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Absence of eigenvalues for the generalized two-dimensional periodic Dirac operator. (English. Russian original) Zbl 1153.35367
St. Petersbg. Math. J. 17, No. 3, 409-433 (2006); translation from Algebra Anal. 17, No. 3, 47-80 (2006).
Summary: A generalized two-dimensional periodic Dirac operator is considered, with $$L^\infty$$-matrix-valued coefficients of the first-order derivatives and with complex matrix-valued potential. It is proved that if the matrix-valued potential has zero bound relative to the free Dirac operator, then the spectrum of the operator in question contains no eigenvalues.

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 47F05 General theory of partial differential operators 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 35B10 Periodic solutions to PDEs
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##### References:
 [1] Peter Kuchment, Floquet theory for partial differential equations, Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. · Zbl 0789.35002 [2] Peter Kuchment and Sergei Levendorskiî, On the structure of spectra of periodic elliptic operators, Trans. Amer. Math. Soc. 354 (2002), no. 2, 537 – 569. · Zbl 1058.35174 [3] L. I. Danilov, On the spectrum of the Dirac operator with periodic potential, Preprint, Fiz.-Tekhn. Inst. Ural. Otdel. Akad. Nauk SSSR, Sverdlovsk, 1987. (Russian) [4] -, A property of the integer lattice in $$\mathbf{R}^3$$ and the spectrum of the Dirac operator with periodic potential, Preprint, Fiz.-Tekhn. Inst. Ural. Otdel. Akad. Nauk SSSR, Sverdlovsk, 1988. (Russian) [5] L. I. Danilov, On the spectrum of the Dirac operator in \?$$^{n}$$ 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). · Zbl 0723.47043 [6] -, The spectrum of the Dirac operator with periodic potential. I, Fiz.-Tekhn. Inst. Ural. Otdel. Akad. Nauk SSSR, Izhevsk, 1991. (Manuscript dep. VINITI 12.12.91, no. 4588-B91.) (Russian) [7] -, The spectrum of the Dirac operator with periodic potential. III, Fiz.-Tekhn. Inst. Ural. Otdel. Ross. Akad. Nauk, Izhevsk, 1992. (Manuscript dep. VINITI 10.07.92, no. 2252-B92.) (Russian) [8] -, The spectrum of the Dirac operator with periodic potential. VI, Fiz.-Tekhn. Inst. Ural. Otdel. Ross. Akad. Nauk, Izhevsk, 1996. (Manuscript dep. VINITI 31.12.96, no. 3855-B96.) (Russian) [9] 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. · Zbl 0855.35105 [10] L. I. Danilov, Absolute continuity of the spectrum of a periodic Dirac operator, Differ. Uravn. 36 (2000), no. 2, 233 – 240, 287 (Russian, with Russian summary); English transl., Differ. Equ. 36 (2000), no. 2, 262 – 271. · Zbl 1041.81026 [11] 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. · Zbl 1086.35506 [12] M. Sh. Birman and T. A. Suslina, The periodic Dirac operator is absolutely continuous, Integral Equations Operator Theory 34 (1999), no. 4, 377 – 395. · Zbl 0937.35032 [13] L. I. Danilov, On the spectrum of the periodic Dirac operator, Teoret. Mat. Fiz. 124 (2000), no. 1, 3 – 17 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 124 (2000), no. 1, 859 – 871. · Zbl 1031.81023 [14] 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. · Zbl 0890.35096 [15] 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. · Zbl 0922.35101 [16] I. S. Lapin, Absolute continuity of the spectra of two-dimensional periodic magnetic Schrödinger operator and Dirac operator with potentials in the Zygmund class, J. Math. Sci. (New York) 106 (2001), no. 3, 2952 – 2974. Function theory and phase transitions. [17] Alexander V. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Invent. Math. 137 (1999), no. 1, 85 – 112. · Zbl 0932.35049 [18] 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. · Zbl 0941.35015 [19] L. I. Danilov, On the absolute continuity of the spectrum of periodic Schrödinger and Dirac operators. I, Fiz.-Tekhn. Inst. Ural. Otdel. Ross. Akad. Nauk, Izhevsk, 2000. (Manuscript dep. VINITI 15.06.00, no. 1683-B00.) (Russian) · Zbl 1041.81026 [20] L. I. Danilov, On the absolute continuity of the spectrum of a periodic Schrödinger operator, Mat. Zametki 73 (2003), no. 1, 49 – 62 (Russian, with Russian summary); English transl., Math. Notes 73 (2003), no. 1-2, 46 – 57. · Zbl 1163.35443 [21] Zhongwei Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 1 (2001), 1 – 31. · Zbl 0998.35007 [22] Zhongwei Shen, Absolute continuity of periodic Schrödinger operators with potentials in the Kato class, Illinois J. Math. 45 (2001), no. 3, 873 – 893. · Zbl 1001.35029 [23] Zhongwei Shen, The periodic Schrödinger operators with potentials in the Morrey class, J. Funct. Anal. 193 (2002), no. 2, 314 – 345. · Zbl 1119.35316 [24] L. Friedlander, On the spectrum of a class of second order periodic elliptic differential operators, Comm. Math. Phys. 229 (2002), no. 1, 49 – 55. · Zbl 1014.35066 [25] T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the Schrödinger operator with the potential concentrated on a periodic system of hypersurfaces, Algebra i Analiz 13 (2001), no. 5, 197 – 240 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 859 – 891. [26] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. · Zbl 0619.47005 [27] 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. · Zbl 0931.35143 [28] L. I. Danilov, On absolute continuity of the spectrum of periodic Schrödinger and Dirac operators. II, Fiz.-Tekhn. Inst. Ural. Otdel. Ross. Akad. Nauk, Izhevsk, 2001. (Manuscript dep. VINITI 09.04.01, no. 916-B2001.) (Russian) [29] -, On the spectrum of the two-dimensional periodic Schrödinger and Dirac operators, Izv. Inst. Mat. i Inform. Udmurt. Univ., vyp. 3 (26), Izhevsk, 2002, pp. 3-98. (Russian) [30] L. I. Danilov, On the spectrum of the two-dimensional periodic Schrödinger operator, Teoret. Mat. Fiz. 134 (2003), no. 3, 447 – 459 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 134 (2003), no. 3, 392 – 403. · Zbl 1178.35275 [31] -, On absolute continuity of the spectrum of periodic Schrödinger and Dirac operators. III, Fiz.-Tekhn. Inst. Ural. Otdel. Ross. Akad. Nauk, Izhevsk, 2002. (Manuscript dep. VINITI 22.10.02, no. 1798-B2002.) (Russian) [32] -, On the absence of eigenvalues in the spectrum of two-dimensional periodic Dirac and Schrödinger operators, Izv. Inst. Mat. i Inform. Udmurt. Univ., vyp. 1 (29), Izhevsk, 2004, pp. 49-84. (Russian) [33] M. Sh. Birman, T. A. Suslina, and R. G. Shterenberg, Absolute continuity of the two-dimensional Schrödinger operator with delta potential concentrated on a periodic system of curves, Algebra i Analiz 12 (2000), no. 6, 140 – 177 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 983 – 1012. [34] R. G. Shterenberg, Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with electric potential of measure derivative type, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 276 – 312, 318 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 115 (2003), no. 6, 2862 – 2882. · Zbl 1040.35049 [35] R. G. Shterenberg, Absolute continuity of the spectrum of the two-dimensional periodic Schrödinger operator with a positive electric potential, Algebra i Analiz 13 (2001), no. 4, 196 – 228 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 4, 659 – 683. · Zbl 1054.35039 [36] -, Absolute continuity of the spectrum of a two-dimensional magnetic periodic Schrödinger operator with positive electric potential, Trudy S.-Peterburg. Mat. Obshch. 9 (2001), 199-233; English transl. in Amer. Math. Soc. Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003. [37] -, Absolute continuity of spectra of two-dimensional periodic Schrödinger operators with strongly subordinate magnetic potentials, Report no. 21, 2002/2003, Mittag-Leffler Inst., Stockholm, 2002. [38] T. A. Suslina and R. G. Shterenberg, Absolute continuity of the spectrum of the magnetic Schrödinger operator with a metric in a two-dimensional periodic waveguide, Algebra i Analiz 14 (2002), no. 2, 159 – 206 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 2, 305 – 343. · Zbl 1057.35024 [39] R. G. Shterenberg, Schrödinger operator in a periodic waveguide on the plane and quasi-conformal mappings, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 295 (2003), 204-243. (Russian) [40] Alexander V. Sobolev and Jonathan Walthoe, Absolute continuity in periodic waveguides, Proc. London Math. Soc. (3) 85 (2002), no. 3, 717 – 741. · Zbl 1247.35077 [41] E. Shargorodsky and A. V. Sobolev, Quasi-conformal mappings and periodic spectral problems in dimension two, LANL Archives: math.SP/0109216 (2001). [42] Lawrence E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335 – 343. [43] I. M. Gel$$^{\prime}$$fand, Expansion in characteristic functions of an equation with periodic coefficients, Doklady Akad. Nauk SSSR (N.S.) 73 (1950), 1117 – 1120 (Russian). [44] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. 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. Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0242.46001 [45] Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. · Zbl 0405.47007
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