On the structure of spectra of periodic elliptic operators

Authors:
Peter Kuchment and Sergei Levendorskiî

Journal:
Trans. Amer. Math. Soc. **354** (2002), 537-569

MSC (2000):
Primary 35P99; Secondary 35J10

DOI:
https://doi.org/10.1090/S0002-9947-01-02878-1

Published electronically:
September 21, 2001

MathSciNet review:
1862558

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Abstract | References | Similar Articles | Additional Information

Abstract: The paper discusses the problem of absolute continuity of spectra of periodic elliptic operators. A new result on absolute continuity for a matrix operator of Schrödinger type is obtained. It is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one. It is also discovered that some natural generalizations stumble upon an obstacle in the form of non-triviality of a certain analytic bundle on the two-dimensional torus.

- M. S. Agranovič and M. I. Višik,
*Elliptic problems with a parameter and parabolic problems of general type*, Uspehi Mat. Nauk**19**(1964), no. 3 (117), 53–161 (Russian). MR**0192188** - Lars V. Ahlfors,
*Lectures on quasiconformal mappings*, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR**0200442** - Lars V. Ahlfors and Leo Sario,
*Riemann surfaces*, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR**0114911** - N.W. Ashcroft and N.D. Mermin,
*Solid State Physics*, Holt, Rinehart and Winston, New York-London 1976. - W. Axmann, P. Kuchment, and L. Kunyansky,
*Asymptotic methods for thin high contrast 2D PBG materials*, Journal of Lightwave Technology,**17**(1999), no.11, 1996–2007. - Anne Berthier,
*On the point spectrum of Schrödinger operators*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 1, 1–15. MR**672473** - M. Sh. Birman and M. Z. Solomyak,
*$L_2$-theory of the Maxwell operator in arbitrary domains*, Uspekhi Mat. Nauk**42**(1987), no. 6(258), 61–76, 247 (Russian). MR**933995** - M. Sh. Birman and T. A. Suslina,
*The two-dimensional periodic magnetic Hamiltonian is absolutely continuous*, Algebra i Analiz**9**(1997), no. 1, 32–48 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**9**(1998), no. 1, 21–32. MR**1458417** - M. Sh. Birman and T. A. Suslina,
*The periodic Dirac operator is absolutely continuous*, Integral Equations Operator Theory**34**(1999), no. 4, 377–395. MR**1702229**, DOI https://doi.org/10.1007/BF01272881 - M. Sh. Birman and T. A. Suslina,
*Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with discontinuous vector potential*, Algebra i Analiz**10**(1998), no. 4, 1–36 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**10**(1999), no. 4, 579–601. MR**1654063** - M. Sh. Birman and T. A. Suslina,
*Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum*, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 13–31. MR**1708785** - M. Sh. Birman and T. A. Suslina,
*A periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity*, Algebra i Analiz**11**(1999), no. 2, 1–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**11**(2000), no. 2, 203–232. MR**1702587** - Rudi Weikard and Gilbert Weinstein (eds.),
*Differential equations and mathematical physics*, AMS/IP Studies in Advanced Mathematics, vol. 16, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. MR**1759613** - M. Sh. Birman, T. A. Suslina, and R. G. Shterenberg,
*Absolute continuity of the spectrum of a two-dimensional Schrödinger operator with potential supported on a periodic system of curves*, Preprint ESI no. 934, http://www.esi.ac.at, 2000. - R. Courant and D. Hilbert,
*Methods of mathematical physics. Vol. II*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR**1013360** - L. Danilov,
*Zone structure of the spectrum of the Dirac operator with periodic potential*, in: Nonlinear Oscillations and Control Theory, Ustinov 1985, 83-90. (Russian) - L. Danilov,
*On the spectrum of the Dirac operator with periodic potential*, Acad. Sci. USSR, Ural Branch, preprint, Sverdlovsk, 1987. (Russian) - L. I. Danilov,
*On the spectrum of the Dirac operator in ${\bf R}^n$ with periodic potential*, Teoret. Mat. Fiz.**85**(1990), no. 1, 41–53 (Russian, with English summary); English transl., Theoret. and Math. Phys.**85**(1990), no. 1, 1039–1048 (1991). MR**1083951**, DOI https://doi.org/10.1007/BF01017245 - L. Danilov,
*The spectrum of the Dirac operator with periodic potential*I, preprint, no.4588-B91, VINITI, Moscow 1991, 36 pp. (Russian) - L. I. Danilov,
*Resolvent estimates and the spectrum of the Dirac operator with a periodic potential*, Teoret. Mat. Fiz.**103**(1995), no. 1, 3–22 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys.**103**(1995), no. 1, 349–365. MR**1470934**, DOI https://doi.org/10.1007/BF02069779 - L. I. Danilov,
*On the spectrum of the two-dimensional periodic Dirac operator*, Teoret. Mat. Fiz.**118**(1999), no. 1, 3–14 (Russian, with Russian summary); English transl., Theoret. and Math. Phys.**118**(1999), no. 1, 1–11. MR**1702856**, DOI https://doi.org/10.1007/BF02557191 - P. Maroni,
*Semi-classical character and finite-type relations between polynomial sequences*, Appl. Numer. Math.**31**(1999), no. 3, 295–330. MR**1711161**, DOI https://doi.org/10.1016/S0168-9274%2898%2900137-8 - L. Danilov,
*On the spectrum of the periodic Dirac operator*, Teoret. i Mat. Fiz.**124**(2000), no.1, 3-17; English transl. in Theoret. Math. Phys.**124**(2000). - L. Danilov,
*On absolute continuity of spectra of periodic Schrödinger and Dirac operators*I, preprint, no. 1683-B00, VINITI, Moscow, 2000. (Russian) - B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,
*Modern geometry—methods and applications. Part I*, 2nd ed., Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1992. The geometry of surfaces, transformation groups, and fields; Translated from the Russian by Robert G. Burns. MR**1138462** - B. A. Dubrovin and S. P. Novikov,
*Fundamental states in a periodic field. Magnetic Bloch functions and vector bundles*, Dokl. Akad. Nauk SSSR**253**(1980), no. 6, 1293–1297 (Russian). MR**583789** - B. A. Dubrovin and S. P. Novikov,
*Ground states of a two-dimensional electron in a periodic magnetic field*(Russian); English transl., Zh. Èksper. Teoret. Fiz.**79**(1980), no. 3, 1006–1016. MR**617904** - M. S. P. Eastham,
*The Spectral Theory of Periodic Differential Equations,*Scottish Acad. Press, Edinburgh-London, 1973. - Leon Ehrenpreis,
*Fourier analysis in several complex variables*, Pure and Applied Mathematics, Vol. XVII, Wiley-Interscience Publishers A Division of John Wiley & Sons, New York-London-Sydney, 1970. MR**0285849** - R. Froese, I. Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof,
*$L^{2}$-lower bounds to solutions of one-body Schrödinger equations*, Proc. Roy. Soc. Edinburgh Sect. A**95**(1983), no. 1-2, 25–38. MR**723095**, DOI https://doi.org/10.1017/S0308210500015778 - Rainer Hempel and Ira Herbst,
*Strong magnetic fields, Dirichlet boundaries, and spectral gaps*, Comm. Math. Phys.**169**(1995), no. 2, 237–259. MR**1329195** - J.D. Joannopoulos, R.D. Meade, and J. N. Winn,
*Photonic Crystals. Molding the Flow of Light,*Princeton Univ. Press, Princeton, NJ, 1995. - Yu. E. Karpeshina,
*Spectrum and eigenfunctions of the Schrödinger operator with point potential of uniform lattice type in a three-dimensional space*, Teoret. Mat. Fiz.**57**(1983), no. 2, 304–313 (Russian, with English summary). MR**734891** - I. E. Egorov and V. E. Fedorov,
*Neklassicheskie uravneniya matematicheskoĭ fiziki vysokogo poryadka*, Ross. Akad. Nauk Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, 1995 (Russian, with English and Russian summaries). MR**1800114** - H. Knörrer and E. Trubowitz,
*A directional compactification of the complex Bloch variety*, Comment. Math. Helv.**65**(1990), no. 1, 114–149. MR**1036133**, DOI https://doi.org/10.1007/BF02566598 - P. A. Kuchment,
*Floquet theory for partial differential equations*, Uspekhi Mat. Nauk**37**(1982), no. 4(226), 3–52, 240 (Russian). MR**667973** - Peter Kuchment,
*Floquet theory for partial differential equations*, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. MR**1232660** - P. Kuchment,
*Mathematics of photonic crystals*, in*Mathematical Modeling in Optical Science*, G. Bao, L. Cowsar, and W. Masters (Editors) SIAM, 2001. - Peter Kuchment and Leonid A. Kunyansky,
*Spectral properties of high contrast band-gap materials and operators on graphs*, Experiment. Math.**8**(1999), no. 1, 1–28. MR**1685034** - P. Kuchment and L. Kunyansky,
*Differential operators on graphs and photonic crystals*, to appear in Adv. Comp. Math. - Peter Kuchment and Sergei Levendorskiĭ,
*On absolute continuity of spectra of periodic elliptic operators*, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 291–297. MR**1708810** - P. Kuchment and Y. Pinchover,
*Integral representations and Liouville theorems for solutions of periodic elliptic equations*, J. Funct. Anal.**181**(2001), 402–446. - Peter Kuchment and Boris Vainberg,
*On embedded eigenvalues of perturbed periodic Schrödinger operators*, Spectral and scattering theory (Newark, DE, 1997) Plenum, New York, 1998, pp. 67–75. MR**1625271** - P. Kuchment and B. Vainberg,
*On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials*, Commun. PDE**25**(2000), no. 9-10, 1809-1826. - B. Malgrange,
*Lectures on the theory of functions of several complex variables*, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 13, Distributed for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1984. Reprint of the 1958 edition; Notes by Raghavan Narasimhan. MR**742775** - V. Meshkov,
*On the possible rate of decay at infinity of solutions of second order partial differential equations*, Mat. Sbornik,**182**(1991), no.3, 364–383. English translation in Math. USSR Sbornik**72**(1992), no.2, 343–351. - Abderemane Morame,
*Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electromagnetic potential*, J. Phys. A**31**(1998), no. 37, 7593–7601. MR**1652918**, DOI https://doi.org/10.1088/0305-4470/31/37/017 - A. Morame,
*The absolute continuity of the spectrum of Maxwell operator in periodic media*, J. Math. Phys.**41**(2000), 7099–7108. - Gen Nakamura, Zi Qi Sun, and Gunther Uhlmann,
*Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field*, Math. Ann.**303**(1995), no. 3, 377–388. MR**1354996**, DOI https://doi.org/10.1007/BF01460996 - S. P. Novikov,
*Two-dimensional Schrödinger operators in periodic fields*, Current problems in mathematics, Vol. 23, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 3–32 (Russian). MR**734312** - V. P. Palamodov,
*Linear differential operators with constant coefficients*, Die Grundlehren der mathematischen Wissenschaften, Band 168, Springer-Verlag, New York-Berlin, 1970. Translated from the Russian by A. A. Brown. MR**0264197** - Victor P. Palamodov,
*Harmonic synthesis of solutions of elliptic equation with periodic coefficients*, Ann. Inst. Fourier (Grenoble)**43**(1993), no. 3, 751–768 (English, with English and French summaries). MR**1242614** - A. Pliś,
*Non-uniqueness in Cauchy’s problem for differential equations of elliptic type*, J. Math. Mech.**9**(1960), 557–562. MR**0121568**, DOI https://doi.org/10.1512/iumj.1960.9.59031 - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419** - Z. Shen,
*On absolute continuity of the periodic Schrödinger operators*, Internat. Math. Res. Notices**2001**, no. 1, 1–31. - Z. Shen,
*The periodic Schrödinger operator with potentials in the C. Fefferman–Phong class*, Preprint #99-15, Math. Dept., Univ. of Kentucky, 1999 and #99-455 in the Texas Math Physics archive, 1999 http://www.ma.utexas.edu/mp_arc. - Z. Shen,
*The periodic Schrödinger operator with potentials in the Morrey-Campanato class*, Preprint 1999. - Z. Shen,
*Absolute continuity of periodic Schrödinger operators with potentials in the Kato class*, Preprint #00-294 in the Texas Math Physics archive, 2000 http://www.ma.utexas.edu/mp_arc. - 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** - Johannes Sjöstrand,
*Microlocal analysis for the periodic magnetic Schrödinger equation and related questions*, Microlocal analysis and applications (Montecatini Terme, 1989) Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 237–332. MR**1178559**, DOI https://doi.org/10.1007/BFb0085125 - M.M. Skriganov,
*Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators*, Trudy Mat. Inst. Steklov.**17**(1985); English transl., Proc. Steklov Inst. Math. 1987, no. 2 (171). ; - G. Springer,
*Introduction to Riemann Surfaces*, Chelsea, New York 1981. (orig. ed.) - Tatiana Suslina,
*Absolute continuity of the spectrum of periodic operators of mathematical physics*, Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000) Univ. Nantes, Nantes, 2000, pp. Exp. No. XVIII, 13. MR**1775694** - Zi Qi Sun,
*An inverse boundary value problem for Schrödinger operators with vector potentials*, Trans. Amer. Math. Soc.**338**(1993), no. 2, 953–969. MR**1179400**, DOI https://doi.org/10.1090/S0002-9947-1993-1179400-1 - Alexander V. Sobolev,
*Absolute continuity of the periodic magnetic Schrödinger operator*, Invent. Math.**137**(1999), no. 1, 85–112. MR**1703339**, DOI https://doi.org/10.1007/s002220050324 - A. Sobolev and J. Walthoe,
*Absolute continuity in periodic waveguides*, Preprint, September 2000. - M. Taylor,
*Partial Differential Equations. I. Basic Theory*, Applied Mathematical Sciences, v. 115. Springer-Verlag, New York, 1996. - Lawrence E. Thomas,
*Time dependent approach to scattering from impurities in a crystal*, Comm. Math. Phys.**33**(1973), 335–343. MR**334766** - I. N. Vekua,
*Generalized analytic functions*, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962. MR**0150320** - M. G. Zaĭdenberg, S. G. Kreĭn, P. A. Kučment, and A. A. Pankov,
*Banach bundles and linear operators*, Uspehi Mat. Nauk**30**(1975), no. 5(185), 101–157 (Russian). MR**0415661** - J. Zak,
*Magnetic translation group*I, II, Phys. Rev.**134**(1964), no. 6A, A1602–A1611. ;

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Additional Information

**Peter Kuchment**

Affiliation:
Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

Address at time of publication:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

MR Author ID:
227235

ORCID:
canUsePostMessage

Email:
kuchment@math.tamu.edu

**Sergei Levendorskiî**

Affiliation:
Department of Mathematics and Statistics, Wichita State University, Wichita, Kansas 67260-0033

Address at time of publication:
Department of Mathematics, Rostov State Academy of Economics, Rostov-on-Don, Russia

Email:
leven@ns.rnd.runnet.ru

Keywords:
Periodic operator,
elliptic operator,
absolutely continuous spectrum,
Schrödinger operator,
magnetic and electric potentials

Received by editor(s):
October 3, 2000

Published electronically:
September 21, 2001

Additional Notes:
The first author was supported in part by an NRC COBASE Grant, NSF Grants DMS 9610444 and DMS 0072248, and by a DEPSCoR Grant

The second author was supported in part by an NRC COBASE Grant.

Article copyright:
© Copyright 2001
American Mathematical Society