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St. Petersburg Mathematical Journal

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On the mathematical work of M. Sh. Birman


Authors: M. Z. Solomyak, T. A. Suslina and D. R. Yafaev
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 23 (2011), nomer 1.
Journal: St. Petersburg Math. J. 23 (2012), 1-38
MSC (2010): Primary 47-02; Secondary 35-02
DOI: https://doi.org/10.1090/S1061-0022-2011-01184-7
Published electronically: November 7, 2011
MathSciNet review: 2760146
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  • 6. -, On minimal functionals for elliptic differential equations of second order, Dokl. Akad. Nauk SSSR (N.S.) 93 (1953), no. 6, 953-956. (Russian) MR 0062923 (16:42f)
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  • 12. -, Method of quadratic forms in problems with small parameter at the highest derivatives, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1957, vyp. 3, 9-12. (Russian) MR 0101398 (21:209)
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  • 16. -, On the spectrum of Schrödinger and Dirac operators, Dokl. Akad. Nauk SSSR 129 (1959), no. 2, 239-241. (Russian) MR 0109934 (22:817)
  • 17. -, On the discrete part of the spectrum of Schrödinger and Dirac operators, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1960, vyp. 2, 167-168. (Russian)
  • 18. M. Sh. Birman and B. S. Pavlov, On the complete continuity of certain imbedding operators, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1961, vyp. 1, 61-74. (Russian) MR 0133491 (24:A3322)
  • 19. M. Sh. Birman, On the spectrum of singular boundary-value-problems, Mat. Sb. (N.S.) 55 (1961), no. 2, 125-174; English transl., 11 Papers on Analysis, Amer. Math. Soc. Transl.(2), vol. 53, Amer. Math. Soc., Providence, RI, 1966, pp. 23-60. MR 0142896 (26:463); MR 0194283 (33:2496)
  • 20. -, Perturbation of the spectrum of a singular elliptic operator under variation of the boundary and boundary conditions, Dokl. Akad. Nauk SSSR 137 (1961), no. 4, 761-763; English transl., Soviet Math. Dokl. 2 (1961), 326-328. MR 0177311 (31:1574)
  • 21. -, On the number of eigenvalues in a quantum scattering problem, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1961, vyp. 3, 163-166. (Russian) MR 0139370 (25:2803)
  • 22. -, Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1962, vyp. 1, 22-55; English transl., Amer. Math. Soc. Transl., Ser. 2 (2008), vol. 225, 19-53. MR 0138874 (25:2314)
  • 23. -, On existence conditions for wave operators, Dokl. Akad. Nauk SSSR 143 (1962), no. 3, 506-509; English transl., Soviet Math. Dokl. 3 (1962), 408-411. MR 0149333 (26:6823)
  • 24. M. Sh. Birman and M. G. Kreĭn, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), no. 3, 475-478; English transl., Soviet Math. Dokl. 3 (1962), 740-744. MR 0139007 (25:2447)
  • 25. M. Sh. Birman and G. E. Skvortsov, On square summability of highest derivatives of the solution of the Dirichlet problem in a domain with piecewise smooth boundary, Izv. Vyssh. Uchebn. Zaved. Mat. 1962, no. 5, 12-21. (Russian) MR 0145196 (26:2731)
  • 26. M. Sh. Birman, On a test for the existence of wave operators, Dokl. Akad. Nauk SSSR 147 (1962), no. 5, 1008-1009; English transl., Soviet Math. Dokl. 3 (1962), 1747-1748. MR 0167841 (29:5107)
  • 27. -, Existence conditions for wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), no. 4, 883-906. (Russian) MR 0161150 (28:4359)
  • 28. M. Sh. Birman and M. G. Kreĭn, Some topics of the theory of the wave and scattering operators, 1963 Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Akad. Nauk SSSR Sibirsk. Otdel., Moscow, 1963, pp. 39-45. MR 0200749 (34:637)
  • 29. M. Sh. Birman and I. M. Glazman, Spectra of singular differential operators, Proc. Fourth All-Union Math. Congr. (Leningrad, 1961). Vol. II, Nauka, Leningrad, 1964, pp. 253-261. (Russian) MR 0220101 (36:3168)
  • 30. M. Sh. Birman and S. B. Èntina, A stationary approach in the abstract theory of scattering, Dokl. Akad. Nauk SSSR 155 (1964), no. 3, 506-508; English transl., Soviet Math. Dokl. 5 (1965), 432-435. MR 0164590 (29:1887)
  • 31. M. Sh. Birman, A local criterion for the existence of wave operators, Dokl. Akad. Nauk SSSR 159 (1964), no. 3, 485-488; English transl., Soviet Math. Dokl. 5 (1965), no. 2, 1505-1509. MR 0173964 (30:4171)
  • 32. M. Sh. Birman and M. Z. Solomyak, On double Stieltjes operator integrals, Dokl. Akad. Nauk SSSR 165 (1965), no. 6, 1223-1226; English transl., Soviet Math. Dokl. 6 (1965), no. 2, 1567-1571. MR 0188802 (32:6234)
  • 33. -, Double Stieltjes operator integrals, Spectral Theory and Wave Processes, Probl. Mat. Fiz., No. 1, Leningrad. Univ., Leningrad, 1966, pp. 33-67; English transl., Topics in Math. Phys., vol. 1, Consultants Bureau Plenum Publ. Corporation, New York, 1967, pp. 25-54. MR 0209872 (35:767b); MR 0210536 (35:1428)
  • 34. -, Approximation of functions of the $ W_p^\alpha$-classes by piecewise-polynomial functions, Dokl. Akad. Nauk SSSR 171 (1966), no. 5, 1015-1018; English transl., Soviet Math. Dokl. 7 (1966), no. 2, 1573-1577. MR 0209733 (35:630)
  • 35. -, Double Stieltjes operator integrals and problems on multipliers, Dokl. Akad. Nauk SSSR 171 (1966), no. 6, 1251-1254; English transl., Soviet Math. Dokl. 7 (1966), no. 2, 1618-1621. MR 0209871 (35:767a)
  • 36. M. Sh. Birman and S. B. Èntina, Stationary approach in abstract scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), no. 2, 401-430; English transl., Math. USSR-Izv. 1 (1967), no. 1, 391-420. MR 0209895 (35:790)
  • 37. M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. II, Spectral Theory, Diffraction Problems, Probl. Mat. Fiz., No. 2, Leningrad. Univ., Leningrad, 1967, pp. 26-60; English transl., Topics in Math. Phys., vol. 2, Consultants Bureau, New York, 1968, pp. 19-46. MR 0234304 (38:2621); MR 0266727 (42:1630)
  • 38. -, Piecewise polynomial approximations of functions of classes $ W_p^\alpha$, Mat. Sb. (N.S.) 73 (115) (1967), no. 3, 331-355; English transl., Math. USSR-Sb. 2 (1967), 295-317. MR 0217487 (36:576)
  • 39. -, Estimates of singular numbers of integral operators. I, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1967, vyp. 2, 43-53. (Russian) MR 0216339 (35:7173)
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  • 41. M. Sh. Birman, A local test for the existence of wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), no. 4, 914-942; English transl., Math. USSR-Izv. 2 (1968), no. 2, 879-906. MR 0248558 (40:1810)
  • 42. A. L. Belopol'skiĭ and M. Sh. Birman, Existence of wave operators in scattering theory for a pair of spaces, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), no. 5, 1162-1175; English transl., Math. USSR-Izv. 2 (1968), no. 2, 1117-1130. MR 0238101 (38:6377)
  • 43. M. Sh. Birman and M. Z. Solomyak, Estimates of singular numbers of integral operators. III. Operators in unbounded domains, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1969, vyp. 1, 35-48. (Russian) MR 0246163 (39:7468)
  • 44. M. Sh. Birman, Some applications of a local criterion for the existence of wave operators, Dokl. Akad. Nauk SSSR 185 (1969), no. 4, 735-738; English transl., Soviet Math. Dokl. 10 (1969), 393-397. MR 0282242 (43:7954)
  • 45. M. Sh. Birman and M. Z. Solomyak, Remarks on the nuclearity of integral operators and the boundedness of pseudodifferential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1969, no. 9, 11-17. (Russian) MR 0254670 (40:7877)
  • 46. M. Sh. Birman, Scattering problems for differential operators with constant coefficients, Funktsional. Anal. i Prilozhen. 3 (1969), no. 3, 1-16; English transl., Funct. Anal. Appl. 3 (1969), no. 3, 167-180.MR 0253083 (40:6298)
  • 47. -, A test of the existence of complete wave operators in scattering theory for a pair of spaces, Spectral Theory. Wave Processes, Probl. Mat. Fiz., No. 4, Leningrad. Univ., Leningrad, 1970, pp. 22-26; English transl., Topics in Math. Phys., vol. 4, Consultants Bureau, New York-London, 1971, pp. 17-21. MR 0276817 (43:2557); MR 0339975 (49:4732)
  • 48. M. Sh. Birman and M. Z. Solomyak, Asymptotics of the spectrum of weakly polar integral operators, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), no. 5, 1142-1158; English transl., Math. USSR-Izv. 4 (1970), 1151-1168 (1971).MR 0279638 (43:5359)
  • 49. -, The principal term of the spectral asymptotics for ``nonsmooth'' elliptic problems, Funktsional. Anal. i Prilozhen. 4 (1970), no. 4, 1-13; English transl., Funct. Anal. Appl. 4 (1970), 265-275 (1971).MR 0278126 (43:3857)
  • 50. -, On the proof of the theorem on the asymptotic behavior of the spectrum for the real part of a Volterra operator with nuclear imaginary component, Mat. Issled. 5 (1970), no. 4, 16-25. (Russian) MR 0284851 (44:2075)
  • 51. M. Sh. Birman, Works on spectral theory of differential operators, Mathematics in Petersburg-Leningrad Univ., Leningrad. Univ., Leningrad, 1970, pp. 129-133. (Russian)
  • 52. -, Scattering problems for differential operators under a perturbation of the space, Izv. Akad.Nauk SSSR Ser. Mat. 35 (1971), no. 2, 440-455; English transl., Math. USSR-Izv. 5 (1971), 459-474 (1972). MR 0291868 (45:957)
  • 53. M. Sh. Birman and M. Z. Solomyak, The asymptotics of the spectrum of ``nonsmooth'' elliptic equations, Funktsional. Anal. i Prilozhen. 5 (1971), no. 1, 69-70; English transl., Funct. Anal. Appl. 5 (1971), 56-57. MR 0283428 (44:659)
  • 54. M. Sh. Birman and V. V. Borzov, The asymptotic behavior of the discrete spectrum of certain singular differential operators, Spectral Theory, Probl. Mat. Fiz., No. 5, Leningrad. Univ., Leningrad, 1971, pp. 24-38; English transl., Topics in Math. Phys., vol. 5, Consultants Bureau, New York-London, 1972, pp. 19-30. MR 0301568 (46:726); MR 0339976 (49:4733)
  • 55. M. Sh. Birman and M. Z. Solomyak, Remarks on the spectral shift function, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 33-46; English transl., J. Soviet Math. 3 (1975), no. 4, 408-419. MR 0315482 (47:4031)
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  • 58. V. M. Babich, M. Sh. Birman, V. I. Smirnov, M. Z. Solomyak, and N. N. Ural'tseva, Ol'ga Aleksandrovna Ladyzhenskaya (on her fiftieth birthday), Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1972, vyp. 2, 159-160. (Russian) MR 0297506 (45:6560)
  • 59. M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. III, Theory of Functions. Spectral Theory. Wave Propagation, Probl. Mat. Fiz., No. 6, Leningrad. Univ., Leningrad, 1973, pp. 27-53. (Russian) MR 0348494 (50:992)
  • 60. -, Spectral asymptotics of nonsmooth elliptic operators. II, Trudy Moskov. Mat. Obshch. 28 (1973), 3-34; English transl., Trans. Moscow Math. Soc. 28 (1973), 1-32 (1975). MR 0364898 (51:1152)
  • 61. -, Equivalent renormings of $ W_p^{\alpha}$ classes in the case of partitioning of the domain, Izv. Vyssh. Uchebn. Zaved. 1973, no. 3, 19-27. (Russian) MR 0344875 (49:9614)
  • 62. -, Quantitative analysis in Sobolev's imbedding theorems and applications to spectral theory, Tenth Mathematical School (Summer School, Kaciveli/Nal'chik, 1972), Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5-189. (Russian) MR 0482138 (58:2224)
  • 63. -, A certain ``model'' nonelliptic spectral problem, Collection of Articles Dedicated to the Memory of Academician V. I. Smirnov, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1975, vyp. 1, 39-45; English transl., Vestnik Leningrad. Univ. Math. 8 (1980), 23-30. MR 0380148 (52:1049)
  • 64. M. Sh. Birman, L. S. Koplienko, and M. Z. Solomyak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1975, no. 3, 3-10; English transl., Soviet Math. (Iz. VUZ) 19 (1975), no. 3, 1-6 (1976). MR 0385597 (52:6458)
  • 65. A. B. Alekseev and M. Sh. Birman, Asymptotic behavior of the spectrum of elliptic boundary value problems with solvable constraints, Dokl. Akad. Nauk SSSR 230 (1976), no. 3, 505-507; English transl., Soviet Math. Dokl. 17 (1976), no. 5, 1319-1322. MR 0420027 (54:8044)
  • 66. -, A variational formulation of the problem of the oscillations of a resonator that is filled with a stratified anisotropic medium, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1977, vyp. 2, 9-15; English transl., Vestnik Leningrad. Univ. Math. 10 (1982), 101-108. MR 0462202 (57:2177)
  • 67. M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1977, vyp. 3, 13-21; English transl., Vestnik Leningrad Univ. Math. 10 (1982), 237-247. MR 0463738 (57:3680)
  • 68. -, Application of interpolational methods to estimates of the spectrum of integral operators, Operator Theory in Function Spaces (Proc. School, Novosibirsk, 1975), Nauka, Sibirsk. Otdel., Novosibirsk, 1977, pp. 42-70; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 127, Amer.Math. Soc., Providence, RI, 1985, pp. 61-82.MR 0502834 (80b:47063)
  • 69. -, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32 (1977), no. 1, 17-84; English transl., Russian Math. Surveys 32 (1977), no. 1, 15-89. MR 0438186 (55:11104)
  • 70. A. B. Alekseev and M. Sh. Birman, Asymptotic behavior of the discrete spectrum of elliptic problems with solvable differential constraints, Uspekhi Mat. Nauk 32 (1977), no. 1, 232-233. (Russian)
  • 71. M. Sh. Birman and M. Z. Solomyak, Asymptotic properties of the spectrum of differential equations, Itogi Nauki i Tekhniki Mat. Anal., vol. 14, VINITI, Moscow, 1977, pp. 5-58; English transl., J. Soviet Math. 12 (1979), no. 3, 247-283. MR 0467022 (57:6892)
  • 72. -, On asymptotic behavior of the spectrum of elliptic variational problems with constraints, Proc.All-Union Conf. on Partial Differential Equations Dedicated to Acad. Petrovskiĭ 75th Birthday, Moskov. Univ., Moscow, 1978, pp. 51-52. (Russian)
  • 73. -, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1979, vyp. 3, 5-10; English transl., Vestnik Leningrad Univ. Math. 12 (1980), 155-161. MR 0555971 (81b:47060)
  • 74. M. Sh. Birman, A. M. Vershik, and M. Z. Solomyak, The product of commuting spectral measures may fail to be countably additive, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 61-62; English transl., Funct. Anal. Appl. 13 (1979), no. 1, 48-49.MR 0527523 (80h:28010)
  • 75. M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Sibirsk. Mat. Zh. 20 (1979), no. 1, 3-22; English transl., Siberian Math. J. 20 (1979), no. 1, 1-15.MR 0523133 (80c:35077)
  • 76. -, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Partial Differential Equations (Proc. Conf., Novosibirsk, 1978), Nauka, Sibirsk. Otdel., Novosibirsk, 1980, pp. 221-223. (Russian) MR 0601449 (82b:35115)
  • 77. -, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations in unbounded domains, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 27-35; English transl., Funct. Anal. Appl. 14 (1980), 267-274. MR 0595726 (82a:35084)
  • 78. M. Sh. Birman and D. R. Yafaev, Asymptotics of the spectrum of the $ s$-matrix in potential scattering, Dokl. Akad. Nauk SSSR 255 (1980), no. 5, 1085-1087. (Russian) MR 0613721 (82j:35116)
  • 79. M. Sh. Birman, Quasiclassical spectral asymptotic behavior of a class of integral operators, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 98 (1980), 22-32; English transl., J. Soviet Math. 21 (1983), 12-20. MR 0591859 (82i:47083)
  • 80. M. Sh. Birman and D. R. Yafaev, Asymptotic behavior of the spectrum of the scattering matrix, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 110 (1981), 3-29; English transl., J. Soviet Math. 25 (1984), 793-814. MR 0643971 (83c:35095)
  • 81. -, Asymptotic behavior of limit phases for scattering by potentials without spherical symmetry, Teoret. Mat. Fiz. 51 (1982), no. 1, 44-53; English transl., Theoret. and Math. Phys. 51 (1982), no. 1, 344-350. MR 0672765 (84a:81037)
  • 82. M. Sh. Birman and M. Z. Solomyak, The asymptotic behavior of the spectrum of variational problems on solutions of elliptic systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 23-39; English transl., J. Soviet Math. 28 (1985), 633-644. MR 0660069 (84c:35082)
  • 83. -, Asymptotic behavior of the spectrum of pseudodifferential variational problems with constraints, Spectral Theory. Wave Processes, Probl. Mat. Fiz., No. 10, Leningrad. Univ., Leningrad, 1982, pp. 20-36; English transl., Selecta Math. Sov. 5 (1986), 245-256. MR 0691142 (84f:35109)
  • 84. -, On subspaces that admit a pseudodifferential projector, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1982, vyp. 1, 18-25; English transl., Vestnik Leningrad Univ. Math. 15 (1983), 17-27. MR 0652058 (83h:58091)
  • 85. -, Compact operators with power asymptotic behavior of the singular numbers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 21-30; English transl., J. Soviet Math. 27 (1984), 2442-2447. MR 0697420 (85e:47027)
  • 86. M. Sh. Birman, Re-expansion operators as objects of spectral analysis, Linear and Complex Analysis Problem Book. 199 Research Problems, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 130-134. MR 0734178 (85k:46001)
  • 87. M. Sh. Birman and M. Z. Solomyak, Some problems on compact operators with power-like behaviour of singular numbers, Linear and Complex Analysis Problem Book. 199 Research Problems, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 217-218. MR 0734178 (85k:46001)
  • 88. M. Sh. Birman, The Maxwell operator in domains with edges, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 147 (1985), 3-9; English transl., J. Soviet Math. 37 (1987), no. 1, 793-797. MR 0821471 (87j:35307)
  • 89. -, The Maxwell operator for a resonator with inward edges, Vestnik Leningrad. Univ. Ser. 1 1986, vyp. 3, 3-8; English transl., Vestnik Leningrad Univ. Math. 19 (1986), no. 3, 1-8. MR 0867387 (87m:35181)
  • 90. M. Sh. Birman and M. Z. Solomyak, The Maxwell operator in domains with a nonsmooth boundary, Sibirsk. Mat. Zh. 28 (1987), no. 1, 23-36; English transl., Siberian Math. J. 28 (1987), 12-24. MR 0886850 (88g:35095)
  • 91. -, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Ser. 1 1987, vyp. 3, 23-28; English transl., Vestnik Leningrad Univ. Math. 20 (1987), no. 3, 15-21. MR 0928156 (89h:35253)
  • 92. -, Construction in a piecewise-smooth domain of a function of the class $ H^2$ from the value of the conormal derivative, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163 (1987), 17-28; English transl., J. Soviet Math. 49 (1990), no. 5, 1128-1136. MR 0918938 (89b:35035)
  • 93. -, $ L_2$-theory of the Maxwell operator in arbitrary domains, Uspekhi Mat. Nauk 42 (1987), no. 6, 61-76; English transl., Russian Math. Surveys 42 (1987), no. 6, 75-96. MR 0933995 (89e:35127)
  • 94. M. Sh. Birman and D. R. Yafaev, A general scheme in the stationary theory of scattering, Wave Propagation. Scattering Theory, Probl. Mat. Fiz., No. 12, Leningrad. Univ., Leningrad, 1987, pp. 89-117; English transl., Amer. Math. Soc. Transl. (2), vol. 157, Amer. Math. Soc., Providence, RI, 1993, pp. 87-112. MR 0923973 (89b:47016); MR 1252226
  • 95. M. Sh. Birman, The Maxwell operator for a periodic resonator with inward edges, Trudy Mat. Inst. Akad. Nauk SSSR 179 (1988), 23-35; English transl., Proc. Steklov Inst. Math. 1989, no. 2, 21-34.MR 0964911 (89m:35160)
  • 96. M. Sh. Birman and M. Z. Solomyak, The $ L_2$-theory of the Maxwell operator in domains with nonsmooth boundary, Symposium ``Partial Differential Equations'' (Holzhau, 1988), Teubner-Texte Math., vol. 112, Teubner, Leipzig, 1989, pp. 33-41. MR 1105797
  • 97. -, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96-110; English transl., Leningrad Math. J. 1 (1990), no. 1, 99-115. MR 1015335 (91e:35197)
  • 98. -, Interpolation estimates for the number of negative eigenvalues of a Schrödinger operator, Schrödinger Operators, Standard and Nonstandard (Dubna, 1988), World Sci. Publishing, Teaneck, NJ, 1989, pp. 2-18.MR 1091987 (92e:35115)
  • 99. M. Sh. Birman and D. R. Yafaev, On the trace-class method in potential scattering theory, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 171 (1989), 12-35; English transl., J. Soviet Math. 56 (1991), no. 2, 2285-2299. MR 1031982 (91g:35198)
  • 100. M. Sh. Birman and M. Z. Solomyak, Operator integration, perturbations and commutators, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), 34-66; English transl., J. Soviet Math. 63 (1993), no. 2, 129-148. MR 1039579 (91b:47086)
  • 101. -, Estimates for the difference of fractional powers of selfadjoint operators under unbounded perturbations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), 120-145; English transl., J. Soviet Math. 61 (1992), no. 2, 2018-2035. MR 1037767 (91d:47006)
  • 102. -, Negative discrete spectrum of the Schrödinger operator with large coupling constant: a qualitative discussion, Order, Disorder and Chaos in Quantum Systems (Dubna, 1989), Oper. Theory Adv. Appl., vol. 46, Birkhäuser, Basel, 1990, pp. 3-16. MR 1124648 (92j:35136)
  • 103. M. Sh. Birman, Discrete spectrum in the gaps of the continuous one in the large-coupling-constant limit, Order, Disorder and Chaos in Quantum Systems (Dubna, 1989), Oper. Theory Adv. Appl., vol. 46, Birkhäuser, Basel, 1990, pp. 17-25. MR 1124649 (92j:47091)
  • 104. M. Sh. Birman and M. Z. Solomyak, The estimates for the number of negative bound states of the Schrödinger operator for large coupling constants, Integral Equations and Inverse Problems (Varna, 1989), Pitman Res. Notes Math. Ser., vol. 235, Longman Sci. Tech., Harlow, 1991, pp. 49-57.MR 1117952 (92g:35159)
  • 105. M. Sh. Birman, S. G. Kreĭn, O. A. Ladyzhenskaya, G. B. Rozenblum, and Yu. G. Safarov, Mikhail Zakharovich Solomyak (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 46 (1991), no. 4, 183-184; English transl., Russian Math. Surveys 46 (1991), no. 4, 217-219. MR 1138976 (92k:01025)
  • 106. M. Sh. Birman, On the discrete spectrum in gaps of a second-order perturbed periodic operator, Funktsional.Anal. i Prilozhen. 25 (1991), no. 2, 89-92; English transl., Funct. Anal. Appl. 25 (1991), no. 2, 158-161.MR 1142222 (92m:47090)
  • 107. -, Discrete spectrum in a gap of perturbed periodic operator at large coupling constants, Rigorous Results in Quantum Dynamics (Liblice, 1990), World Sci. Publishing, River Edge, NJ, 1991, pp. 16-24. MR 1243026
  • 108. M. Sh. Birman and M. Z. Solomyak, Discrete negative spectrum under nonregular perturbations (polyharmonic operators, Schrödinger operators with a magnetic field, periodic operators), Rigorous Results in Quantum Dynamics (Liblice, 1990), World Sci. Publishing, River Edge, NJ, 1991, pp. 25-36. MR 1243027
  • 109. -, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1-55. MR 1306507 (95h:35161)
  • 110. M. Sh. Birman, Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 57-73. MR 1306508 (95h:47009)
  • 111. M. Sh. Birman and G. D. Raikov, Discrete spectrum in the gaps for perturbations of the magnetic Schrödinger operator, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 75-84. MR 1306509 (95h:35153)
  • 112. M. Sh. Birman, G. E. Karadzhov, and M. Z. Solomyak, Boundedness conditions and spectrum estimates for the operators $ b(X)a(D)$ and their analogs, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85-106. MR 1306510 (95g:47075)
  • 113. M. Sh. Birman and M. Z. Solomyak, Schrödinger operator. Estimates for number of bound states as function-theoretical problem, Spectral Theory of Operators (Novgorod, 1989), Amer. Math. Soc. Transl. Ser. 2, vol. 150, Amer. Math. Soc., Providence, RI, 1992, pp. 1-54. MR 1157648 (93j:35120)
  • 114. M. Sh. Birman, Three problems in continuum theory in polyhedra, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 200 (1992), 27-37; English transl., J. Math. Sci. (N.Y.) 77 (1995), no. 3, 3153-3160. MR 1192110 (94b:35204)
  • 115. M. Sh. Birman and D. R. Yafaev, The spectral shift function. The papers of M. G. Kreĭn and their further development, Algebra i Analiz 4 (1992), no. 5, 1-44; English transl., St. Petersburg Math. J. 4 (1993), no. 5, 833-870.MR 1202723 (94g:47002)
  • 116. -, Spectral properties of the scattering matrix, Algebra i Analiz 4 (1992), no. 6, 1-27; English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1055-1079. MR 1199632 (94g:47003)
  • 117. M. Sh. Birman and M. Z. Solomyak, On the main singularities of the electric component of the electro-magnetic field in regions with screens, Algebra i Analiz 5 (1993), no. 1, 143-159; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 125-139. MR 1220492 (94f:35137)
  • 118. M. Sh. Birman, Discrete spectrum of the periodic Schrödinger operator for non-negative perturbations, Mathematical Results in Quantum Mechanics (Blossin, 1993), Oper. Theory Adv. Appl., vol. 70, Birkhäuser, Basel, 1994, pp. 3-7. MR 1308998
  • 119. M. Sh. Birman and T. Weidl, The discrete spectrum in a gap of the continuous one for compact supported perturbations, Mathematical Results in Quantum Mechanics (Blossin, 1993), Oper. Theory Adv. Appl., vol. 70, Birkhäuser, Basel, 1994, pp. 9-12. MR 1308999
  • 120. M. Sh. Birman and A. Laptev, Discrete spectrum of the perturbed Dirac operator, Mathematical Results in Quantum Mechanics (Blossin, 1993), Oper. Theory Adv. Appl., vol. 70, Birkhäuser, Basel, 1994, pp. 55-59. MR 1309006
  • 121. M. Sh. Birman and D. R. Yafaev, Scattering matrix for the perturbation of a periodic Schrödinger operator by a decaying potential, Algebra i Analiz 6 (1994), no. 3, 17-39; English transl., St. Petersburg Math. J. 6 (1995), no. 3, 453-474. MR 1301828 (95i:47086)
  • 122. M. Sh. Birman, Discrete spectrum of the periodic elliptic operator with a differential perturbation, Journées ``Équations aux Dérivées Partielles'' (Saint-Jean-de-Monts, 1994), Exp. No. XIV, École Polytech., Palaiseau, 1994, 4 pp. MR 1298685
  • 123. M. Sh. Birman and A. Laptev, Discrete spectrum of the perturbed Dirac operator, Ark. Mat. 32 (1994), no. 1, 13-32. MR 1277918 (95h:35162)
  • 124. M. Sh. Birman, Ilya Bakelman in Russia. A tribute to Ilya Bakelman (College Station, TX, 1993), Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 2-4. MR 1423361
  • 125. V. I. Arnol'd, M. Sh. Birman, I. M. Gel'fand et al., Anatoliĭ Moiseevich Vershik (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 49 (1994), no. 3, 195-204; English transl., Russian Math. Surveys 49 (1994), no. 3, 207-221. MR 1289402
  • 126. M. Sh. Birman, Spectral shift function and double operator integrals, Linear and Complex Analysis Problem Book 3. Part I, Lecture Notes in Math., vol. 1573, Springer-Verlag, Berlin, 1994, pp. 272-273. MR 1334345 (96c:00001a)
  • 127. -, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations, Boundary Value Problems, Schrödinger Operators, Deformation Quantization, Math. Top., vol. 8, Akademie Verlag, Berlin, 1995, pp. 334-352. MR 1389015 (97d:47055)
  • 128. -, Discrete spectrum of the periodic Schrödinger operator perturbed by a decaying potential, Algebra i Analiz 8 (1996), no. 1, 3-20; English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1-14. MR 1392011 (97h:47047)
  • 129. M. Sh. Birman and M. Solomyak, Tensor product of a finite number of spectral measures is always a spectral measure, Integral Equations Operator Theory 24 (1996), no. 2, 179-187. MR 1371945 (96m:47038)
  • 130. M. Sh. Birman and A. Laptev, The negative discrete spectrum of a two-dimensional Schrödinger operator, Comm. Pure Appl. Math. 49 (1996), no. 9, 967-997.MR 1399202 (97i:35131)
  • 131. M. Sh. Birman and T. A. Suslina, Two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algera i Analiz 9 (1997), no. 1, 32-48; English transl., St. Petersburg Math. J. 9 (1998), no. 1, 21-32. MR 1458417 (98g:47038)
  • 132. M. Sh. Birman, A. Laptev, and M. Solomyak, The negative discrete spectrum of the operator $ (-\Delta)^l - \alpha V$ in $ L_2(\mathbf{R}^d)$ for $ d$ even and $ 2l\geq d$, Ark. Mat. 35 (1997), no. 1, 87-126. MR 1443037 (98d:35167)
  • 133. M. Sh. Birman and A. Laptev, ``Non-standard'' spectral asymptotics for a two-dimensional Schrödinger operator, Partial Differential Equations and their Applications (Toronto, ON, 1995), CRM Proc. Lecture Notes, vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 9-16. MR 1479234
  • 134. M. Sh. Birman, Discrete spectrum in the gaps of a perturbed periodic Schrödinger operator. II. Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62-89; English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073-1095. MR 1610239 (99h:47054)
  • 135. M. Sh. Birman and A. B. Pushnitskiĭ, Discrete spectrum in the gaps of the perturbed pseudorelativistic magnetic Hamiltonian, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov.(POMI) 249 (1997), 102-117; English transl., J. Math. Sci. (N.Y.) 101 (2000), no. 5, 3437-3447. MR 1698515 (2000h:35123)
  • 136. -, Spectral shift function, amazing and multifaceted, Integral Equations Operator Theory 30 (1998), no. 2, 191-199. MR 1607900 (98m:47012)
  • 137. M. Sh. Birman, A. Laptev, and M. Solomyak, On the eigenvalue behaviour for a class of differential operators on semiaxis, Math. Nachr. 195 (1998), 17-46. MR 1654744 (99m:34194)
  • 138. M. Sh. Birman and T. A. Suslina, Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, Algebra i Analiz 10 (1998), no. 4, 1-36; English transl., St. Petersburg Math. J. 10 (1999), no. 4, 579-601. MR 1654063 (99k:81060)
  • 139. -, Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum, Mathematical Results in Quantum Mechanics (Prague, 1998), Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 13-31. MR 1708785 (2000g:81049)
  • 140. -, The periodic Dirac operator is absolutely continuous, Integral Equations Operator Theory 34 (1999), no. 4, 377-395. MR 1702229 (2000h:47068)
  • 141. -, Periodic magnetic Hamiltonian with variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no. 2, 1-40; English transl., St. Petersburg Math. J. 11 (2000), no. 2, 203-232. MR 1702587 (2000i:35026)
  • 142. 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, Algebra i Analiz 12 (2000), no. 6, 140-177; English transl., St. Petersburg Math. J. 12 (2001), no. 6, 983-1012.MR 1816514 (2002k:35227)
  • 143. M. Sh. Birman and T. A. Suslina, On the absolute continuity of the periodic Schrödinger and Dirac operators with magnetic potential, Differential Equations and Mathematical Physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., vol. 16, Amer. Math. Soc., Providence, RI, 2000, pp. 41-49. MR 1764740
  • 144. M. Sh. Birman, A. Laptev, and T. A. Suslina, Discrete spectrum of a two-dimensional periodic elliptic second order operator perturbed by a decaying potential. I. Semibounded gap, Algebra i Analiz 12 (2000), no. 4, 36-78; English transl., St. Petersburg Math. J. 12 (2001), no. 4, 535-567. MR 1793617 (2003b:47078)
  • 145. M. Sh. Birman and T. A. Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71-107. MR 1882692 (2003f:35220)
  • 146. M. Sh. Birman and M. Z. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math. 83 (2001), 337-391. MR 1828497 (2002k:35226)
  • 147. B. A. Amosov, M. Sh. Birman, M. I. Vishik et al., Mikhail Semenovich Agranovich (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 56 (2001), no. 4, 163-168; English transl., Russian Math. Surveys 56 (2001), no. 4, 777-784. MR 1861459 (2002h:01026)
  • 148. M. Sh. Birman and T. A. Suslina, Absolute continuity of the spectrum of the periodic operator of elasticity theory for constant shear modulus, Nonlinear Problems in Mathematical Physics and Related Topics, II, Mezhdunar. Mat. Ser., vol. 2, Tamara Rozhkovskaya, Novosibirsk, 2002, pp. 65-70; English transl., Int. Math. Ser., vol. 2, Kluwer/Plenum, New York etc., 2002, pp. 69-74. MR 1971990 (2004c:35296)
  • 149. A. A. Arkhipova, M. Sh. Birman, V. S. Buslaev et al., On the jubilee of Ol'ga Aleksandrovna Ladyzhenskaya, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 5-13; English transl., J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4523-4526. MR 1923542
  • 150. M. Sh. Birman, On homogenization procedure for periodic operators near the edge of an internal gap, Algebra i Analiz 15 (2003), no. 4, 61-71; English transl., St. Petersburg Math. J. 15 (2004), no. 4, 507-513. MR 2068979 (2006i:35010)
  • 151. M. Sh. Birman and T. A. Suslina, Second order periodic differential operators. Threshold properties and homogenization, Algebra i Analiz 15 (2003), no. 5, 1-108; English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639-714. MR 2068790 (2005k:47097)
  • 152. M. Sh. Birman and M. Z. Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), 131-168. MR 2002663 (2004f:47029)
  • 153. M. Sh. Birman and T. A. Suslina, Homogenization of multidimensional periodic elliptic operator in a neighborhood of an edge of an inner gap, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318 (2004), 60-74; English transl., J. Math. Sci. (N.Y.) 136 (2006), no. 2, 3682-3690. MR 2120232 (2005j:35006)
  • 154. V. I. Arnol'd, M. Sh. Birman, A. M. Vershik et al., Ol'ga Aleksandrovna Ladyzhenskaya, Uspekhi Mat. Nauk 59 (2004), no. 3, 151-152; English transl., Russian Math. Surveys 59 (2004), no. 3, 553-555. MR 2116538
  • 155. M. Sh. Birman and T. A. Suslina, Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family, Algebra i Analiz 17 (2005), no. 5, 69-90; English transl., St. Petersburg Math. J. 17 (2006), no. 5, 745-762. MR 2241423 (2008d:47047)
  • 156. -, Homogenization with corrector term for periodic elliptic differential operators, Algebra i Analiz 17 (2005), no. 6, 1-104; English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897-973. MR 2202045 (2006k:35011)
  • 157. A. B. Alekseev, M. Sh. Birman, and N. D. Filonov, Spectrum asymptotics for one ``nonsmooth'' variational problem with solvable constraint, Algebra i Analiz 18 (2006), no. 5, 1-22; English transl., St. Petersburg Math. J. 18 (2007), no. 5, 681-697. MR 2301038 (2008b:35193)
  • 158. M. Sh. Birman and T. A. Suslina, Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class $ H^1(\mathbb{R}^d)$, Algebra i Analiz 18 (2006), no. 6, 1-130; English transl., St. Petersburg Math. J. 18 (2007), no. 6, 857-955. MR 2307356 (2008d:35008)
  • 159. -, Homogenization of a stationary periodic Maxwell system in the case of constant magnetic permeability, Funktsional. Anal. i Prilozhen. 41 (2007), no. 2, 3-23; English transl., Funct. Anal. Appl. 41 (2007), no. 2, 81-98. MR 2345036 (2008i:35014)
  • 160. M. Sh. Birman and N. D. Filonov, Weyl asymptotics of the spectrum of the Maxwell operator with non-smooth coefficients in Lipschitz domains, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 27-44.MR 2343605 (2009a:35243)
  • 161. M. Sh. Birman and T. A. Suslina, Operator error estimates in the homogenization problem for nonstationary periodic equations, Algebra i Analiz 20 (2008), no. 6, 30-107; English transl., St. Petersburg Math. J. 20 (2009), no. 6, 873-928. MR 2530894 (2010f:35017)
  • 162. -, The limiting absorption principle and the homogenization procedure for periodic elliptic operators, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 105-108; English transl., Funct. Anal. Appl. 42 (2008), no. 4, 336-339. MR 2492431 (2010f:35016)
  • 163. M. Sh. Birman and T. A. Suslina, Homogenization of periodic differential operators as a spectral threshold effect, New Trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics, Springer, Berlin, 2009, pp. 667-683.
  • 164. M. Sh. Birman and V. A. Sloushch, Two-sided estimates for the trace of the difference of a pair of semigroups, Funktsional. Anal. i Prilozhen. 43 (2009), no. 3, 26-32; English transl., Funct. Anal. Appl. 43 (2009), no. 3, 184-189. MR 2583637 (2010k:47082)
  • 165. -, Discrete spectrum of the periodic Schrödinger operator with a variable metric perturbed by a nonnegative potential, Math. Model. Nat. Phenom. 5 (2010), no. 4, 32-53. MR 2662449 (2011e:47085)
  • 166. M. Sh. Birman and T. A. Suslina, The analog of the limiting absorption principle in homogenization of periodic elliptic operators, Operator Theory and its Applications: In Memory of V. B. Lidskii (1924-2008), Amer. Math. Soc. Transl. Ser. 2, vol. 231, Amer. Math. Soc., Providence, RI, 2010, pp. 35-57. MR 2509773

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Additional Information

M. Z. Solomyak
Affiliation: Department of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel
Email: michail.solomyak@weizmann.ac.il

T. A. Suslina
Affiliation: St. Petersburg State University, Department of Physics, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: suslina@list.ru

D. R. Yafaev
Affiliation: IRMAR, University Rennes-1, Rennes 35042, France
Email: dimitri.yafaev@univ-rennes1.fr

DOI: https://doi.org/10.1090/S1061-0022-2011-01184-7
Keywords: Spectral theory, spectral asymptotics, function spaces, scattering theory
Received by editor(s): October 31, 2010
Published electronically: November 7, 2011
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society