Abstract
The goal is to investigate spectral properties of the operator H=(−i∇ +a(x))2+a0(x) in the two-dimensional situation, a(x), a0(x)) being periodic. We construct asymptotic formulae for Bloch eigenvalues and eigenfunctions in the high-energy region, describe properties of isoenergetic curves in the space of quasimomenta and give a new proof of the Bethe-Sommerfeld conjecture.
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References
Bethe, G., Sommerfeld, A.: Elektronentheorie der Metalle, Berlin-New York: Springer Verlag, 1967
Birman, M.Sh., Suslina, T.A.: The two-dimensional periodic magnetic Hamiltonian is absolutely continuous. Algebra i Analiz 9(1), 32 –48 (1997); translation in St.Petersburg Math. J. 9(1), 21 –32 (1998)
Birman, M.Sh., Suslina, T.A.: Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential. Algebra i Analiz 10(4), (1998) translation in St.Petersburg Math. J 10(4), 1–26 (1999)
Birman, M.Sh., Suslina, T.A.: Periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity. Algebra i Analiz 11, 2 (1999) translation in St.Petersburg Math. J. 11, 2 (2000)
Brüning, J., Sunada, T.: On the spectrum of periodic elliptic operators. Nagoya Math. J. 126, 159–171 (1992)
Dahlberg, B.E.J., Trubowitz, E.: A Remark on Two Dimensional Periodic Potentials. Comment. Math. Helv. 57, 130 –134 (1982)
Dubrovin, D.A., Novikov, S.P.: Ground states in a periodc field. Magnetic Bloch functions and vector bundles. Soviet Math. Dokl. 22, 240 – 244 (1980)
Friedlander, L.: On the spectrum of a class of second order periodic elliptic differential operators. Commun. Math. Phys. 229(1), 49–55 (2002)
Gel’fand, I.M.: Expansion in Eigenfunctions of an Equation with Periodic Coefficients. Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950) (in Russian)
Helffer, B., Mohamed, A.: Asymptotic of the density of states for the Schrödinger Operator with Periodic Electric Potential. Duke Math. J. 92(1), 1–60 (1998)
Hempel, R., Herbst, I.: Strong magnetic fields, Dirichlet boundaries and spectral gaps. Commun. Math. Phys. 164, 237 – 259 (1995)
Hempel, R., Herbst, I.: Bands and gaps for periodic magnetic Hamiltonians. In: Partial differential operators and mathematical physics (Holzhau, 1994), Oper. Theory Adv. Appl. 78, Basel: Birkhauser, 1995, pp. 175–184
Iwatsuka, A.: On Schrödinger operators with magnetic fields. In: Lecture Notes in Mathematics 1450, Berlin: Springer-Verlag, 1990, pp. 157–172
Kato, T.: Perturbation Theory for Linear Operators. Berlin-Heidelberg-New York: Springer-Verlag, 1966
Karpeshina, Yu. E.: Analytic Perturbation Theory for a Periodic Potential. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 45–65 (1989); English transl.: Math. USSR Izv. 34(1), 43–63 (1990)
Karpeshina, Yu.E.: Perturbation Theory for the Schrödinger Operator with a Periodic Potential. Trudy Mat. Inst. Steklov 188, 88 – 116 (1990); Engl. transl.: Proceedings of the Steklov Institute of Mathematics, 1991, Issue 3, pp. 109–145
Karpeshina, Yu.E.: Perturbation Theory for the Schrödinger operator with a periodic potential. Lecture Notes in Mathematics, # 1663, Berlin-Heidelberg-New York: Springer-Verlag, 1997, pp. 352
Karpeshina, Yu.E.: System of Basic Functions for the Two-Dimensional Periodic Magnetic Schrödinger Operator. http:// rene.ma.Utexas.edu/mp-arc-bin/mpa? Yn=01-340,2001
Karpeshina Yu.E.: On the Periodic Magnetic Schrödinger Operator in Rd. Eigenvalues and Model Functions. Operator Theory: Advances and Applications. 132, Basel: Birkhäuser, 2002, pp. 219–231
Kuchment, P.: Floquet theory for partial differential equations. Basel: Birkhäuser, 1993
Kuchment, P., Levendorskii, S.: On absolute continuity of spectra of periodic elliptic operators. In: Mathematical Results in Quantum Mechanics, J. Dittrich, P.Exner et al (eds.), Operator Theory. Advances and Applications 108, Basel: Bürkhäuser, 1999, pp. 291–297
Landau, L.D., Livshitz, E.M.: Quantum mechanics, London: Pergamon Press, 1958
Madelung, O.: Introduction to Solid State Theory. Berlin, New-York: Springer-Verlag, 1978
Mohamed, A.: Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential. J. Math. Phys. 38(8), 4023–4051 (1997)
Parnovski, L., Sobolev, A. V.: Lattice points, perturbation theory and the periodic polyharmonic operator. Ann. Henri Poincare 2(3), 573–581 (2001)
Parnovski, L., Sobolev, A. V.: On the Bethe-Sommerfeld conjecture for the polyharmonic operator. Duke Math. J. 107(2), 209–238 (2001)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics., Vol IV, New York-San Francisco-London: Academic Press, 3rd ed., 1987
Sjöstrand, J.: Microlocal analysis for periodic magnetic Schrödinger equations and related questions. In: Microlocal Analysis and Applications, Lecture Notes in Physics, 1495, Berlin: Springer – Verlag, 1991, pp. 237–332
Skriganov, M.M.: Proof of the Bethe-Sommerfeld Conjecture in Dimension Two. DAN SSSR, 248, 1, 49–52 (1979); English transl. in Soviet Math. Dokl. 20(5), 956 – 959 (1979)
Skriganov, M.M.: Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators. Trudy Mat. Inst. Steklov. 171, 1–121 (1985); Engl. transl.: Proc. Steklov Inst. Math. 171, 2 (1987)
Skriganov, M.M.: The Spectrum Band Structure of the Three-Dimensional Schrödinger Operator with a Periodic Potential. Invent. Math. 80, 107–121 (1985)
Sobolev, A.: Absolute continuity of the periodic magnetic Schrödinger operator. Invent. Math. 137(1), 85–112 (1999)
Thomas, L.E.: Time-dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33, 335–343 (1973)
Veliev, O.A.: Asymptotic Formulae for Eigenvalues of a Periodic Schrödinger Operator and Bethe-Sommerfeld Conjecture. Functional. Anal. i Prilozhen. 21(2), 1–15 (1987); Engl. transl.: Funct. Anal. Appl. 21, 87–99 (1987)
Veliev, O.A.: Asymptotic Formulae for Bloch Functions of Multidimensional Periodic Schrödinger operator and Some of Their Applications. In: Spectral Theory of Operators and its Applications, 9, Baku, 1989, pp. 59 –76 (in Russian)
Veliev, O.A.: The Periodic Multidimensional Schrodinger Operator, Part 2, Asymptotic Formulae for Bloch Functions and Fermi Surfaces. http://rene.ma.Uteras.edu/mp-arc-bin/mpa? yn=01-463, 2001
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Communicated by B. Simon
Research partially supported by USNSF Grant DMS-0201383.
Acknowledgements The author is thankful to Konstantin Makarov for very useful discussions and to Young-Ran Lee for her great help with pictures.
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Karpeshina, Y. Spectral Properties of the Periodic Magnetic Schrödinger Operator in the High-Energy Region. Two-Dimensional Case. Commun. Math. Phys. 251, 473–514 (2004). https://doi.org/10.1007/s00220-004-1129-0
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DOI: https://doi.org/10.1007/s00220-004-1129-0