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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]**Joseph Avron and Barry Simon,*Almost periodic Schrödinger operators. I. Limit periodic potentials*, Comm. Math. Phys.**82**(1981/82), no. 1, 101–120. MR**638515****[2]**V. A. Chulaevskiĭ,*Perturbations of a Schrödinger operator with periodic potential*, Uspekhi Mat. Nauk**36**(1981), no. 5(221), 203–204 (Russian). MR**637459****[3]**Jürgen Moser,*An example of a Schroedinger equation with almost periodic potential and nowhere dense spectrum*, Comment. Math. Helv.**56**(1981), no. 2, 198–224. MR**630951**, https://doi.org/10.1007/BF02566210**[4]**Barry Simon,*Almost periodic Schrödinger operators: a review*, Adv. in Appl. Math.**3**(1982), no. 4, 463–490. MR**682631**, https://doi.org/10.1016/S0196-8858(82)80018-3**[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), no. 5, 1050–1053 (Russian). MR**796729****[6]**L. A. Pastur and V. A. Tkachenko,*Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials*, Trudy Moskov. Mat. Obshch.**51**(1988), 114–168, 258 (Russian); English transl., Trans. Moscow Math. Soc. (1989), 115–166. MR**983634****[7]**Leonid Pastur and Alexander Figotin,*Spectra of random and almost-periodic operators*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297, Springer-Verlag, Berlin, 1992. MR**1223779****[8]**Joseph E. Avron and Barry Simon,*Transient and recurrent spectrum*, J. Funct. Anal.**43**(1981), no. 1, 1–31. MR**639794**, https://doi.org/10.1016/0022-1236(81)90034-3**[9]**S. A. Molchanov and V. A. Chulaevskiĭ,*The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator*, Funktsional. Anal. i Prilozhen.**18**(1984), no. 4, 90–91 (Russian). MR**775943****[10]**Leonid Zelenko,*On a generic topological structure of the spectrum to one-dimensional Schrödinger operators with complex limit-periodic potentials*, Integral Equations Operator Theory**50**(2004), no. 3, 393–430. MR**2104262**, https://doi.org/10.1007/s00020-003-1239-7**[11]**M. A. Šubin,*The density of states of selfadjoint elliptic operators with almost periodic coefficients*, Trudy Sem. Petrovsk.**3**(1978), 243–275 (Russian). MR**0499819****[12]**M. A. Šubin,*Spectral theory and the index of elliptic operators with almost-periodic coefficients*, Uspekhi Mat. Nauk**34**(1979), no. 2(206), 95–135 (Russian). MR**535710****[13]**Joseph Avron and Barry Simon,*Almost periodic Schrödinger operators. II. The integrated density of states*, Duke Math. J.**50**(1983), no. 1, 369–391. MR**700145**, https://doi.org/10.1215/S0012-7094-83-05016-0

P. Deift and B. Simon,*Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension*, Comm. Math. Phys.**90**(1983), no. 3, 389–411. MR**719297****[14]***Partial differential equations. VII*, Encyclopaedia of Mathematical Sciences, vol. 64, Springer-Verlag, Berlin, 1994. Spectral theory of differential operators; A translation of Current problems in mathematics. Fundamental directions. Vol. 64 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 [ MR1033499 (91h:35229a)]; Translation by T. Zastawniak; Translation edited by M. A. Shubin. MR**1313735****[15]**Yu. P. Chuburin,*On the multidimensional discrete Schrödinger equation with a limit-periodic potential*, Teoret. Mat. Fiz.**102**(1995), no. 1, 74–82 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys.**102**(1995), no. 1, 53–59. MR**1348621**, https://doi.org/10.1007/BF01017455**[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]**Giovanni Gallavotti,*Perturbation theory for classical Hamiltonian systems*, Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982) Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, pp. 359–426. MR**733479****[18]**Lawrence E. Thomas and Stephen R. Wassell,*Stability of Hamiltonian systems at high energy*, J. Math. Phys.**33**(1992), no. 10, 3367–3373. MR**1182907**, https://doi.org/10.1063/1.529937**[19]**Lawrence E. Thomas and Stephen R. Wassell,*Semiclassical approximation for Schrödinger operators at high energy*, Schrödinger operators (Aarhus, 1991) Lecture Notes in Phys., vol. 403, Springer, Berlin, 1992, pp. 194–210. MR**1181249**, https://doi.org/10.1007/3-540-55490-4_13**[20]**Michael Reed and Barry Simon,*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419**

Michael Reed and Barry Simon,*Methods of modern mathematical physics. II. Fourier analysis, self-adjointness*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0493420**

Michael Reed and Barry Simon,*Methods of modern mathematical physics. IV. Analysis of operators*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0493421****[21]**Yulia E. Karpeshina,*Perturbation theory for the Schrödinger operator with a periodic potential*, Lecture Notes in Mathematics, vol. 1663, Springer-Verlag, Berlin, 1997. MR**1472485****[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.

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (2000):
81Q15,
81Q10

Retrieve articles in all journals with MSC (2000): 81Q15, 81Q10

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.