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Volumn 9, Issue 1, 2002, Pages 14-49

The spectral asymptotics of the two-dimensional Schrödinger operator with a strong magnetic field. I

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EID: 0036433126     PISSN: 10619208     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (26)

References (130)
  • 1
    • 0003308724 scopus 로고
    • Lectures on exponential decay of solutions of second-order elliptic equations
    • Princeton: Princeton Univ. Press
    • Agmon, S., Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations, Math. Notes, vol. 29, Princeton: Princeton Univ. Press, 1982.
    • (1982) Math. Notes , vol.29
    • Agmon, S.1
  • 6
    • 0009041768 scopus 로고
    • Modes and quasimodes
    • Arnold, V.I., Modes and Quasimodes, Funkts. Anal. Prilozh., 1972, vol. 6, no. 2, pp. 12-20; English transl.: Funct. Anal. Appl., 1972, vol. 6, pp. 94-101.
    • (1972) Funkts. Anal. Prilozh. , vol.6 , Issue.2 , pp. 12-20
    • Arnold, V.I.1
  • 7
    • 0009041768 scopus 로고
    • English transl.
    • Arnold, V.I., Modes and Quasimodes, Funkts. Anal. Prilozh., 1972, vol. 6, no. 2, pp. 12-20; English transl.: Funct. Anal. Appl., 1972, vol. 6, pp. 94-101.
    • (1972) Funct. Anal. Appl. , vol.6 , pp. 94-101
  • 8
    • 0002432075 scopus 로고
    • On the quantum hall effect
    • Avron, J. and Seiler, R., On the Quantum Hall Effect, J. Geom. Phys., 1984, vol. 1, no. 3, pp. 13-23.
    • (1984) J. Geom. Phys. , vol.1 , Issue.3 , pp. 13-23
    • Avron, J.1    Seiler, R.2
  • 9
    • 21344496244 scopus 로고
    • Charge transport, and comparison of dimensions
    • Avron, J., Seiler, R., and Simon, B., Charge Transport, and Comparison of Dimensions, Commun. Math. Phys., 1994, vol. 159, pp. 399-422.
    • (1994) Commun. Math. Phys. , vol.159 , pp. 399-422
    • Avron, J.1    Seiler, R.2    Simon, B.3
  • 10
    • 0001507295 scopus 로고
    • Energy spectrum of a condition electron in a magnetic field
    • Azbel, M.Ya., Energy Spectrum of a Condition Electron in a Magnetic Field, Zh. Exp. Teor. Fiz., 1964, vol. 46, pp. 929-938; English transl.: Sov. Phys. JETP, 1964, vol. 19, pp. 634-645.
    • (1964) Zh. Exp. Teor. Fiz. , vol.46 , pp. 929-938
    • Azbel, M.Ya.1
  • 11
    • 0000653017 scopus 로고
    • English transl.
    • Azbel, M.Ya., Energy Spectrum of a Condition Electron in a Magnetic Field, Zh. Exp. Teor. Fiz., 1964, vol. 46, pp. 929-938; English transl.: Sov. Phys. JETP, 1964, vol. 19, pp. 634-645.
    • (1964) Sov. Phys. JETP , vol.19 , pp. 634-645
  • 13
    • 37149037543 scopus 로고
    • Quantum transport in semiconductor nanostructures
    • Ehrenstein, H. and Turnhall, D., Eds., San Diego: Academic Press
    • Beenakker, C.W.J. and van Houten, H., Quantum Transport in Semiconductor Nanostructures, in Solid State Phys., Ehrenstein, H. and Turnhall, D., Eds., San Diego: Academic Press, 1991, vol. 44, p. 1.
    • (1991) Solid State Phys. , vol.44 , pp. 1
    • Beenakker, C.W.J.1    Van Houten, H.2
  • 14
    • 0003164337 scopus 로고
    • Heinsenberg's picture and non-commutative geometry in the semiclassical limit in quantum mechanics
    • Bellisard, J. and Vittot, M., Heinsenberg's Picture and Non-Commutative Geometry in the SemiClassical Limit in Quantum Mechanics, Ann. Inst. Henri Poincare, 1990, vol. 52, pp. 175-235.
    • (1990) Ann. Inst. Henri Poincare , vol.52 , pp. 175-235
    • Bellisard, J.1    Vittot, M.2
  • 15
    • 0003004075 scopus 로고
    • Irreducible representations of translational symmetry operators for the bloch electron Hamiltonian in magnetic field
    • Belokolos, E.D., Irreducible Representations of Translational Symmetry Operators for the Bloch Electron Hamiltonian in Magnetic Field, Teor. Mat. Fiz., 1971, vol. 7, no. 1, pp. 61-71.
    • (1971) Teor. Mat. Fiz. , vol.7 , Issue.1 , pp. 61-71
    • Belokolos, E.D.1
  • 16
    • 0000432034 scopus 로고
    • Quasi-classical maslov asymptotics with complex phases. I: General approach
    • Belov, V.V. and Dobrokhotov, S.Yu., Quasi-Classical Maslov Asymptotics with Complex Phases. I. General Approach, Teoret. Mat. Fiz., 1992, vol. 92, no. 2, pp. 215-254; English transl.: Theor. Math. Phys., 1992, vol. 92, no. 2, pp. 843-868.
    • (1992) Teoret. Mat. Fiz. , vol.92 , Issue.2 , pp. 215-254
    • Belov, V.V.1    Dobrokhotov, S.Yu.2
  • 17
    • 0007115752 scopus 로고
    • English transl.
    • Belov, V.V. and Dobrokhotov, S.Yu., Quasi-Classical Maslov Asymptotics with Complex Phases. I. General Approach, Teoret. Mat. Fiz., 1992, vol. 92, no. 2, pp. 215-254; English transl.: Theor. Math. Phys., 1992, vol. 92, no. 2, pp. 843-868.
    • (1992) Theor. Math. Phys. , vol.92 , Issue.2 , pp. 843-868
  • 18
    • 0001905748 scopus 로고    scopus 로고
    • Two-dimensional periodic magnetic Hamiltonian is absolutely continuous
    • Birman, M.Sh. and Suslina, T.A., Two-Dimensional Periodic Magnetic Hamiltonian Is Absolutely Continuous, Algebra i Analiz, 1997, vol. 9, no. 1, pp. 32-48; English transl.: St. Petersburg Math. J., 1998, vol. 9, no. 1, pp. 21-32.
    • (1997) Algebra i Analiz , vol.9 , Issue.1 , pp. 32-48
    • Birman, M.Sh.1    Suslina, T.A.2
  • 19
    • 0003191142 scopus 로고    scopus 로고
    • English transl.
    • Birman, M.Sh. and Suslina, T.A., Two-Dimensional Periodic Magnetic Hamiltonian Is Absolutely Continuous, Algebra i Analiz, 1997, vol. 9, no. 1, pp. 32-48; English transl.: St. Petersburg Math. J., 1998, vol. 9, no. 1, pp. 21-32.
    • (1998) St. Petersburg Math. J. , vol.9 , Issue.1 , pp. 21-32
  • 20
    • 34249773743 scopus 로고
    • Semiclassical quantization rules near separatrices
    • Bleher, P.M., Semiclassical Quantization Rules near Separatrices, Commun. Math. Phys., 1994, vol. 165, pp. 621-640.
    • (1994) Commun. Math. Phys. , vol.165 , pp. 621-640
    • Bleher, P.M.1
  • 22
    • 0343358890 scopus 로고
    • English transl., Berlin: Springer
    • Bogaevski, V.N. and Povzner, A. Ya., Algebraic Methods in Nonlinear Perturbation Theory, Moscow: Nauka, 1987; English transl.: Applied Mathematical Sciences, vol. 88, Berlin: Springer, 1991.
    • (1991) Applied Mathematical Sciences , vol.88
  • 24
    • 0003450054 scopus 로고
    • English transl., Jerusalem: Israel Program for Scientific Translations
    • Bogolyubov, N.N. and Mitropolski, Yu.A., Asymptotic Methods in Oscillation Theory, Moscow: Nauka, 1974; English transl.: Problems of the Asymptotic Theory of Nonstationary Vibrations, Jerusalem: Israel Program for Scientific Translations, 1965.
    • (1965) Problems of the Asymptotic Theory of Nonstationary Vibrations
  • 25
    • 0002663934 scopus 로고
    • The methods of asymptotic approximation for systems with rotating phase and its application to the motion of charged particles in a magnetic field
    • Bogolyubov, N.N. and Zubarev, D.N., The Methods of Asymptotic Approximation for Systems with Rotating Phase and Its Application to the Motion of Charged Particles in a Magnetic Field, Ukr. Mat. Zh., 1955, vol. 7, no. 1, pp. 5-17.
    • (1955) Ukr. Mat. Zh. , vol.7 , Issue.1 , pp. 5-17
    • Bogolyubov, N.N.1    Zubarev, D.N.2
  • 28
    • 0004233461 scopus 로고    scopus 로고
    • Quasi-periodic motions in families of dynamical systems: Order admits chaos
    • Berlin: Springer
    • Broer, H.W., Huitema, G.B., and Sevrjuk, M.B., Quasi-Periodic Motions in Families of Dynamical Systems: Order Admits Chaos, Lecture Notes in Mathematics, vol. 1645, Berlin: Springer, 1996.
    • (1996) Lecture Notes in Mathematics , vol.1645
    • Broer, H.W.1    Huitema, G.B.2    Sevrjuk, M.B.3
  • 29
    • 0035547892 scopus 로고    scopus 로고
    • A global semiclassical description of the spectrum of the two-dimensional magnetic Schrödinger operator with periodic electric potential
    • Brüning, J. and Dobrokhotov, S.Yu., A Global Semiclassical Description of the Spectrum of the Two-dimensional Magnetic Schrödinger Operator with Periodic Electric Potential, Dokl. Ross. Akad. Nauk, 2001, vol. 379, no. 3, pp. 313-317; English transl.: Doklady Math., 2001, vol. 64, no. 1, pp. 131-136.
    • (2001) Dokl. Ross. Akad. Nauk , vol.379 , Issue.3 , pp. 313-317
    • Brüning, J.1    Dobrokhotov, S.Yu.2
  • 30
    • 0001373781 scopus 로고    scopus 로고
    • English transl.
    • Brüning, J. and Dobrokhotov, S.Yu., A Global Semiclassical Description of the Spectrum of the Two-dimensional Magnetic Schrödinger Operator with Periodic Electric Potential, Dokl. Ross. Akad. Nauk, 2001, vol. 379, no. 3, pp. 313-317; English transl.: Doklady Math., 2001, vol. 64, no. 1, pp. 131-136.
    • (2001) Doklady Math. , vol.64 , Issue.1 , pp. 131-136
  • 31
    • 0000264040 scopus 로고    scopus 로고
    • Averaging of Hamiltonian systems with one fast phase and small amplitude
    • Brüning, J., Dobrokhotov, S.Yu., and Poteryakhin, M.A., Averaging of Hamiltonian Systems with One Fast Phase and Small Amplitude, Mat. Zametki, 2001, vol. 70, no. 5, pp. 660-669; English transl.: Math. Notes, 2001, vol. 70, no. 5, pp. 599-607.
    • (2001) Mat. Zametki , vol.70 , Issue.5 , pp. 660-669
    • Brüning, J.1    Dobrokhotov, S.Yu.2    Poteryakhin, M.A.3
  • 32
    • 24244440306 scopus 로고    scopus 로고
    • English transl.
    • Brüning, J., Dobrokhotov, S.Yu., and Poteryakhin, M.A., Averaging of Hamiltonian Systems with One Fast Phase and Small Amplitude, Mat. Zametki, 2001, vol. 70, no. 5, pp. 660-669; English transl.: Math. Notes, 2001, vol. 70, no. 5, pp. 599-607.
    • (2001) Math. Notes , vol.70 , Issue.5 , pp. 599-607
  • 33
    • 0002726473 scopus 로고
    • Bloch solutions of difference equations
    • Buslaev, V. and Fedotov, A., Bloch Solutions of Difference Equations, Algebra i Analiz, 1995, vol. 7, no. 4, pp. 74-121; English transl.: St. Petersburg Math. J., 1996, vol. 7, no. 4, pp. 561-594.
    • (1995) Algebra i Analiz , vol.7 , Issue.4 , pp. 74-121
    • Buslaev, V.1    Fedotov, A.2
  • 34
    • 0000168768 scopus 로고    scopus 로고
    • English transl.
    • Buslaev, V. and Fedotov, A., Bloch Solutions of Difference Equations, Algebra i Analiz, 1995, vol. 7, no. 4, pp. 74-121; English transl.: St. Petersburg Math. J., 1996, vol. 7, no. 4, pp. 561-594.
    • (1996) St. Petersburg Math. J. , vol.7 , Issue.4 , pp. 561-594
  • 35
    • 0002373689 scopus 로고    scopus 로고
    • Harper equation: Monodromization without semiclassics
    • Buslaev, V. and Fedotov, A., Harper Equation: Monodromization without Semiclassics, Algebra i Analiz, 1996, vol. 8, no. 2, pp. 75-109; English transl.: St. Petersburg Math. J., 1997, vol. 8, no. 2, pp. 231-254.
    • (1996) Algebra i Analiz , vol.8 , Issue.2 , pp. 75-109
    • Buslaev, V.1    Fedotov, A.2
  • 36
    • 0000616250 scopus 로고    scopus 로고
    • English transl.
    • Buslaev, V. and Fedotov, A., Harper Equation: Monodromization without Semiclassics, Algebra i Analiz, 1996, vol. 8, no. 2, pp. 75-109; English transl.: St. Petersburg Math. J., 1997, vol. 8, no. 2, pp. 231-254.
    • (1997) St. Petersburg Math. J. , vol.8 , Issue.2 , pp. 231-254
  • 38
    • 0001777364 scopus 로고
    • Spectre conjoint d'opérateurs pseudo-diffèrentiels qui commmutent. II: Le cas intégrable
    • Colin de Verdière, Y., Spectre conjoint d'opérateurs pseudo-diffèrentiels qui commmutent. II. Le cas intégrable, Math. Z., 1980, vol. 171, no. 1, pp. 51-73.
    • (1980) Math. Z. , vol.171 , Issue.1 , pp. 51-73
    • Colin de Verdière, Y.1
  • 40
    • 0001511248 scopus 로고
    • Eigenvalues of the Schrödinger operator for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis
    • Connor, J.N.L., Uzer, T., Marcus, R.A., and Smith, A.D., Eigenvalues of the Schrödinger Operator for a Periodic Potential with Nonperiodic Boundary Conditions: a Uniform Semiclassical Analysis, J. Chem. Phys., 1984, vol. 80, no. 10, pp. 5095-5106.
    • (1984) J. Chem. Phys. , vol.80 , Issue.10 , pp. 5095-5106
    • Connor, J.N.L.1    Uzer, T.2    Marcus, R.A.3    Smith, A.D.4
  • 41
    • 0001639158 scopus 로고    scopus 로고
    • Some questions of spectral theory of discrete operators with quasi-periodic coefficients
    • Dinaburg, E.I., Some Questions of Spectral Theory of Discrete Operators with Quasi-Periodic Coefficients, Uspekhi Mat. Nauk, 1997, vol. 52, no. 3, pp. 3-52; English transl.: Russ. Math. Surv., 1997, vol. 52, no. 3, pp. 451-499.
    • (1997) Uspekhi Mat. Nauk , vol.52 , Issue.3 , pp. 3-52
    • Dinaburg, E.I.1
  • 42
    • 0001639157 scopus 로고    scopus 로고
    • English transl.
    • Dinaburg, E.I., Some Questions of Spectral Theory of Discrete Operators with Quasi-Periodic Coefficients, Uspekhi Mat. Nauk, 1997, vol. 52, no. 3, pp. 3-52; English transl.: Russ. Math. Surv., 1997, vol. 52, no. 3, pp. 451-499.
    • (1997) Russ. Math. Surv. , vol.52 , Issue.3 , pp. 451-499
  • 43
    • 0031518869 scopus 로고    scopus 로고
    • Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations
    • Dinaburg, E.I., Sinai, Ya.G., and Soshnikov, A.B., Splitting of the Low Landau Levels into a Set of Positive Lebesgue Measure under Small Periodic Perturbations, Commun. Math. Phys., 1997, vol. 189, pp. 559-575.
    • (1997) Commun. Math. Phys. , vol.189 , pp. 559-575
    • Dinaburg, E.I.1    Sinai, Ya.G.2    Soshnikov, A.B.3
  • 44
    • 0002812374 scopus 로고
    • Quantization of the Bellman equation, exponential asymptotics and tunneling
    • AMS
    • Dobrokhotov, S.Yu., Kolokoltsov, V.N., and Maslov, V.P., Quantization of the Bellman Equation, Exponential Asymptotics and Tunneling, Adv. Sov. Math., vol. 13, AMS, 1992, pp. 1-46.
    • (1992) Adv. Sov. Math. , vol.13 , pp. 1-46
    • Dobrokhotov, S.Yu.1    Kolokoltsov, V.N.2    Maslov, V.P.3
  • 45
    • 0002969521 scopus 로고    scopus 로고
    • Closed trajectories and two-dimensional tort in the quantum spectral problem for three-dimensional anharmonic oscillator
    • Dobrokhotov, S.Yu. and Martínez-Olivé, V., Closed Trajectories and Two-Dimensional Tort in the Quantum Spectral Problem for Three-Dimensional Anharmonic Oscillator, Trudy Mosk. Matem. Ob., 1997, vol. 58, pp. 3-87; English transl.: Trans. Moscow Math. Soc., 1997, vol. 58, pp. 1-73.
    • (1997) Trudy Mosk. Matem. Ob. , vol.58 , pp. 3-87
    • Dobrokhotov, S.Yu.1    Martínez-Olivé, V.2
  • 46
    • 0012290156 scopus 로고    scopus 로고
    • English transl.
    • Dobrokhotov, S.Yu. and Martínez-Olivé, V., Closed Trajectories and Two-Dimensional Tort in the Quantum Spectral Problem for Three-Dimensional Anharmonic Oscillator, Trudy Mosk. Matem. Ob., 1997, vol. 58, pp. 3-87; English transl.: Trans. Moscow Math. Soc., 1997, vol. 58, pp. 1-73.
    • (1997) Trans. Moscow Math. Soc. , vol.58 , pp. 1-73
  • 47
    • 0000961844 scopus 로고    scopus 로고
    • "Momentum" tunneling between toff and the splitting of eigenvalues of the beltrami-laplace operator on Liouville surfaces
    • Dobrokhotov, S.Yu. and Shafarevich, A.I., "Momentum" Tunneling between Toff and the Splitting of Eigenvalues of the Beltrami-Laplace Operator on Liouville Surfaces, Math. Phys. Anal. Geom., 1999, vol. 2, pp. 141-177.
    • (1999) Math. Phys. Anal. Geom. , vol.2 , pp. 141-177
    • Dobrokhotov, S.Yu.1    Shafarevich, A.I.2
  • 48
    • 0001772195 scopus 로고
    • Nonlinear KdV-type equations, finite-gap linear operators and abelian manifolds
    • Dubrovin, B.A., Matveev, V.B., and Novikov, S.P., Nonlinear KdV-Type Equations, Finite-Gap Linear Operators and Abelian Manifolds, Uspekhi Mat. Nauk, 1976, vol. 31, pp. 55-136; English transl.: Russ. Math. Surv., 1976, vol. 31, pp. 59-146.
    • (1976) Uspekhi Mat. Nauk , vol.31 , pp. 55-136
    • Dubrovin, B.A.1    Matveev, V.B.2    Novikov, S.P.3
  • 49
    • 0003178926 scopus 로고
    • English transl.
    • Dubrovin, B.A., Matveev, V.B., and Novikov, S.P., Nonlinear KdV-Type Equations, Finite-Gap Linear Operators and Abelian Manifolds, Uspekhi Mat. Nauk, 1976, vol. 31, pp. 55-136; English transl.: Russ. Math. Surv., 1976, vol. 31, pp. 59-146.
    • (1976) Russ. Math. Surv. , vol.31 , pp. 59-146
  • 50
    • 0000769217 scopus 로고
    • Ground states in a periodic field. Magnetic bloch function and vector bundles
    • Dubrovin, B.A. and Novikov, S.P., Ground States in a Periodic Field. Magnetic Bloch Function and Vector Bundles, Dokl. AN SSSR, 1980, vol. 253, pp. 1293-1297; English transl.: Sov. Math. Doklady, 1980, vol. 22, no. 1, pp. 240-244.
    • (1980) Dokl. AN SSSR , vol.253 , pp. 1293-1297
    • Dubrovin, B.A.1    Novikov, S.P.2
  • 51
    • 0000798363 scopus 로고
    • English transl.
    • Dubrovin, B.A. and Novikov, S.P., Ground States in a Periodic Field. Magnetic Bloch Function and Vector Bundles, Dokl. AN SSSR, 1980, vol. 253, pp. 1293-1297; English transl.: Sov. Math. Doklady, 1980, vol. 22, no. 1, pp. 240-244.
    • (1980) Sov. Math. Doklady , vol.22 , Issue.1 , pp. 240-244
  • 52
    • 0000095445 scopus 로고
    • Ground states of a two-dimensional electron in a periodic magnetic field
    • Dubrovin, B.A. and Novikov, S.P., Ground States of a Two-Dimensional Electron in a Periodic Magnetic Field, Zh. Exper. Teor. Fiz., 1980, vol. 79, no. 3, pp. 1006-1016; English transl.: Sov. Phys. JETP, 1980, vol. 52, no. 3, pp. 511-516.
    • (1980) Zh. Exper. Teor. Fiz. , vol.79 , Issue.3 , pp. 1006-1016
    • Dubrovin, B.A.1    Novikov, S.P.2
  • 53
    • 0001225877 scopus 로고
    • English transl.
    • Dubrovin, B.A. and Novikov, S.P., Ground States of a Two-Dimensional Electron in a Periodic Magnetic Field, Zh. Exper. Teor. Fiz., 1980, vol. 79, no. 3, pp. 1006-1016; English transl.: Sov. Phys. JETP, 1980, vol. 52, no. 3, pp. 511-516.
    • (1980) Sov. Phys. JETP , vol.52 , Issue.3 , pp. 511-516
  • 55
    • 0034723117 scopus 로고    scopus 로고
    • Topological properties of quantum periodic Hamiltonians
    • Faure, F., Topological Properties of Quantum Periodic Hamiltonians, J. Phys. A, vol. 33, pp. 531-555.
    • J. Phys. A , vol.33 , pp. 531-555
    • Faure, F.1
  • 56
    • 0034342141 scopus 로고    scopus 로고
    • Semiclassical study of the origin of quantized hall conductance in periodic potential
    • Faure, P. and Parisse, B., Semiclassical Study of the Origin of Quantized Hall Conductance in Periodic Potential, J. Math. Phys., 2000, vol. 41, no. 1, pp. 62-75.
    • (2000) J. Math. Phys. , vol.41 , Issue.1 , pp. 62-75
    • Faure, P.1    Parisse, B.2
  • 58
    • 0003597264 scopus 로고
    • English transl., Berlin: Springer
    • Fedoryuk, M.V., Asymptotic Methods for Ordinary Differential Equations, Moscow: Nauka, 1983; English transl.: Asymptotic Analysis: Linear Ordinary Differential Equations, Berlin: Springer, 1993.
    • (1993) Asymptotic Analysis: Linear Ordinary Differential Equations
  • 59
    • 0000939872 scopus 로고
    • Topological classification of integrable systems
    • Providence, Rhode Island: AMS
    • Fomenko, A.T., Topological Classification of Integrable Systems, Adv. Sov. Math., vol. 6, Providence, Rhode Island: AMS, 1991.
    • (1991) Adv. Sov. Math. , vol.6
    • Fomenko, A.T.1
  • 60
    • 0001123378 scopus 로고
    • Expansion by eigenfunctions of equations with periodic coefficients
    • Gelfand, I.M., Expansion by Eigenfunctions of Equations with Periodic Coefficients, Dokl. Akad. Nauk SSSR, 1950, vol. 73, no. 6, pp. 1117-1120.
    • (1950) Dokl. Akad. Nauk SSSR , vol.73 , Issue.6 , pp. 1117-1120
    • Gelfand, I.M.1
  • 62
    • 0000273567 scopus 로고
    • The two-dimensional Schrödinger operator with a uniform magnetic field, and its perturbation by periodic zero-range potentials
    • Geyler, V.A., The Two-Dimensional Schrödinger Operator with a Uniform Magnetic Field, and Its Perturbation by Periodic Zero-Range Potentials, Algebra i Analiz, 1991, no. 3, pp. 1-48; English transl.: St. Petersburg Math. J., 1992, no. 3, pp. 489-532.
    • (1991) Algebra i Analiz , Issue.3 , pp. 1-48
    • Geyler, V.A.1
  • 63
    • 0000782854 scopus 로고
    • English transl.
    • Geyler, V.A., The Two-Dimensional Schrödinger Operator with a Uniform Magnetic Field, and Its Perturbation by Periodic Zero-Range Potentials, Algebra i Analiz, 1991, no. 3, pp. 1-48; English transl.: St. Petersburg Math. J., 1992, no. 3, pp. 489-532.
    • (1992) St. Petersburg Math. J. , Issue.3 , pp. 489-532
  • 65
    • 0003278519 scopus 로고
    • Semi-classical analysis for the Schrödinger operator and applications
    • Berlin: Springer
    • Helffer, B., Semi-Classical Analysis for the Schrödinger Operator and Applications, Lect. Notes Math., vol. 1336, Berlin: Springer, 1988.
    • (1988) Lect. Notes Math. , vol.1336
    • Helffer, B.1
  • 66
    • 0003307540 scopus 로고
    • Analyse semi-classique pour l'équation de Harper (avec application à l'équation de Schrödinger avec champs magnétiques)
    • Helffer, B. and Sjöstrand, J., Analyse Semi-classique pour l'équation de Harper (avec application à l'équation de Schrödinger avec champs magnétiques), Mém. Soc. Math. Fr., Nouv. Sér., vol. 34, 1988.
    • (1988) Mém. Soc. Math. Fr., Nouv. Sér. , vol.34
    • Helffer, B.1    Sjöstrand, J.2
  • 67
    • 0000301201 scopus 로고
    • Équation de Schrödinger avec champs magnétiques et équation de Harper
    • Helffer, B. and Sjöstrand, J., Équation de Schrödinger avec champs magnétiques et équation de Harper, Lect. Notes Phys., 1988, vol. 345, pp. 118-197.
    • (1988) Lect. Notes Phys. , vol.345 , pp. 118-197
    • Helffer, B.1    Sjöstrand, J.2
  • 68
    • 0002330387 scopus 로고
    • Semi-classical analysis for Harper's equation. III: Cantor structure of the spectrum
    • Helffer, B. and Sjöstrand, J., Semi-Classical Analysis for Harper's Equation. III. Cantor Structure of the Spectrum, Mem. Soc. Math. Fr., Nouv. Ser., vol. 39, 1989.
    • (1989) Mem. Soc. Math. Fr., Nouv. Ser. , vol.39
    • Helffer, B.1    Sjöstrand, J.2
  • 69
    • 0001032458 scopus 로고
    • Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields
    • Hofstadter, D., Energy Levels and Wave Functions of Bloch Electrons in Rational and Irrational Magnetic Fields, Phys. Rev. B, 1976, vol. 14, pp. 2239-2249.
    • (1976) Phys. Rev. B , vol.14 , pp. 2239-2249
    • Hofstadter, D.1
  • 71
    • 0011182321 scopus 로고
    • English transl., Providence, Rhode Island: AMS
    • Il'in, A.M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Moscow: Nauka, 1984; English transl.: Transl. Math. Monogr., 102, Providence, Rhode Island: AMS, 1992.
    • (1992) Transl. Math. Monogr. , pp. 102
  • 72
    • 0040507795 scopus 로고    scopus 로고
    • On normal behavior of partially elliptic lower-dimensional tori of Hamiltonian systems
    • Jorba, A. and Villanueva, J., On Normal Behavior of Partially Elliptic Lower-Dimensional Tori of Hamiltonian Systems, Nonlinearity, 1997, vol. 10, pp. 783-822.
    • (1997) Nonlinearity , vol.10 , pp. 783-822
    • Jorba, A.1    Villanueva, J.2
  • 73
    • 0000136142 scopus 로고
    • Asymptotic and geometric quantization
    • Karasev, M.V. and Maslov, V.P., Asymptotic and Geometric Quantization, Uspekhi Mat. Nauk, 1984, vol. 39, no. 6, pp. 115-173; English transl.: Russ. Math. Surv., 1994, vol. 39, no. 6, pp. 133-205.
    • (1984) Uspekhi Mat. Nauk , vol.39 , Issue.6 , pp. 115-173
    • Karasev, M.V.1    Maslov, V.P.2
  • 74
    • 0000643984 scopus 로고
    • English transl.
    • Karasev, M.V. and Maslov, V.P., Asymptotic and Geometric Quantization, Uspekhi Mat. Nauk, 1984, vol. 39, no. 6, pp. 115-173; English transl.: Russ. Math. Surv., 1994, vol. 39, no. 6, pp. 133-205.
    • (1994) Russ. Math. Surv. , vol.39 , Issue.6 , pp. 133-205
  • 75
    • 0001904713 scopus 로고
    • Statistical properties of the spectra of the laplace-beltrami operators on Liouville surfaces
    • Kosygin, D.V., Minasov, A.A., and Sinai, Ya.G., Statistical Properties of the Spectra of the Laplace-Beltrami Operators on Liouville Surfaces, Uspekhi Mat. Nauk, 1993, vol. 48, no. 4, pp. 3-130; English transl.: Russ. Math. Surv., 1993, vol. 48, no. 4, pp. 1-142.
    • (1993) Uspekhi Mat. Nauk , vol.48 , Issue.4 , pp. 3-130
    • Kosygin, D.V.1    Minasov, A.A.2    Sinai, Ya.G.3
  • 76
    • 0010963512 scopus 로고
    • English transl.
    • Kosygin, D.V., Minasov, A.A., and Sinai, Ya.G., Statistical Properties of the Spectra of the Laplace-Beltrami Operators on Liouville Surfaces, Uspekhi Mat. Nauk, 1993, vol. 48, no. 4, pp. 3-130; English transl.: Russ. Math. Surv., 1993, vol. 48, no. 4, pp. 1-142.
    • (1993) Russ. Math. Surv. , vol.48 , Issue.4 , pp. 1-142
  • 77
    • 0030491042 scopus 로고    scopus 로고
    • Models of the Hofstadter-type
    • Kreft, Ch. and Seller, R., Models of the Hofstadter-Type, J. Math. Phys., 1996, vol. 37, no. 10, pp. 5207-5243.
    • (1996) J. Math. Phys. , vol.37 , Issue.10 , pp. 5207-5243
    • Kreft, Ch.1    Seller, R.2
  • 78
    • 0001123379 scopus 로고
    • A generalization of some investigations of A.M. Lyapunov on linear differential equations with periodic coefficients
    • Krein, M.G., A Generalization of Some Investigations of A.M. Lyapunov on Linear Differential Equations with Periodic Coefficients, Dokl. AN SSSR, 1950, vol. 73, pp. 445-448.
    • (1950) Dokl. AN SSSR , vol.73 , pp. 445-448
    • Krein, M.G.1
  • 79
    • 0002340021 scopus 로고
    • Perturbation of quasi-periodic solutions of infinite-dimensional Hamiltonian systems
    • Kuksin, S.B., Perturbation of Quasi-Periodic Solutions of Infinite-Dimensional Hamiltonian Systems, Izv. Akad. Nauk SSSR, Ser. Mat., 1988, vol. 52, pp. 41-63; English transl.: Math. USSR Izv., 1989, vol. 32, no. 1, pp. 39-62.
    • (1988) Izv. Akad. Nauk SSSR, Ser. Mat. , vol.52 , pp. 41-63
    • Kuksin, S.B.1
  • 80
    • 0003174915 scopus 로고
    • English transl.
    • Kuksin, S.B., Perturbation of Quasi-Periodic Solutions of Infinite-Dimensional Hamiltonian Systems, Izv. Akad. Nauk SSSR, Ser. Mat., 1988, vol. 52, pp. 41-63; English transl.: Math. USSR Izv., 1989, vol. 32, no. 1, pp. 39-62.
    • (1989) Math. USSR Izv. , vol.32 , Issue.1 , pp. 39-62
  • 85
    • 0001151901 scopus 로고
    • Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and weak potential
    • Lyskova, A.S., Topological Characteristics of the Spectrum of the Schrödinger Operator in a Magnetic Field and Weak Potential, Teor. Mat. Fiz., 1985, vol. 65, no. 3, pp. 368-378.
    • (1985) Teor. Mat. Fiz. , vol.65 , Issue.3 , pp. 368-378
    • Lyskova, A.S.1
  • 87
    • 0011538016 scopus 로고
    • English transl, Basel: Birkhäuser
    • Marchenko, V.A., Sturm-Liouville Operators and Their Applications, Kiev: Naukova Dumka, 1977; English transl, in Operator Theory: Adv. Appl., vol. 22, Basel: Birkhäuser, 1986.
    • (1986) Operator Theory: Adv. Appl. , vol.22
  • 88
    • 84974743838 scopus 로고    scopus 로고
    • Spectral asymptotics for Hill's equation near potential maximum
    • März, Ch., Spectral Asymptotics for Hill's Equation near Potential Maximum, Asymp. Anal., vol. 5, no. 3, pp. 221-267.
    • Asymp. Anal. , vol.5 , Issue.3 , pp. 221-267
    • März, Ch.1
  • 89
    • 0004313287 scopus 로고
    • Moscow: Izdat. Mosk. Gos. Univ.
    • Maslov, V.P., Perturbation Theory and Asymptotic Methods, Moscow: Izdat. Mosk. Gos. Univ., 1965; Asymptotic Methods and Perturbation Theory (Continuation), Moscow: Nauka, 1988; French transl, of the first book: Paris: Dunod, 1972.
    • (1965) Perturbation Theory and Asymptotic Methods
    • Maslov, V.P.1
  • 90
    • 0003824317 scopus 로고
    • Moscow: Nauka, French transl, of the first book: Paris: Dunod, 1972
    • Maslov, V.P., Perturbation Theory and Asymptotic Methods, Moscow: Izdat. Mosk. Gos. Univ., 1965; Asymptotic Methods and Perturbation Theory (Continuation), Moscow: Nauka, 1988; French transl, of the first book: Paris: Dunod, 1972.
    • (1988) Asymptotic Methods and Perturbation Theory (Continuation)
  • 94
    • 0003599656 scopus 로고
    • English transl., Dordrecht: Reidel
    • Maslov, V.P. and Fedoryuk, M.V., Semiclassical Approximation for the Equations of Quantum Me-chanics, Moscow: Nauka, 1976; English transl.: Semiclassical Approximation in Quantum Mechanics, Dordrecht: Reidel, 1981.
    • (1981) Semiclassical Approximation in Quantum Mechanics
  • 96
    • 0021660205 scopus 로고
    • The separation of motions in systems with rapidly rotating phase
    • Neishtadt, A.I., The Separation of Motions in Systems with Rapidly Rotating Phase, Prikl. Mat. Mekh., 1984, vol. 48, no. 2, pp. 197-204; English transl.: J. Appl. Math. Mech., 1985, vol. 48, no. 2, pp. 133-139.
    • (1984) Prikl. Mat. Mekh. , vol.48 , Issue.2 , pp. 197-204
    • Neishtadt, A.I.1
  • 97
    • 0021660205 scopus 로고
    • English transl.
    • Neishtadt, A.I., The Separation of Motions in Systems with Rapidly Rotating Phase, Prikl. Mat. Mekh., 1984, vol. 48, no. 2, pp. 197-204; English transl.: J. Appl. Math. Mech., 1985, vol. 48, no. 2, pp. 133-139.
    • (1985) J. Appl. Math. Mech. , vol.48 , Issue.2 , pp. 133-139
  • 99
    • 0001763296 scopus 로고
    • Magnetic bloch function and vector bundles. Typical dispersion laws and their quantum numbers
    • Novikov, S.P., Magnetic Bloch Function and Vector Bundles. Typical Dispersion Laws and Their Quantum Numbers, Dokl. Akad. Nauk SSSR, 1980, vol. 257, pp. 538-543; English transl.: Sov. Math. Doklady, 1981, vol. 23, no. 2, pp. 298-303.
    • (1980) Dokl. Akad. Nauk SSSR , vol.257 , pp. 538-543
    • Novikov, S.P.1
  • 100
    • 0001505369 scopus 로고
    • English transl.
    • Novikov, S.P., Magnetic Bloch Function and Vector Bundles. Typical Dispersion Laws and Their Quantum Numbers, Dokl. Akad. Nauk SSSR, 1980, vol. 257, pp. 538-543; English transl.: Sov. Math. Doklady, 1981, vol. 23, no. 2, pp. 298-303.
    • (1981) Sov. Math. Doklady , vol.23 , Issue.2 , pp. 298-303
  • 101
    • 0002233428 scopus 로고
    • Two-dimensional Schrödinger operators in a periodic field
    • Moscow: VINITI
    • Novikov, S.P., Two-Dimensional Schrödinger Operators in a Periodic Field, Itogi Nauki i Tekhniki: Sovrem. Problemy Matem., vol. 23, Moscow: VINITI, 1983, pp. 3-23; English transl.: J. Sov. Math., 1985, vol. 28, no. 1, pp. 1-20.
    • (1983) Itogi Nauki i Tekhniki: Sovrem. Problemy Matem. , vol.23 , pp. 3-23
    • Novikov, S.P.1
  • 102
    • 0003181719 scopus 로고
    • English transl.
    • Novikov, S.P., Two-Dimensional Schrödinger Operators in a Periodic Field, Itogi Nauki i Tekhniki: Sovrem. Problemy Matem., vol. 23, Moscow: VINITI, 1983, pp. 3-23; English transl.: J. Sov. Math., 1985, vol. 28, no. 1, pp. 1-20.
    • (1985) J. Sov. Math. , vol.28 , Issue.1 , pp. 1-20
  • 103
    • 0001412487 scopus 로고    scopus 로고
    • Topological phenomena in normal metals
    • Novikov, S.P. and Maltsev, A.Ya., Topological Phenomena in Normal Metals, Uspekhi Fiz. Nauk, 1998, vol. 168, no. 3, pp. 249-258; English transl.: Physics-Uspekhi, 1998, vol. 41, no. 3, pp. 231-239.
    • (1998) Uspekhi Fiz. Nauk , vol.168 , Issue.3 , pp. 249-258
    • Novikov, S.P.1    Maltsev, A.Ya.2
  • 104
    • 54649084927 scopus 로고    scopus 로고
    • English transl.
    • Novikov, S.P. and Maltsev, A.Ya., Topological Phenomena in Normal Metals, Uspekhi Fiz. Nauk, 1998, vol. 168, no. 3, pp. 249-258; English transl.: Physics-Uspekhi, 1998, vol. 41, no. 3, pp. 231-239.
    • (1998) Physics-Uspekhi , vol.41 , Issue.3 , pp. 231-239
  • 105
    • 0000064362 scopus 로고
    • Exactly solvable periodic 2-D Schrödinger operators
    • Novikov, S.P. and Veselov, A.P., Exactly Solvable Periodic 2-D Schrödinger Operators, Uspekhi Mat. Nauk, 1995, vol. 50, no. 6, pp. 171-172; English transl.: Russ. Math. Surv., 1995, vol. 50, no. 6.
    • (1995) Uspekhi Mat. Nauk , vol.50 , Issue.6 , pp. 171-172
    • Novikov, S.P.1    Veselov, A.P.2
  • 106
    • 0011655613 scopus 로고
    • English transl.
    • Novikov, S.P. and Veselov, A.P., Exactly Solvable Periodic 2-D Schrödinger Operators, Uspekhi Mat. Nauk, 1995, vol. 50, no. 6, pp. 171-172; English transl.: Russ. Math. Surv., 1995, vol. 50, no. 6.
    • (1995) Russ. Math. Surv. , vol.50 , Issue.6
  • 107
    • 0001528282 scopus 로고    scopus 로고
    • Exactly solvable two-dimensional Schrödinger operators and laplace transformations
    • Solitons, Geometry and Topology: on the Crossroad, Buchstaber, V.M. et al., Eds., Rhode Island: AMS
    • Novikov, S.P. and Veselov, A.P., Exactly Solvable Two-Dimensional Schrödinger Operators and Laplace Transformations, in Solitons, Geometry and Topology: on the Crossroad, Buchstaber, V.M. et al., Eds., AMS Transl., Ser. 2., vol. 179, Rhode Island: AMS, 1997, pp. 109-132.
    • (1997) AMS Transl., Ser. 2. , vol.179 , pp. 109-132
    • Novikov, S.P.1    Veselov, A.P.2
  • 108
  • 111
    • 0000285454 scopus 로고    scopus 로고
    • Invariant tort in non-degenerate nearly integrable Hamiltonian systems
    • Rüssman, H., Invariant Tort in Non-Degenerate Nearly Integrable Hamiltonian Systems, Regul. Chaotic Dyn., 2001, vol. 6, no. 2, pp. 119-204.
    • (2001) Regul. Chaotic Dyn. , vol.6 , Issue.2 , pp. 119-204
    • Rüssman, H.1
  • 113
    • 46149151422 scopus 로고    scopus 로고
    • Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds
    • Shubin, M.A., Essential Self-Adjointness for Magnetic Schrödinger Operators on Non-Compact Manifolds, Seminaire Équations aux Dérivées Partielles, 1998-1999, no. 24.
    • (1998) Seminaire Équations aux Dérivées Partielles , Issue.24
    • Shubin, M.A.1
  • 114
    • 0001149991 scopus 로고
    • Semiclassical analysis of low lying eigenvalues. I: Non-degenerate minima: Asymptotic expansions
    • Simon, B., Semiclassical Analysis of Low Lying Eigenvalues. I. Non-Degenerate Minima: Asymptotic Expansions, Ann. Inst. Henri Poincaré, 1983, vol. A38, pp. 295-308.
    • (1983) Ann. Inst. Henri Poincaré , vol.A38 , pp. 295-308
    • Simon, B.1
  • 115
    • 0001983592 scopus 로고
    • Semiclassical analysis of low lying eigenvalues. II: Tunneling
    • Simon, B., Semiclassical Analysis of Low Lying Eigenvalues. II. Tunneling, Ann. Math., 1984, vol. 120, pp. 89-118.
    • (1984) Ann. Math. , vol.120 , pp. 89-118
    • Simon, B.1
  • 116
    • 21844520222 scopus 로고
    • Quantum averaging. I. Poincaré-von Zeipel is Rayleigh-Schrödinger
    • Scherer, W., Quantum Averaging. I. Poincaré-von Zeipel Is Rayleigh-Schrödinger, J. Phys. A, 1994, vol. 27, pp. 8231-8246.
    • (1994) J. Phys. A , vol.27 , pp. 8231-8246
    • Scherer, W.1
  • 117
    • 0031582174 scopus 로고    scopus 로고
    • Quantum averaging. II: Kolmogorov's algorithm
    • Scherer, W., Quantum Averaging. II. Kolmogorov's Algorithm, J. Phys. A, 1997, vol. 30, no. 8, pp. 2825-2851.
    • (1997) J. Phys. A , vol.30 , Issue.8 , pp. 2825-2851
    • Scherer, W.1
  • 118
    • 0001967757 scopus 로고
    • Geometric and arithmetic methods in spectral problems of multi-dimensional periodic operators
    • Leningrad: Nauka
    • Skriganov, M.M., Geometric and Arithmetic Methods in Spectral Problems of Multi-Dimensional Periodic Operators, Trudy Mat. Inst. ira. Steklova, vol. 171, Leningrad: Nauka, 1985; English transl.: Proc. Steklov Inst. Math., vol. 171, 1987.
    • (1985) Trudy Mat. Inst. ira. Steklova , vol.171
    • Skriganov, M.M.1
  • 119
    • 0011534830 scopus 로고
    • English transl.
    • Skriganov, M.M., Geometric and Arithmetic Methods in Spectral Problems of Multi-Dimensional Periodic Operators, Trudy Mat. Inst. ira. Steklova, vol. 171, Leningrad: Nauka, 1985; English transl.: Proc. Steklov Inst. Math., vol. 171, 1987.
    • (1987) Proc. Steklov Inst. Math. , vol.171
  • 121
    • 3442880129 scopus 로고
    • Quantized hall conductance in a two-dimensional periodic potential
    • Thouless, D.J., Khomoto, M., Nightingale, M.P., and den Nijs, M., Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett., 1982, vol. 49, pp. 405-408.
    • (1982) Phys. Rev. Lett. , vol.49 , pp. 405-408
    • Thouless, D.J.1    Khomoto, M.2    Nightingale, M.P.3    Den Nijs, M.4
  • 122
    • 0023428688 scopus 로고
    • Asymptotic behavior of stability regions for Hill's equation
    • Weinstein, M. and Keller, J., Asymptotic Behavior of Stability Regions for Hill's Equation, SIAM J. Appl. Math., 1987, vol. 47, no. 5, pp. 941-958.
    • (1987) SIAM J. Appl. Math. , vol.47 , Issue.5 , pp. 941-958
    • Weinstein, M.1    Keller, J.2
  • 123
    • 0022043563 scopus 로고
    • Hill's equation with a large potential
    • Weinstein, M. and Keller, J., Hill's Equation with a Large Potential, SIAM J. Appl. Math., 1985, vol. 45, no. 2, pp. 200-214.
    • (1985) SIAM J. Appl. Math. , vol.45 , Issue.2 , pp. 200-214
    • Weinstein, M.1    Keller, J.2
  • 125
    • 0001409871 scopus 로고
    • An example of phase holonomy and WKB theory
    • Wilkinson, M., An Example of Phase Holonomy and WKB Theory, J. Phys. A, 1984, vol. 17, pp. 3459-3476.
    • (1984) J. Phys. A , vol.17 , pp. 3459-3476
    • Wilkinson, M.1
  • 126
    • 0021757421 scopus 로고
    • Critical properties of electron eigenstates in incommensurate systems
    • Wilkinson, M., Critical Properties of Electron Eigenstates in Incommensurate Systems, Proc. Royal Soc. London, 1984, vol. A391, pp. 303-350.
    • (1984) Proc. Royal Soc. London , vol.A391 , pp. 303-350
    • Wilkinson, M.1
  • 127
    • 36149008019 scopus 로고
    • Magnetic translation group. I, II
    • Zak, J., Magnetic Translation Group. I, II, Phys. Rev., 1964, vol. 134, pp. A1602-A1606, A1607-A1613.
    • (1964) Phys. Rev. , vol.134
    • Zak, J.1
  • 128
    • 4243702385 scopus 로고
    • Perturbed cristalls in the kq representation. III
    • Zak, J., Perturbed Cristalls in the kq Representation. III, Phys. Rev. B., 1977, vol. 16, pp. 4154-4163.
    • (1977) Phys. Rev. B. , vol.16 , pp. 4154-4163
    • Zak, J.1
  • 129
    • 0001565081 scopus 로고
    • Proper functions for a bloch electron in a magnetic field
    • Zak, J., Proper Functions for a Bloch Electron in a Magnetic Field, Phys. Rev., 1965, vol. 139, pp. A1159-A1162.
    • (1965) Phys. Rev. , vol.139
    • Zak, J.1


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