Transactions of the American Mathematical Society

Author(s): Peter Kuchment; Sergei Levendorskiî
Journal: Trans. Amer. Math. Soc. 354 (2002), 537-569.
MSC (2000): Primary 35P99; Secondary 35J10
Posted: September 21, 2001
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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.

1.: M.S. Agranovich and M.I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Russian Math. Surveys 19 (1964), no.3, 53-157. MR 33:415
2.: L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies, No. 10, Van Nostrand, New York, 1966. MR 34:336
3.: L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Univ. Press, Princeton NJ 1960. MR 22:5729
4.: N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York-London 1976.
5.: 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.
6.: A. Berthier, On the point spectrum of Schrödinger operators, Ann. Scient. Ec. Norm. Super., 4 serie, 15 (1982), 1-15. MR 84i:35106
7.: M. Birman and M. Solomyak, -theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys 42 (1987), no 6: 75-96. MR 89e:35127
8.: M. Sh. Birman and T. A. Suslina, The two-dimensional periodic magnetic Hamiltonian is absolutely continuous. (Russian) Algebra i Analiz 9 (1997), no. 1, 32-48; translation in St. Petersburg Math. J. 9 (1998), no. 1, 21-32. MR 98g:47038
9.: M. Sh. Birman and T. A. Suslina, The periodic Dirac operator is absolutely continuous, Integr. Equat. and Operator Theory 34 (1999), 377-395. MR 2000h:47068
10.: M. Sh. Birman and T. A. Suslina, Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential. (Russian) Algebra i Analiz 10 (1998), no. 4, 1-36; translation in St. Petersburg Math. J. 10 (1999), no. 4, 579-601. MR 99k:81060
11.: M. Sh. Birman and T. A. Suslina, Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum, in Mathematical Results in Quantum Mechanics (QMath7, Prague, June 22-26,1998), J. Dittrich, P. Exner, et al (Editors), 13-31, Operator Theory, Adv. and Appl., v. 108, Birkhäuser, Basel, 1999. MR 2000g:81049
12.: M. Sh. Birman and T. A. Suslina, Periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no.2. English translation in St. Petersburg Math J. 11 (2000), no.2, 203-232. MR 2000i:35026
13.: M. Sh. Birman and T. A. Suslina, On the absolute continuity of the periodic Schrödinger and Dirac operators with magnetic potential, in Differential Equations and Mathematical Physics (Birmingham, AL, 1999), AMSIP Stud. Adv. Math., vol. 16, Amer. Math. Soc., Providence, RI, 2000, pp. 41-49. MR 2000m:00026
14.: 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.
15.: R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II. Partial differential equations. John Wiley & Sons, Inc., New York, 1989. MR 90k:35001
16.: 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)
17.: L. Danilov, On the spectrum of the Dirac operator with periodic potential, Acad. Sci. USSR, Ural Branch, preprint, Sverdlovsk, 1987. (Russian)
18.: L. Danilov, On the spectrum of the Dirac operator in $\mathbb{R}\mathbf{^n}$ with periodic potential. (Russian) Teoret. Mat. Fiz. 85 (1990), no. 1,41-53; translation in Theoret. and Math. Phys. 85 (1990), no. 1, 1039-1048 (1991). MR 92a:35119
19.: L. Danilov, The spectrum of the Dirac operator with periodic potential I, preprint, no.4588-B91, VINITI, Moscow 1991, 36 pp. (Russian)
20.: L. Danilov, Resolvent estimates and the spectrum of the Dirac operator with a periodic potential. (Russian) Teoret. Mat. Fiz. 103 (1995), no. 1, 3-22; translation in Theoret. and Math. Phys. 103 (1995), no. 1, 349-365. MR 98f:35112
21.: L. Danilov, On the spectrum of the two-dimensional periodic Dirac operator. (Russian) Teoret. Mat. Fiz. 118 (1999), no. 1, 3-14; translation in Theoret. and Math. Phys. 118 (1999), no. 1, 1-11. MR 2000h:35117
22.: L. Danilov, Absolute continuity of the spectrum of the periodic Dirac operator, Diff. Uravneniya, 36 (2000), no.2, 233-240; English transl., Differential Equations 36 (2000), 262-271. MR 2001f:42038
23.: 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). CMP 2001:10
24.: L. Danilov, On absolute continuity of spectra of periodic Schrödinger and Dirac operators I, preprint, no. 1683-B00, VINITI, Moscow, 2000. (Russian)
25.: B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry - Methods and Applications, Part I, 2nd ed., Springer Verlag, New York, 1992. MR 92h:53001
26.: B. A. Dubrovin and S. P. Novikov, Ground states in a periodic field. Magnetic Bloch functions and vector bundles, Soviet Math. Dokl. 22 (1980), 240-244. MR 82c:81181
27.: B. A. Dubrovin and S. P. Novikov, Ground states of a two-dimensional electron in a periodic magnetic field, Soviet Physics JETF 52 (1980), 511-516. MR 82i:81018
28.: M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Acad. Press, Edinburgh-London, 1973.
29.: L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, New York-London 1970. MR 44:3066
30.: R. Froese, I. Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof, -lower bounds to solutions of one-body Schrödinger equations, Proc. Royal Soc. Edinburgh, 95A (1983), 25-38. MR 86a:35044
31.: R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 164 (1995), 237-259. MR 96a:81026
32.: J.D. Joannopoulos, R.D. Meade, and J. N. Winn, Photonic Crystals. Molding the Flow of Light, Princeton Univ. Press, Princeton, NJ, 1995.
33.: Yu. Karpeshina, Spectrum and eigenfunctions of Schrödinger operator with zero-range potential of the homoeneous lattice type in three dimensional space, Ter. i Mat. Fiz. 57 (1983), no.2, 304-313; Engl. ransl. in Theor. and Math. Phys. 57 (1983), 1156-1162. MR 86b:81016
34.: Yu. Karpeshina, Perturbation Theory for the Schrödinger Operator with a Periodic Potential. Lecture Notes in Mathematics, v. 1663. Springer-Verlag, Berlin, 1997. MR 2001i:35002
35.: H. Knörrer and E. Trubowitz, A directional compactification of the complex Bloch variety, Comment. Math. Helv. 65 (1990), no. 1, 114-149. MR 91k:58134
36.: P. Kuchment, Floquet Theory for Partial Differential Equations, Russian Math. Surveys, 37 (1982), no.4, 1-60. MR 84b:35018
37.: P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser Verlag, Basel 1993. MR 94h:35002
38.: P. Kuchment, Mathematics of photonic crystals, in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters (Editors) SIAM, 2001.
39.: P. Kuchment and L. Kunyansky, Spectral properties of high-contrast band-gap materials and operators on graphs, Experimental Mathematics 8 (1999), no.1, 1-28. MR 2000c:78027
40.: P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, to appear in Adv. Comp. Math.
41.: P. Kuchment and S. Levendorskiî, On the absolute continuity of spectra of periodic elliptic operators, in Mathematical Results in Quantum Mechanics (QMath7, Prague, June 22-26,1998) , J. Dittrich, P. Exner, et al (Editors), 291-297, Operator Theory, Adv. and Appl., v. 108, Birkhäuser, Basel, 1999. MR 2001c:35166
42.: P. Kuchment and Y. Pinchover, Integral representations and Liouville theorems for solutions of periodic elliptic equations, J. Funct. Anal. 181 (2001), 402-446. CMP 2001:10
43.: P. Kuchment and B. Vainberg, On embedded eigenvalues of perturbed periodic Schrödinger operators, in Spectral and Scattering Theory (Newark, DE, 1997), 67-75, Plenum, New York, 1998. MR 99e:35032
44.: 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. CMP 2000:17
45.: B. Malgrange, Lectures on the theory of functions of several complex variables, Tata Institute of Fundamental Research, Bombay, Springer-Verlag, Berlin 1958. MR 85c:32001 (reprint)
46.: 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. MR 92d:335032
47.: A. Morame, Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential, J. Phys. A: Math. Gen. 31 (1998), 7593-7601. MR 99i:81039
48.: A. Morame, The absolute continuity of the spectrum of Maxwell operator in periodic media, J. Math. Phys. 41 (2000), 7099-7108. CMP 2001:01
49.: G. Nakamura, Z. Sun, and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann. 303 (1995), no. 3, 377-388. MR 96m:35336
50.: S.Novikov, Two-dimensional Schrödinger operators in the periodic fields, Itogi Nauki i Tekhn.: Sovremennye Problemy Mat., vol. 23, VINITI, Moscow, 1983, 3-32; English transl., J. Soviet Math. 28 (1985), 1-19. MR 85i:81020
51.: V. P. Palamodov, Linear Differential Equations with Constant Coefficients, Springer-Verlag, Berlin 1970. MR 41:8793
52.: V. Palamodov, Harmonic synthesis of solutions of elliptic equations with periodic coefficients, Ann. Inst. Fourier 43 (1993). 751-768. MR 95f:35037
53.: A. Plis, Non-uniqueness in Cauchy's problem for differential equations of elliptic type, J. Math. Mech. 9 (1960), 557-562. MR 22:12305
54.: M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol.IV: Analysis of Operators, Academic Press, New York, 1978. MR 58:12429c
55.: Z. Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 2001, no. 1, 1-31.
56.: 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.
57.: Z. Shen, The periodic Schrödinger operator with potentials in the Morrey-Campanato class, Preprint 1999.
58.: 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.
59.: M. Shubin, Spectral theory and the index of elliptic operators with almost periodic coefficients, Russian Math. Surveys 34 (1979), no. 2, 109-157. MR 81f:35090
60.: J. Sjostrand, Microlocal analysis for the periodic magnetic Schrödinger equation and related questions, in : Microlocal Analysis and Applications, Lect. Notes in Math., v.1495, 237-332, Springer-Verlag, Berlin, 1991. MR 94f:35119
61.: 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). MR 87h:47110; MR 88g:47038
62.: G. Springer, Introduction to Riemann Surfaces, Chelsea, New York 1981. MR 19:1169g (orig. ed.)
63.: T. Suslina, Absolute continuity of the spectrum of periodic operators of mathematical physics, Journees Equations aux Derivees Partielles, Nantes, 5-9 Juin 2000, GDR 1151 (CNRS). MR 2001f:35295
64.: Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc. 338 (1993), no. 2, 953-969. MR 93j:35066
65.: A. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Inventiones Mathematicae 137 (1999), 85-112. MR 2000g:35028
66.: A. Sobolev and J. Walthoe, Absolute continuity in periodic waveguides, Preprint, September 2000.
67.: M. Taylor, Partial Differential Equations. I. Basic Theory, Applied Mathematical Sciences, v. 115. Springer-Verlag, New York, 1996. MR 98b:35002
68.: L. E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335-343. MR 48:13084
69.: I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London; Addison Wesley Pub. Co., Inc., Reading, Mass. 1962. MR 27:321
70.: M. Zaidenberg, S. Krein, P. Kuchment, and A. Pankov, Banach bundles and linear operators, Russian Math. Surveys 30 (1975), no.5, 115-175. MR 54:3741
71.: J. Zak, Magnetic translation group I, II, Phys. Rev. 134 (1964), no. 6A, A1602-A1611. MR 31:2031; MR 31:2032

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35P99, 35J10

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
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

DOI: 10.1090/S0002-9947-01-02878-1
PII: S 0002-9947(01)02878-1
Keywords: Periodic operator, elliptic operator, absolutely continuous spectrum, Schr\"{o}dinger operator, magnetic and electric potentials
Received by editor(s): October 3, 2000
Posted: 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.
Copyright of article: Copyright 2001, American Mathematical Society

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