On polyharmonic operators with limit-periodic potential in dimension two
Authors:
Yulia Karpeshina and Young-Ran Lee
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 113-120
MSC (2000):
Primary 81Q15; Secondary 81Q10
DOI:
https://doi.org/10.1090/S1079-6762-06-00167-3
Published electronically:
August 11, 2006
MathSciNet review:
2237275
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Abstract: This is an announcement of the following results. We consider a polyharmonic operator $H=(-\Delta )^l+V(x)$ in dimension two with $l\geq 6$ and $V(x)$ being a limit-periodic potential. We prove that the spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves at the high-energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor-type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.
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[1]1 J. Avron and B. Simon, Almost periodic Schrödinger operators I: Limit periodic potentials. Comm. Math. Phys. 82 (1981), 101–120.
[2]3 V. A. Chulaevskii, On perturbation of a Schrödinger operator with periodic potential. Russian Math. Surv. 36(5) (1981), 143–144.
[3]4 J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comm. Math. Helv. 56 (1981), 198–224.
[4]5 B. Simon, Almost periodic Schrödinger operators: a review. Advances in Applied Mathematics 3 (1982), 463–490.
[5]6 L. A. Pastur and V. A. Tkachenko, On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential. Dokl. Akad. Nauk SSSR 279 (1984), 1050–1053; English transl., Soviet Math. Dokl. 30 (1984), no. 3, 773–776.
[6]8 L. A. Pastur and V. A. Tkachenko, Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials. Trans. Moscow Math. Soc. 51 (1989), 115–166.
[7]9 L. Pastur and A. Figotin, Spectra of random and almost-periodic operators. Springer-Verlag, Berlin, 1992, 583pp.
[8]2 J. Avron and B. Simon, Cantor sets and Schrödinger operators: transient and recurrent spectrum. J. Func. Anal. 43 (1981), 1–31.
[9]7 S. A. Molchanov and V. A. Chulaevskii, Structure of the spectrum of a lacunary limit-periodic Schrödinger operator. Functional Anal. Appl. 18 (1984), 343–344.
[10]10 L. Zelenko, On a generic topological structure of the spectrum to one-dimensional Schrödinger operators with complex limit-periodic potentials. Integral Equations and Operator Theory 50 (2004), 393–430.
[11]11 M. A. Shubin, The density of states for selfadjoint elliptic operators with almost periodic coefficients. Trudy Sem. Petrovsk. 3 (1978), 243–275. (Russian)
[12]12 M. A. Shubin, Spectral theory and index of elliptic operators with almost periodic coefficients. Russ. Math. Surveys 34(2) (1979), 109–157.
[13]13 J. Avron and B. Simon, Almost periodic Schrödinger operators. II: The integrated density of states. Duke Math. J. 50 (1983), 1, 369–391.
[14]14 G. V. Rozenblum, M. A. Shubin, and M. Z. Solomyak, Spectral theory of differential operators. Encyclopaedia of Mathematical Sciences, 64, Springer-Verlag, Berlin, 1994.
[15]15 Yu. P. Chuburin, On the multidimensional discrete Schrödinger equation with a limit peridic potential. Theoretical and Mathematical Physics 102 (1995), no. 1, 53–59.
[16]20 M. M. Skriganov and A. V. Sobolev, On the spectrum of polyharmonic operators with limit-periodic potentials, St. Petersburg Mathematical Journal (to appear).
[17]21 G. Gallavotti, Perturbation theory for classical Hamiltonian systems. Scaling and self-similarity. Progr. Phys. 7, edited by J. Froehlich, Birkhäuser, Basel, 1983, pp. 359–424.
[18]22 L. E. Thomas and S. R. Wassel, Stability of Hamiltonian systems at high evergy. J. Math. Phys. 33(10) (1992), 3367–3373.
[19]24 L. E. Thomas and S. R. Wassel, Semiclassical approximation for Schrödinger operators at high energy, Lecture Notes in Physics, 403, edited by E. Balslev, Springer-Verlag, Berlin, 1992, pp. 194–210.
[20]RS M. Reed and B. Simon, Methods of modern mathematical physics., Vol IV, 3rd ed., Academic Press, New York–San Francisco–London, 1987, 396 pp.
[21]19 Yu. Karpeshina, Perturbation theory for the Schrödinger operator with a periodic potential. Lecture Notes in Mathematics, 1663, Springer-Verlag, 1997, 352 pp.
[22]17 Young-Ran Lee, Spectral properties of a polyharmonic operator with limit periodic potential in dimension two. PhD thesis, defended at UAB, Dept. of Mathematics, on May 12, 2004.
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Additional Information
Yulia Karpeshina
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, Birmingham, Alabama 35294
MR Author ID:
189810
Email:
karpeshi@math.uab.edu
Young-Ran Lee
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email:
yrlee4@math.uiuc.edu
Keywords:
Limit-periodic potential
Received by editor(s):
January 4, 2006
Published electronically:
August 11, 2006
Additional Notes:
Research partially supported by USNSF Grant DMS-0201383.
Dedicated:
In memory of our colleague and friend Robert M. Kauffman.
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
