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 in dimension two with and being a limit-periodic potential. We prove that the spectrum of 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.

**[1]**J. Avron and B. Simon,*Almost periodic Schrödinger operators I: Limit periodic potentials.*Comm. Math. Phys.**82**(1981), 101-120. MR**0638515 (84i:34023)****[2]**V. A. Chulaevskii,*On perturbation of a Schrödinger operator with periodic potential.*Russian Math. Surv.**36(5)**(1981), 143-144. MR**0637459 (83m:34019)****[3]**J. Moser,*An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum.*Comm. Math. Helv.**56**(1981), 198-224. MR**0630951 (82k:34029)****[4]**B. Simon,*Almost periodic Schrödinger operators: a review.*Advances in Applied Mathematics**3**(1982), 463-490. MR**0682631 (85d:34030)****[5]**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. MR**0796729 (86j:34026)****[6]**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. MR**0983634 (90h:47090)****[7]**L. Pastur and A. Figotin,*Spectra of random and almost-periodic operators.*Springer-Verlag, Berlin, 1992, 583pp. MR**1223779 (94h:47068)****[8]**J. Avron and B. Simon,*Cantor sets and Schrödinger operators: transient and recurrent spectrum.*J. Func. Anal.**43**(1981), 1-31. MR**0639794 (83c:47008)****[9]**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. MR**0775943 (86k:34022)****[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. MR**2104262 (2005k:47095)****[11]**M. A. Shubin,*The density of states for selfadjoint elliptic operators with almost periodic coefficients.*Trudy Sem. Petrovsk.**3**(1978), 243-275. (Russian) MR**0499819 (58:17587)****[12]**M. A. Shubin,*Spectral theory and index of elliptic operators with almost periodic coefficients.*Russ. Math. Surveys**34(2)**(1979), 109-157. MR**0535710 (81f:35090)****[13]**J. Avron and B. Simon,*Almost periodic Schrödinger operators. II: The integrated density of states.*Duke Math. J.**50**(1983), 1, 369-391. MR**0700145 (85i:34009a)****[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. MR**1313735 (95j:35156)****[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. MR**1348621 (96h:47040)****[16]**M. M. Skriganov and A. V. Sobolev,*On the spectrum of polyharmonic operators with limit-periodic potentials*, St. Petersburg Mathematical Journal (to appear).**[17]**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. MR**0733479 (85e:58127)****[18]**L. E. Thomas and S. R. Wassel,*Stability of Hamiltonian systems at high evergy.*J. Math. Phys.**33(10)**(1992), 3367-3373. MR**1182907 (94c:58178)****[19]**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. MR**1181249 (93i:81037)****[20]**M. Reed and B. Simon,*Methods of modern mathematical physics.*, Vol IV, 3rd ed., Academic Press, New York-San Francisco-London, 1987, 396 pp. MR**0493421 (58:12429c)****[21]**Yu. Karpeshina,*Perturbation theory for the Schrödinger operator with a periodic potential.*Lecture Notes in Mathematics,**1663**, Springer-Verlag, 1997, 352 pp. MR**1472485 (2000i:35002)****[22]**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

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

DOI:
https://doi.org/10.1090/S1079-6762-06-00167-3

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.