 143-156. MR 2190166 (2006m:35048)
  • 7. -, Schrödinger operators and associated hyperbolic pencils, J. Funct. Anal. 254 (2008), 2186-2226. MR 2402106 (2009g:47111)
  • 8. R. L. Frank and O. Safronov, Absolutely continuous spectrum of a class of random nonergodic Schrödinger operators, Int. Math. Res. Notices 2005, no. 42, 2559-2577. MR 2182706 (2006i:47075)
  • 9. S. Denisov and A. Kiselev, Spectral properties of the Schrödinger operators with decaying potentials, Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon's 60th Birthday, Proc. Sympos. Pure Math., vol. 76, part 2, Amer. Math. Soc., Providence, RI, 2007, pp.565-589. MR 2307748 (2008j:35029)
  • 10. A. Laptev, S. Naboko, and O. Safronov, Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials, Comm. Math. Phys. 253 (2005), no. 3, 611-631. MR 2116730 (2005h:81126)
  • 11. R. Killip and B. Simon, Sum rules for Jacobi matrices and their application to spectral theory, Ann. of Math. (2) 158 (2003), 253-321. MR 1999923 (2004f:47040)
  • 12. A. Pushnitski, Spectral shift function of the Schrodinger operator in the large coupling constant limit, Comm. Partial Differential Equations 25 (2000), no. 3-4, 703-736. MR 1748353 (2001h:47076)
  • 13. O. Safronov, On the absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials, Comm. Math. Phys. 254 (2005), no. 2, 361-366. MR 2117630 (2005i:35048)
  • 14. -, Multi-dimensional Schrödinger operators with some negative spectrum, J. Funct. Anal. 238 (2006), no. 1, 327-339. MR 2253019 (2007i:35038)
  • 15. -, Multi-dimensional Schrödinger operators with no negative spectrum, Ann. Henri Poincaré 7 (2006), no. 4, 781-789. MR 2232372 (2007b:81078)
  • 16. -, Absolutely continuous spectrum of one random elliptic operator, J. Funct. Anal. 255 (2008), no. 3, 755-767. MR 2426436 (2009d:47040)
  • 17. -, Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law, Probl. Mat. Anal. No. 45, J. Math. Sci. (N.Y.) 166 (2010), no. 3, 300-318. MR 2839034 (2012j:35059)
  • 18. -, Absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying potentials, Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 205-214. MR 2509785 (2010i:35266)
  • 19. O. Safronov and G. Stolz, Absolutely continuous spectrum of Schrödinger operators with potentials slowly decaying inside a cone, J. Math. Anal. Appl. 326 (2007), 192-208. MR 2277776 (2007i:35039)
  • 20. O. Safronov and B. Vainberg, Estimates for negative eigenvalues of a random Schrödinger operator, Proc. Amer. Math. Soc. 136 (2008), no. 11, 3921-3929. MR 2425732 (2009h:47068)
  • 21. B. Simon, Schrödinger operators in the twenty-first century, Mathematical Physics 2000, Imp. Coll. Press, London, 2000, pp. 283-288. MR 1773049 (2001g:81071)
  • 22. B. R. Vainberg, The analytic properties of the resolvent for a certain class of bundles of operators, Mat. Sb. (N. S.) 77 (119) (1968), no. 2, 259-296; English transl. in Math. USSR-Sb. 6 (1968). MR 0240447 (39:1795)
  • 23. D. R. Yafaev, Scattering theory: some old and new problems, Lecture Notes in Math., vol. 1735, Springer-Verlag, Berlin, 2000. MR 1774673 (2001j:81248)

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Additional Information

O. Safronov
Affiliation: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, North Carolina
Email: osafrono@uncc.edu

DOI: https://doi.org/10.1090/S1061-0022-2013-01275-1
Keywords: Schr\"odinger operator, spectral measure, Fourier transform, selfadjoint operator
Received by editor(s): June 7, 2011
Published electronically: September 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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