메뉴 건너뛰기




Volumn 354, Issue 2, 2002, Pages 537-569

On the structure of spectra of periodic elliptic operators

Author keywords

Absolutely continuous spectrum; Elliptic operator; Magnetic and electric potentials; Periodic operator; Schr dinger operator

Indexed keywords


EID: 0036035440     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-01-02878-1     Document Type: Article
Times cited : (39)

References (71)
  • 1
    • 0002088332 scopus 로고
    • Elliptic problems with a parameter and parabolic problems of general type
    • MR 33:415
    • 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.
    • (1964) Russian Math. Surveys , vol.19 , Issue.3 , pp. 53-157
    • Agranovich, M.S.1    Vishik, M.I.2
  • 2
    • 0003360861 scopus 로고
    • Lectures on quasiconformal mappings
    • Van Nostrand, New York. MR 34:336
    • L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies, No. 10, Van Nostrand, New York, 1966. MR 34:336.
    • (1966) Van Nostrand Mathematical Studies , vol.10
    • Ahlfors, L.V.1
  • 3
    • 84954243832 scopus 로고
    • Princeton Univ. Press, Princeton NJ. MR 22:5729
    • L.V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Univ. Press, Princeton NJ 1960. MR 22:5729.
    • (1960) Riemann Surfaces
    • Ahlfors, L.V.1    Sario, L.2
  • 5
    • 0033340072 scopus 로고    scopus 로고
    • Asymptotic methods for thin high contrast 2D PBG materials
    • 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.
    • (1999) Journal of Lightwave Technology , vol.17 , Issue.11 , pp. 1996-2007
    • Axmann, W.1    Kuchment, P.2    Kunyansky, L.3
  • 6
    • 0005146065 scopus 로고
    • On the point spectrum of Schrödinger operators
    • MR 84i:35106
    • A. Berthier, On the point spectrum of Schrödinger operators, Ann. Scient. Ec. Norm. Super., 4 serie, 15 (1982), 1-15. MR 84i:35106.
    • (1982) Ann. Scient. Ec. Norm. Super., 4 serie , vol.15 , pp. 1-15
    • Berthier, A.1
  • 7
    • 0001878229 scopus 로고
    • 2-theory of the Maxwell operator in arbitrary domains
    • MR 89e:35127
    • 2-theory of the Maxwell operator in arbitrary domains, Russian Math. Surveys 42 (1987), no 6: 75-96. MR 89e:35127.
    • (1987) Russian Math. Surveys , vol.42 , Issue.6 , pp. 75-96
    • Birman, M.1    Solomyak, M.2
  • 8
    • 0001905748 scopus 로고    scopus 로고
    • The two-dimensional periodic magnetic Hamiltonian is absolutely continuous
    • Russian; translation in St. Petersburg Math. J. 9 (1998), no. 1, 21-32. MR 98g:47038
    • 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.
    • (1997) Algebra i Analiz , vol.9 , Issue.1 , pp. 32-48
    • Birman, M.Sh.1    Suslina, T.A.2
  • 9
    • 0005184084 scopus 로고    scopus 로고
    • The periodic Dirac operator is absolutely continuous
    • MR 2000h:47068
    • 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.
    • (1999) Integr. Equat. and Operator Theory , vol.34 , pp. 377-395
    • Birman, M.Sh.1    Suslina, T.A.2
  • 10
    • 0002626473 scopus 로고    scopus 로고
    • Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential
    • Russian; translation in St. Petersburg Math. J. 10 (1999), no. 4, 579-601. MR 99k:81060
    • 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.
    • (1998) Algebra i Analiz , vol.10 , Issue.1 , pp. 1-36
    • Birman, M.Sh.1    Suslina, T.A.2
  • 11
    • 0005142359 scopus 로고    scopus 로고
    • Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum
    • Mathematical Results in Quantum Mechanics (QMath7, Prague, June 22-26), J. Dittrich, P. Exner, et al. (Editors), Birkhäuser, Basel. MR 2000g:81049
    • 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.
    • (1998) Operator Theory, Adv. and Appl. , vol.108 , pp. 13-31
    • Birman, M.Sh.1    Suslina, T.A.2
  • 12
    • 0001890867 scopus 로고    scopus 로고
    • Periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity
    • English translation in St. Petersburg Math J. 11 (2000), no. 2, 203-232. MR 2000i:35026
    • 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.
    • (1999) Algebra i Analiz , vol.11 , Issue.2
    • Birman, M.Sh.1    Suslina, T.A.2
  • 13
    • 0005179579 scopus 로고    scopus 로고
    • On the absolute continuity of the periodic Schrödinger and Dirac operators with magnetic potential
    • Differential Equations and Mathematical Physics (Birmingham, AL), Amer. Math. Soc., Providence, RI. MR 2000m:00026
    • 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.
    • (1999) AMSIP Stud. Adv. Math. , vol.16 , pp. 41-49
    • Birman, M.Sh.1    Suslina, T.A.2
  • 15
    • 0003346454 scopus 로고
    • Methods of mathematical physics
    • John Wiley & Sons, Inc., New York. MR 90k:35001
    • R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II. Partial differential equations. John Wiley & Sons, Inc., New York, 1989. MR 90k:35001.
    • (1989) Partial differential equations , vol.2
    • Courant, R.1    Hilbert, D.2
  • 16
    • 0005215568 scopus 로고
    • Zone structure of the spectrum of the Dirac operator with periodic potential
    • Ustinov. Russian
    • 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).
    • (1985) Nonlinear Oscillations and Control Theory , pp. 83-90
    • Danilov, L.1
  • 17
    • 0005220503 scopus 로고
    • On the spectrum of the Dirac operator with periodic potential
    • Ural Branch, preprint, Sverdlovsk. Russian
    • L. Danilov, On the spectrum of the Dirac operator with periodic potential, Acad. Sci. USSR, Ural Branch, preprint, Sverdlovsk, 1987. (Russian).
    • (1987) Acad. Sci. USSR
    • Danilov, L.1
  • 18
    • 0001258350 scopus 로고
    • n with periodic potential
    • Russian; translation in Theoret. and Math. Phys. 85 (1990), no. 1, 1039-1048 (1991). MR 92a:35119
    • 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.
    • (1990) Teoret. Mat. Fiz. , vol.85 , Issue.1 , pp. 41-53
    • Danilov, L.1
  • 20
    • 21844489758 scopus 로고
    • Resolvent estimates and the spectrum of the Dirac operator with a periodic potential
    • Russian; translation in Theoret. and Math. Phys. 103 (1995), no. 1, 349-365. MR 98f:35112
    • 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.
    • (1995) Teoret. Mat. Fiz. , vol.103 , Issue.1 , pp. 3-22
    • Danilov, L.1
  • 21
    • 0033245430 scopus 로고    scopus 로고
    • On the spectrum of the two-dimensional periodic Dirac operator
    • Russian; translation in Theoret. and Math. Phys. 118 (1999), no. 1, 1-11. MR 2000h:35117
    • 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.
    • (1999) Teoret. Mat. Fiz. , vol.118 , Issue.1 , pp. 3-14
    • Danilov, L.1
  • 22
    • 0005179581 scopus 로고    scopus 로고
    • Absolute continuity of the spectrum of the periodic Dirac operator
    • English transl., Differential Equations 36 (2000), 262-271. MR 2001f:42038
    • 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.
    • (2000) Diff. Uravneniya , vol.36 , Issue.2 , pp. 233-240
    • Danilov, L.1
  • 23
    • 0005146815 scopus 로고    scopus 로고
    • On the spectrum of the periodic Dirac operator
    • English transl. in Theoret. Math. Phys. 124 (2000). CMP 2001:10
    • 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.
    • (2000) Teoret. i Mat. Fiz. , vol.124 , Issue.1 , pp. 3-17
    • Danilov, L.1
  • 26
    • 0000798363 scopus 로고
    • Ground states in a periodic field. Magnetic Bloch functions and vector bundles
    • MR 82c:81181
    • 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.
    • (1980) Soviet Math. Dokl. , vol.22 , pp. 240-244
    • Dubrovin, B.A.1    Novikov, S.P.2
  • 27
    • 0001225877 scopus 로고
    • Ground states of a two-dimensional electron in a periodic magnetic field
    • MR 82i:81018
    • 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.
    • (1980) Soviet Physics JETF , vol.52 , pp. 511-516
    • Dubrovin, B.A.1    Novikov, S.P.2
  • 31
    • 21844520045 scopus 로고
    • Strong magnetic fields, Dirichlet boundaries, and spectral gaps
    • MR 96a:81026
    • R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 164 (1995), 237-259. MR 96a:81026.
    • (1995) Comm. Math. Phys. , vol.164 , pp. 237-259
    • Hempel, R.1    Herbst, I.2
  • 33
    • 0000886914 scopus 로고
    • Spectrum and eigenfunctions of Schrödinger operator with zero-range potential of the homoeneous lattice type in three dimensional space
    • Engl. transl. in Theor. and Math. Phys. 57 (1983), 1156-1162. MR 86b:81016
    • 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. transl. in Theor. and Math. Phys. 57 (1983), 1156-1162. MR 86b:81016.
    • (1983) Ter. i Mat. Fiz. , vol.57 , Issue.2 , pp. 304-313
    • Karpeshina, Yu.1
  • 34
    • 0003245560 scopus 로고    scopus 로고
    • Perturbation theory for the Schrödinger operator with a periodic potential
    • Springer-Verlag, Berlin. MR 2001i:35002
    • 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.
    • (1997) Lecture Notes in Mathematics , vol.1663
    • Karpeshina, Yu.1
  • 35
    • 51249175788 scopus 로고
    • A directional compactification of the complex Bloch variety
    • MR 91k:58134
    • 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.
    • (1990) Comment. Math. Helv. , vol.65 , Issue.1 , pp. 114-149
    • Knörrer, H.1    Trubowitz, E.2
  • 36
    • 0002461297 scopus 로고
    • Floquet theory for partial differential equations
    • MR 84b:35018
    • P. Kuchment, Floquet Theory for Partial Differential Equations, Russian Math. Surveys, 37 (1982), no. 4, 1-60. MR 84b:35018.
    • (1982) Russian Math. Surveys , vol.37 , Issue.4 , pp. 1-60
    • Kuchment, P.1
  • 38
    • 0001325714 scopus 로고    scopus 로고
    • Mathematics of photonic crystals
    • G. Bao, L. Cowsar, and W. Masters (Editors) SIAM
    • P. Kuchment, Mathematics of photonic crystals, in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters (Editors) SIAM, 2001.
    • (2001) Mathematical Modeling in Optical Science
    • Kuchment, P.1
  • 39
    • 0033245667 scopus 로고    scopus 로고
    • Spectral properties of high-contrast band-gap materials and operators on graphs
    • MR 2000c:78027
    • 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.
    • (1999) Experimental Mathematics , vol.8 , Issue.1 , pp. 1-28
    • Kuchment, P.1    Kunyansky, L.2
  • 40
    • 0036109923 scopus 로고    scopus 로고
    • Differential operators on graphs and photonic crystals
    • P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, to appear in Adv. Comp. Math.
    • Adv. Comp. Math.
    • Kuchment, P.1    Kunyansky, L.2
  • 41
    • 0005234447 scopus 로고    scopus 로고
    • On the absolute continuity of spectra of periodic elliptic operators
    • Mathematical Results in Quantum Mechanics (QMath7, Prague, June 22-26), J. Dittrich, P. Exner, et al. (Editors), Birkhäuser, Basel. MR 2001c:35166
    • 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.
    • (1998) Operator Theory, Adv. and Appl. , vol.108 , pp. 291-297
    • Kuchment, P.1    Levendorskiî, S.2
  • 42
    • 0005228574 scopus 로고    scopus 로고
    • Integral representations and Liouville theorems for solutions of periodic elliptic equations
    • CMP 2001:10
    • 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.
    • (2001) J. Funct. Anal. , vol.181 , pp. 402-446
    • Kuchment, P.1    Pinchover, Y.2
  • 43
    • 0002503979 scopus 로고    scopus 로고
    • On embedded eigenvalues of perturbed periodic Schrödinger operators
    • Newark, DE. MR 99e:35032
    • 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.
    • (1997) Spectral and Scattering Theory , pp. 67-75
    • Kuchment, P.1    Vainberg, B.2
  • 44
    • 0034353901 scopus 로고    scopus 로고
    • On absence of embedded eigenvalues for Schrödinger operators with perturbed periodic potentials
    • CMP 2000:17
    • P. Kuehment 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.
    • (2000) Commun. PDE , vol.25 , Issue.9-10 , pp. 1809-1826
    • Kuehment, P.1    Vainberg, B.2
  • 46
    • 0000134587 scopus 로고
    • On the possible rate of decay at infinity of solutions of second order partial differential equations
    • English translation in Math. USSR Sbornik 72 (1992), no. 2, 343-351. MR 92d:335032
    • 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.
    • (1991) Mat. Sbornik , vol.182 , Issue.3 , pp. 364-383
    • Meshkov, V.1
  • 47
    • 0032544287 scopus 로고    scopus 로고
    • Absence of singular spectrum for a perturbation of a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic potential
    • MR 99i:81039
    • A. Moraine, 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.
    • (1998) J. Phys. A: Math. Gen. , vol.31 , pp. 7593-7601
    • Moraine, A.1
  • 48
    • 0034288254 scopus 로고    scopus 로고
    • The absolute continuity of the spectrum of maxwell operator in periodic media
    • CMP 2001:01
    • A. Moraine, The absolute continuity of the spectrum of Maxwell operator in periodic media, J. Math. Phys. 41 (2000), 7099-7108. CMP 2001:01.
    • (2000) J. Math. Phys. , vol.41 , pp. 7099-7108
    • Moraine, A.1
  • 49
    • 0000264485 scopus 로고
    • Global identifiability for an inverse problem for the schrödinger equation in a magnetic field
    • MR 96m:35336
    • 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.
    • (1995) Math. Ann. , vol.303 , Issue.3 , pp. 377-388
    • Nakamura, G.1    Sun, Z.2    Uhlmann, G.3
  • 50
    • 0002233428 scopus 로고
    • Two-dimensional Schrödinger operators in the periodic fields
    • VINITI, Moscow; English transl., J. Soviet Math. 28 (1985), 1-19. MR 85i:81020
    • 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.
    • (1983) Itogi Nauki i Tekhn.: Sovremennye Problemy Mat. , vol.23 , pp. 3-32
    • Novikov, S.1
  • 52
    • 0005234449 scopus 로고
    • Harmonic synthesis of solutions of elliptic equations with periodic coefficients
    • MR 95f:35037
    • V. Palamodov, Harmonic synthesis of solutions of elliptic equations with periodic coefficients, Ann. Inst. Fourier 43 (1993). 751-768. MR 95f:35037.
    • (1993) Ann. Inst. Fourier , vol.43 , pp. 751-768
    • Palamodov, V.1
  • 53
    • 0005211998 scopus 로고
    • Non-uniqueness in Cauchy's problem for differential equations of elliptic type
    • MR 22:12305
    • A. Plis, Non-uniqueness in Cauchy's problem for differential equations of elliptic type, J. Math. Mech. 9 (1960), 557-562. MR 22:12305.
    • (1960) J. Math. Mech. , vol.9 , pp. 557-562
    • Plis, A.1
  • 55
    • 0002680721 scopus 로고    scopus 로고
    • On absolute continuity of the periodic Schrödinger operators
    • Z. Shen, On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Notices 2001, no. 1, 1-31.
    • (2001) Internat. Math. Res. Notices , Issue.1 , pp. 1-31
    • Shen, Z.1
  • 59
    • 84957149530 scopus 로고
    • Spectral theory and the index of elliptic operators with almost periodic coefficients
    • MR 81f:35090
    • 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.
    • (1979) Russian Math. Surveys , vol.34 , Issue.2 , pp. 109-157
    • Shubin, M.1
  • 60
    • 0000313557 scopus 로고
    • Microlocal analysis for the periodic magnetic Schrödinger equation and related questions
    • Microlocal Analysis and Applications, Springer-Verlag, Berlin. MR 94f:35119
    • 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.
    • (1991) Lect. Notes in Math. , vol.1495 , pp. 237-332
    • Sjostrand, J.1
  • 61
    • 0001967757 scopus 로고
    • Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators
    • English transl., Proc. Steklov Inst. Math. 1987, no. 2 (171). MR 87h:47110; MR 88g:47038
    • 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.
    • (1985) Trudy Mat. Inst. Steklov. , vol.17
    • Skriganov, M.M.1
  • 63
    • 77950788431 scopus 로고    scopus 로고
    • Absolute continuity of the spectrum of periodic operators of mathematical physics
    • Nantes, 5-9 Juin, GDR 1151 (CNRS). MR 2001f:35295
    • 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.
    • (2000) Journees Equations aux Derivees Partielles
    • Suslina, T.1
  • 64
    • 84968484652 scopus 로고
    • An inverse boundary value problem for Schrödinger operators with vector potentials
    • MR 93j:35066
    • 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.
    • (1993) Trans. Amer. Math. Soc. , vol.338 , Issue.2 , pp. 953-969
    • Sun, Z.1
  • 65
    • 0033242631 scopus 로고    scopus 로고
    • Absolute continuity of the periodic magnetic Schrödinger operator
    • MR 2000g:35028
    • A. Sobolev, Absolute continuity of the periodic magnetic Schrödinger operator, Inventiones Mathematicae 137 (1999), 85-112. MR 2000g:35028.
    • (1999) Inventiones Mathematicae , vol.137 , pp. 85-112
    • Sobolev, A.1
  • 67
    • 0003278079 scopus 로고    scopus 로고
    • Partial differential equations. I. Basic theory
    • Springer-Verlag, New York. MR 98b:35002
    • M. Taylor, Partial Differential Equations. I. Basic Theory, Applied Mathematical Sciences, v. 115. Springer-Verlag, New York, 1996. MR 98b:35002.
    • (1996) Applied Mathematical Sciences , vol.115
    • Taylor, M.1
  • 68
    • 0000485145 scopus 로고
    • Time dependent approach to scattering from impurities in a crystal
    • MR 48:13084
    • L.E. Thomas, Time dependent approach to scattering from impurities in a crystal, Comm. Math. Phys. 33 (1973), 335-343. MR 48:13084.
    • (1973) Comm. Math. Phys. , vol.33 , pp. 335-343
    • Thomas, L.E.1
  • 69
    • 0004034302 scopus 로고
    • Pergamon Press, London; Addison Wesley Pub. Co., Inc., Reading, Mass., MR 27:321
    • I.N. Vekua, Generalized Analytic Functions, Pergamon Press, London; Addison Wesley Pub. Co., Inc., Reading, Mass. 1962. MR 27:321.
    • (1962) Generalized Analytic Functions
    • Vekua, I.N.1
  • 71
    • 36149008019 scopus 로고
    • Magnetic translation group I, II
    • MR 31:2031; MR 31:2032
    • J. Zak, Magnetic translation group I, II, Phys. Rev. 134 (1964), no. 6A, A1602-A1611. MR 31:2031; MR 31:2032.
    • (1964) Phys. Rev. , vol.134 , Issue.6 A
    • Zak, J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.