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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

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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  • 1. M. Š. Birman, Some estimates for the method of steepest descent, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 3(37), 152–155 (Russian). MR 0035922
    M. Sh. Birman, Some estimates for the method of steepest descent, Four articles on numerical matrix methods, NBS Rep. 2007, U. S. Department of Commerce, National Bureau of Standards, Washington, D. C., 1952, pp. 51–56. Translated by C. D. Benster. MR 0050985
  • 2. -, On computation of eigenvalues by the method of steepest descent, Zap. Leningrad. Gornogo Inst. 27 (1952), no. 1, 209-216. (Russian)
  • 3. -, On a variant of the method of successive approximations, Vestnik Leningrad. Univ. Ser. Mat. Phys. Chem. 1952, no. 9, 69-76. (Russian)
  • 4. M. Š. Birman, On the theory of self-adjoint extensions of positive definite operators, Doklady Akad. Nauk SSSR (N.S.) 91 (1953), 189–191 (Russian). MR 0058130
  • 5. M. Š. Birman, On the theory of general boundary problems for elliptic differential equations, Doklady Akad. Nauk SSSR (N.S.) 92 (1953), 205–208 (Russian). MR 0062922
  • 6. M. Š. Birman, On minimal functionals for elliptic differential equations of second order, Doklady Akad. Nauk SSSR (N.S.) 93 (1953), 953–956 (Russian). MR 0062923
  • 7. M. Š. Birman, On the spectrum of singular boundary problems for elliptic differential equations, Doklady Akad. Nauk SSSR (N.S.) 97 (1954), 5–7 (Russian). MR 0064283
  • 8. M. Š. Birman, On Trefftz’s variational method for the equation Δ²𝑢=𝑓, Dokl. Akad. Nauk SSSR (N.S.) 101 (1955), 201–204 (Russian). MR 0070831
  • 9. -, On the Friedrichs extension procedure of a positive operator to the selfadjoint one, Zap. Leningrad. Gornogo Inst. 33 (1956), no. 3, 132-136.
  • 10. M. Š. Birman, On the theory of self-adjoint extensions of positive definite operators, Mat. Sb. N.S. 38(80) (1956), 431–450 (Russian). MR 0080271
  • 11. M. Š. Birman, Variational methods of solution of boundary problems analogous to the method of Trefftz, Vestnik Leningrad. Univ. 11 (1956), no. 13, 69–89 (Russian). MR 0080867
  • 12. M. Š. Birman, Method of quadratic forms in problems with small parameter in the highest derivatives, Vestnik Leningrad. Univ. 12 (1957), no. 13, 9–12 (Russian, with English summary). MR 0101398
  • 13. -, A characterization of elliptic differential operators with maximal domain, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1957, vyp. 4, 177-183. (Russian)
  • 14. -, On multi-dimensional boundary value problems with a small parameter at higher derivatives, Uspekhi Mat. Nauk (N.S.) 12 (1957), no. 6, 212-213. (Russian)
  • 15. M. Š. Birman, Perturbations of quadratic forms and the spectrum of singular boundary value problems, Dokl. Akad. Nauk SSSR 125 (1959), 471–474 (Russian). MR 0107816
  • 16. M. Š. Birman, On the spectrum of Schrödinger and Dirac operators, Dokl. Akad. Nauk SSSR 129 (1959), 239–241 (Russian). MR 0109934
  • 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. Š. Birman and B. S. Pavlov, On the complete continuity of certain imbedding operators, Vestnik Leningrad. Univ. 16 (1961), no. 1, 61–74 (Russian, with English summary). MR 0133491
  • 19. M. Š. Birman, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.) 55 (97) (1961), 125–174 (Russian). MR 0142896
    American Mathematical Society Translations. Series 2, Vol. 53: 11 papers on analysis, American Mathematical Society, Providence, R.I., 1966. MR 0194283
  • 20. M. Š. Birman, Perturbation of the spectrum of a singular elliptic operator under variation of the boundary and boundary conditions, Soviet Math. Dokl. 2 (1961), 326–328. MR 0177311
  • 21. M. Š. Birman, On the number of eigenvalues in a quantum scattering problem, Vestnik Leningrad. Univ. 16 (1961), no. 13, 163–166 (Russian, with English summary). MR 0139370
  • 22. M. Š. Birman, Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions, Vestnik Leningrad. Univ. 17 (1962), no. 1, 22–55 (Russian, with English summary). MR 0138874
  • 23. M. Š. Birman, On existence conditions for wave operators, Dokl. Akad. Nauk SSSR 143 (1962), 506–509 (Russian). MR 0149333
  • 24. M. Š. Birman and M. G. Kreĭn, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), 475–478 (Russian). MR 0139007
  • 25. M. Š. Birman and G. E. Skvorcov, On square summability of highest derivatives of the solution of the Dirichlet problem in a domain with piecewise smooth boundary, Izv. Vysš. Učebn. Zaved. Matematika 1962 (1962), no. 5 (30), 11–21 (Russian). MR 0145196
  • 26. M. Š. Birman, On a test for the existence of wave operators, Dokl. Akad. Nauk SSSR 147 (1962), 1008–1009 (Russian). MR 0167841
  • 27. M. Š. Birman, Existence conditions for wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 883–906 (Russian). MR 0161150
  • 28. M. Š. Birman and M. G. Kreĭn, Some topics of the theory of the wave and scattering operators, Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), Acad. Sci. USSR Siberian Branch, Moscow, 1963, pp. 39–45. MR 0200749
  • 29. M. Š. Birman and I. M. Glazman, Spectra of singular differential operators, Proc. Fourth All-Union Math. Congr. (Leningrad, 1961) Izdat. “Nauka”, Leningrad, 1964, pp. 253–261 (Russian). MR 0220101
  • 30. M. Š. Birman and S. B. Èntina, A stationary approach in the abstract theory of scattering, Dokl. Akad. Nauk SSSR 155 (1964), 506–508 (Russian). MR 0164590
  • 31. M. Š. Birman, A local criterion for the existence of wave operators, Dokl. Akad. Nauk SSSR 159 (1964), 485–488 (Russian). MR 0173964
  • 32. M. Š. Birman and M. Z. Solomjak, On double Stieltjes operator integrals, Dokl. Akad. Nauk SSSR 165 (1965), 1223–1226 (Russian). MR 0188802
  • 33. M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals, Probl. Math. Phys., No. I, Spectral Theory and Wave Processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 33–67 (Russian). MR 0209872
    Spectral theory and wave processes, Edited by M. Sh. Birman. Translated from the Russian. Topics in Mathematical Physics, Vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967. MR 0210536
  • 34. M. Š. Birman and M. Z. Solomjak, Approximation of functions of the 𝑊_{𝑝}^{𝛼}-classes by piece-wise-polynomial functions, Dokl. Akad. Nauk SSSR 171 (1966), 1015–1018 (Russian). MR 0209733
  • 35. M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals and problems on multipliers, Dokl. Akad. Nauk SSSR 171 (1966), 1251–1254 (Russian). MR 0209871
  • 36. M. Š. Birman and S. B. Èntina, Stationary approach in abstract scattering theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 401–430 (Russian). MR 0209895
  • 37. M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals. II, Problems of Mathematical Physics, No. 2, Spectral Theory, Diffraction Problems (Russian), Izdat. Leningrad. Univ., Leningrad, 1967, pp. 26–60 (Russian). MR 0234304
    Spectral theory and problems in diffraction, Edited by M. Sh. Birman. Translated from Russian. Topics in Mathematical Physics, Vol. 2, Consultants Bureau, New York, 1968. MR 0266727
  • 38. M. Š. Birman and M. Z. Solomjak, Piecewise polynomial approximations of functions of classes 𝑊_{𝑝}^{𝛼}, Mat. Sb. (N.S.) 73 (115) (1967), 331–355 (Russian). MR 0217487
  • 39. M. Š. Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. I, Vestnik Leningrad. Univ. 22 (1967), no. 7, 43–53 (Russian, with English summary). MR 0216339
  • 40. M. Š. Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. II, Vestnik Leningrad. Univ. 22 (1967), no. 13, 21–28 (Russian, with English summary). MR 0217650
  • 41. M. Š. Birman, A local test for the existence of wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 914–942 (Russian). MR 0248558
  • 42. A. L. Belopol′skiĭ and M. Š. Birman, Existence of wave operators in scattering theory for a pair of spaces., Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1162–1175 (Russian). MR 0238101
  • 43. M. Š. Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. III. Operators in unbounded domains, Vestnik Leningrad. Univ. 24 (1969), no. 1, 35–48 (Russian, with English summary). MR 0246163
  • 44. M. Š. Birman, Some applications of a local criterion for the existence of wave operators, Soviet Math. Dokl. 10 (1969), 393–397. MR 0282242
  • 45. M. Š. Birman and M. Z. Solomjak, Remarks on the nuclearity of integral operators and the boundedness of pseudodifferential operators, Izv. Vysš. Učebn. Zaved. Matematika 1969 (1969), no. 9 (88), 11–17 (Russian). MR 0254670
  • 46. M. Š. Birman, Scattering problems for differential operators with constant coefficients, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 1–16 (Russian). MR 0253083
  • 47. M. Š. Birman, A test of the existence of complete wave operators in scattering theory for a pair of spaces, Problems of Mathematical Physics, No. 4: Spectral Theory. Wave Process (Russian), Izdat. Leningrad. Univ., Leningrad, 1970, pp. 22–26 (Russian). MR 0276817
    Spectral theory and wave processes, Consultants Bureau, New York-London, 1971. Edited by M. Sh. Birman; Translated from the Russian by J. R. Schulenberger; Topics in Mathematical Physics, Vol. 4. MR 0339975
  • 48. M. Š. Birman and M. Z. Solomjak, Asymptotics of the spectrum of weakly polar integral operators, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1142–1158 (Russian). MR 0279638
  • 49. M. Š. Birman and M. Z. Solomjak, The principal term of the spectral asymptotics for “non-smooth” elliptic problems, Funkcional. Anal. i Priložen. 4 (1970), no. 4, 1–13 (Russian). MR 0278126
  • 50. M. Š. Birman and M. Z. Solomjak, 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. vyp. 4 (18), 16–25 (Russian). MR 0284851
  • 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. M. Š. Birman, Scattering problems for differential operators under a perturbation of the space, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 440–455 (Russian). MR 0291868
  • 53. M. Š. Birman and M. Z. Solomjak, The asymptotics of the spectrum of “nonsmooth” elliptic equations., Funkcional. Anal. i Priložen. 5 (1971), no. 1, 69–70 (Russian). MR 0283428
  • 54. M. Š. Birman and V. V. Borzov, The asymptotic behavior of the discrete spectrum of certain singular differential operators, Problems of mathematical physics, No. 5: spectral theory (Russian), Izdat. Leningrad. Univ., Leningrad, 1971, pp. 24–38 (Russian). MR 0301568
    Spectral theory, Consultants Bureau, New York-London, 1972. Edited by M. Sh. Birman; Translated from the Russian by J. R. Schulenberger; Topics in Mathematical Physics, Vol. 5. MR 0339976
  • 55. M. Š. Birman and M. Z. Solomjak, Remarks on the spectral shift function, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 33–46 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR 0315482
  • 56. M. Š. Birman and M. Z. Solomjak, Spectral asymptotic behavior of nonsmooth elliptic operators, Dokl. Akad. Nauk SSSR 205 (1972), 267–270 (Russian). MR 0305148
  • 57. M. Š. Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. Obšč. 27 (1972), 3–52; ibid. 28 (1973), 3–34 (Russian). MR 0364898
  • 58. V. M. Babič, M. Š. Birman, V. I. Smirnov, M. Z. Solomjak, and N. N. Ural′ceva, Ol′ga Aleksandrovna Ladyženskaja (on her fiftieth birthday), Vestnik Leningrad. Univ. 7 (1972), 159–160 (Russian). MR 0297506
  • 59. M. Š. Birman and M. Z. Solomjak, Double Stieltjes operator integrals. III, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 27–53 (Russian). MR 0348494
  • 60. M. Š. Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. Obšč. 27 (1972), 3–52; ibid. 28 (1973), 3–34 (Russian). MR 0364898
  • 61. M. Š. Birman and M. Z. Solomjak, Equivalent renormings of 𝑊^{𝑟}_{𝑝} classes in the case of partitioning of the domain, Izv. Vysš. Učebn. Zaved. Matematika 3(130) (1973), 19–27 (Russian). MR 0344875
  • 62. M. Š. Birman and M. Z. Solomjak, Quantitative analysis in Sobolev’s imbedding theorems and applications to spectral theory, Tenth Mathematical School (Summer School, Kaciveli/Nalchik, 1972) Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5–189 (Russian). MR 0482138
  • 63. M. Š. Birman and M. Z. Solomjak, A certain “model” nonelliptic spectral problem, Vestnik Leningrad. Univ. 1 Mat. Meh. Astronom. vyp. 1 (1975), 39–45, 145 (Russian, with English summary). Collection of articles dedicated to the memory of Academician V. I. Smirnov. MR 0380148
  • 64. M. Š. Birman, L. S. Koplienko, and M. Z. Solomjak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. Vysš. Učebn. Zaved. Matematika 3(154) (1975), 3–10 (Russian). MR 0385597
  • 65. A. B. Alekseev and M. Š. Birman, Asymptotic behavior of the spectrum of elliptic boundary value problems with solvable constraints, Dokl. Adad. Nauk SSSR 230 (1976), no. 3, 505–507 (Russian). MR 0420027
  • 66. A. B. Alekseev and M. Š. Birman, A variational formulation of the problem of the oscillations of a resonator that is filled with a stratified anisotropic medium, Vestnik Leningrad. Univ. 7 Mat. Meh. Astronom., vyp. 2 (1977), 9–15, 161 (Russian, with English summary). MR 0462202
  • 67. M. Š. Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, Vestnik Leningrad. Univ. 13 Mat. Meh. Astronom. vyp. 3 (1977), 13–21, 169 (Russian, with English summary). MR 0463738
  • 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. M. Š. Birman and M. Z. Solomjak, Estimates for the singular numbers of integral operators, Uspehi Mat. Nauk 32 (1977), no. 1(193), 17–84, 271 (Russian). MR 0438186
  • 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. Š. Birman and M. Z. Solomjak, Asymptotic properties of the spectrum of differential equations, Mathematical analysis, Vol. 14 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow, 1977, pp. 5–58, i. (loose errata) (Russian). MR 0467022
  • 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 scattering theory [ MR0923973 (89b:47016)], Wave propagation. Scattering theory, Amer. Math. Soc. Transl. Ser. 2, vol. 157, Amer. Math. Soc., Providence, RI, 1993, pp. 87–112. MR 1252226, https://doi.org/10.1090/trans2/157/07
  • 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. Solomjak, The 𝐿₂-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. M. Sh. Birman and M. Z. Solomyak, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96–110 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 99–115. MR 1015335
  • 98. M. Sh. Birman and M. Z. Solomyak, Interpolation estimates for the number of negative eigenvalues of a Schroedinger operator, Schrödinger operators, standard and nonstandard (Dubna, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 2–18. MR 1091987
  • 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), no. Kraev. Zadachi Mat. Fiz. i Smezh. Voprosy Teor. Funktsiĭ. 20, 12–35, 183 (Russian, with English summary); English transl., J. Soviet Math. 56 (1991), no. 2, 2285–2299. MR 1031982, https://doi.org/10.1007/BF01671931
  • 100. N. G. Makarov, The size of the set of singular points on the boundary of a non-Smirnov domain, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 17, 176–183, 323 (Russian, with English summary); English transl., J. Soviet Math. 63 (1993), no. 2, 212–216. MR 1039579, https://doi.org/10.1007/BF01099312
  • 101. M. Sh. Birman and M. Z. Solomyak, Estimates for the difference of fractional powers of selfadjoint operators under unbounded perturbations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 18, 120–145, 185 (Russian, with English summary); English transl., J. Soviet Math. 61 (1992), no. 2, 2018–2035. MR 1037767, https://doi.org/10.1007/BF01095665
  • 102. M. Sh. Birman and M. Z. Solomyak, Negative discrete spectrum of the Schroedinger 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
  • 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
  • 104. M. Sh. Birman and M. Z. Solomjak, 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
  • 105. M. Sh. Birman, S. G. Kreĭn, O. A. Ladyzhenskaya, G. V. Rozenblyum, and Yu. G. Safarov, Mikhail Zakharovich Solomyak (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 46 (1991), no. 4(280), 183–184 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 4, 217–219. MR 1138976, https://doi.org/10.1070/RM1991v046n04ABEH002828
  • 106. M. Sh. Birman, On a discrete spectrum in gaps of a second-order perturbed periodic operator, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 89–92 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 2, 158–161. MR 1142222, https://doi.org/10.1007/BF01079605
  • 107. M. Sh. Birman, Discrete spectrum in a gap of perturbed periodic operator at large coupling constants, Rigorous results in quantum dynamics (Liblice, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 16–24. MR 1243026, https://doi.org/10.1021/cen-v069n011.p016
  • 108. M. Sh. Birman and M. Z. Solomyak, Discrete negative spectrum under nonregular perturbations (polyharmonic operators, Schroedinger operators with a magnetic field, periodic operators), Rigorous results in quantum dynamics (Liblice, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 25–36. MR 1243027, https://doi.org/10.1117/12.24776
  • 109. M. Sh. Birman and M. Z. Solomyak, 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
  • 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
  • 111. M. Sh. Birman and G. D. Raĭkov, 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
  • 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
  • 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, https://doi.org/10.1090/trans2/150/01
  • 114. M. Sh. Birman, Three problems in continuum theory in polyhedra, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 200 (1992), no. Kraev. Zadachi Mat. Fiz. Smezh. Voprosy Teor. Funktsiĭ. 24, 27–37, 187 (Russian, with English and Russian summaries); English transl., J. Math. Sci. 77 (1995), no. 3, 3153–3160. MR 1192110, https://doi.org/10.1007/BF02364704
  • 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 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 5, 833–870. MR 1202723
  • 116. M. Sh. Birman and D. R. Yafaev, Spectral properties of the scattering matrix, Algebra i Analiz 4 (1992), no. 6, 1–27 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 6, 1055–1079. MR 1199632
  • 117. M. Sh. Birman and M. Z. Solomyak, Principal singularities of the electric component of an electromagnetic field in regions with screens, Algebra i Analiz 5 (1993), no. 1, 143–159 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 125–139. MR 1220492
  • 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, https://doi.org/10.1007/978-3-0348-8545-4_1
  • 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, https://doi.org/10.1007/978-3-0348-8545-4_2
  • 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, https://doi.org/10.1007/978-3-0348-8545-4_9
  • 121. M. Sh. Birman and D. R. Yafaev, The scattering matrix for a perturbation of a periodic Schrödinger operator by decreasing potential, Algebra i Analiz 6 (1994), no. 3, 17–39 (Russian); English transl., St. Petersburg Math. J. 6 (1995), no. 3, 453–474. MR 1301828
  • 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) École Polytech., Palaiseau, 1994, pp. Exp. No. XIV, 4. MR 1298685
  • 123. Mikhail Sh. Birman and Ari Laptev, Discrete spectrum of the perturbed Dirac operator, Ark. Mat. 32 (1994), no. 1, 13–32. MR 1277918, https://doi.org/10.1007/BF02559521
  • 124. Mikhail S. 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(297), 195–204 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 3, 207–221. MR 1289402, https://doi.org/10.1070/RM1994v049n03ABEH002272
  • 126. V. P. Havin and N. K. Nikolski (eds.), Linear and complex analysis. Problem book 3. Part I, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. MR 1334345
  • 127. M. Sh. Birman, 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
  • 128. M. Sh. Birman, The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential, Algebra i Analiz 8 (1996), no. 1, 3–20 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 1, 1–14. MR 1392011
  • 129. M. 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, https://doi.org/10.1007/BF01193459
  • 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, https://doi.org/10.1002/(SICI)1097-0312(199609)49:9<967::AID-CPA3>3.3.CO;2-O
  • 131. M. Sh. Birman and T. A. Suslina, The two-dimensional periodic magnetic Hamiltonian is absolutely continuous, Algebra i Analiz 9 (1997), no. 1, 32–48 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 1, 21–32. MR 1458417
  • 132. Mikhail Sh. Birman, Ari Laptev, and Michael Solomyak, The negative discrete spectrum of the operator (-Δ)^{𝑙}-𝛼𝑉 in 𝐿₂(𝑅^{𝑑}) for 𝑑 even and 2𝑙≥𝑑, Ark. Mat. 35 (1997), no. 1, 87–126. MR 1443037, https://doi.org/10.1007/BF02559594
  • 133. Mikhail Sh. Birman and Ari 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, The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations, Algebra i Analiz 9 (1997), no. 6, 62–89 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1073–1095. MR 1610239
  • 135. M. Sh. Birman and A. B. Pushnitskiĭ, The discrete spectrum in the gaps of the perturbed pseudo-relativistic magnetic Hamiltonian, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 29, 102–117, 315 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 101 (2000), no. 5, 3437–3447. MR 1698515, https://doi.org/10.1007/BF02680144
  • 136. M. Sh. Birman and A. B. Pushnitski, Spectral shift function, amazing and multifaceted, Integral Equations Operator Theory 30 (1998), no. 2, 191–199. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). MR 1607900, https://doi.org/10.1007/BF01238218
  • 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, https://doi.org/10.1002/mana.19981950103
  • 138. M. Sh. Birman and T. A. Suslina, Absolute continuity of a two-dimensional periodic magnetic Hamiltonian with discontinuous vector potential, Algebra i Analiz 10 (1998), no. 4, 1–36 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 10 (1999), no. 4, 579–601. MR 1654063
  • 139. M. Sh. Birman and T. A. Suslina, 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
  • 140. M. Sh. Birman and T. A. Suslina, The periodic Dirac operator is absolutely continuous, Integral Equations Operator Theory 34 (1999), no. 4, 377–395. MR 1702229, https://doi.org/10.1007/BF01272881
  • 141. M. Sh. Birman and T. A. Suslina, A periodic magnetic Hamiltonian with a variable metric. The problem of absolute continuity, Algebra i Analiz 11 (1999), no. 2, 1–40 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 2, 203–232. MR 1702587
  • 142. M. Sh. Birman, T. A. Suslina, and R. G. Shterenberg, Absolute continuity of the two-dimensional Schrödinger operator with delta potential concentrated on a periodic system of curves, Algebra i Analiz 12 (2000), no. 6, 140–177 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 983–1012. MR 1816514
  • 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, The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap, Algebra i Analiz 12 (2000), no. 4, 36–78 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 4, 535–567. MR 1793617
  • 145. Michael Birman and Tatyana 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
  • 146. M. Sh. Birman and M. 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, https://doi.org/10.1007/BF02790267
  • 147. B. A. Amosov, M. S. Birman, M. I. Vishik et al., Mikhail Semenovich Agranovich (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 56 (2001), no. 4(340), 163–168 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 4, 777–784. MR 1861459, https://doi.org/10.1070/RM2001v056n04ABEH000434
  • 148. Michael Sh. Birman and Tat′yana 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, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 69–74. MR 1971990, https://doi.org/10.1007/978-1-4615-0701-7_4
  • 149. A. A. Arkhipova, M. S. 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), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32, 5–13 (Russian); English transl., J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4523–4526. MR 1923542, https://doi.org/10.1023/B:JOTH.0000041473.41796.59
  • 150. M. Sh. Birman, On the averaging procedure for periodic operators in a neighborhood of an edge of an internal gap, Algebra i Analiz 15 (2003), no. 4, 61–71 (Russian); English transl., St. Petersburg Math. J. 15 (2004), no. 4, 507–513. MR 2068979, https://doi.org/10.1090/S1061-0022-04-00819-2
  • 151. M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, https://doi.org/10.1090/S1061-0022-04-00827-1
  • 152. Mikhail Sh. Birman and Michael Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), no. 2, 131–168. MR 2002663, https://doi.org/10.1007/s00020-003-1157-8
  • 153. M. Sh. Birman and T. A. Suslina, Homogenization of a 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), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 36 [35], 60–74, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 136 (2006), no. 2, 3682–3690. MR 2120232, https://doi.org/10.1007/s10958-006-0192-9
  • 154. V. I. Arnol′d, M. Sh. Birman, A. M. Vershik et al., Ol′ga Aleksandrovna Ladyzhenskaya, Uspekhi Mat. Nauk 59 (2004), no. 3(357), 151–152 (Russian); English transl., Russian Math. Surveys 59 (2004), no. 3, 553–555. MR 2116538, https://doi.org/10.1070/RM2004v059n03ABEH000739
  • 155. M. Sh. Birman and T. A. Suslina, Threshold approximations for the resolvent of a factorized selfadjoint family taking a corrector into account, Algebra i Analiz 17 (2005), no. 5, 69–90 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 5, 745–762. MR 2241423, https://doi.org/10.1090/S1061-0022-06-00927-7
  • 156. M. Sh. Birman and T. A. Suslina, Averaging of periodic elliptic differential operators taking a corrector into account, Algebra i Analiz 17 (2005), no. 6, 1–104 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897–973. MR 2202045, https://doi.org/10.1090/S1061-0022-06-00935-6
  • 157. A. B. Alekseev, M. Sh. Birman, and N. D. Filonov, Asymptotics of the spectrum of a “nonsmooth” variational problem with a solvable constraint, Algebra i Analiz 18 (2006), no. 5, 1–22 (Russian); English transl., St. Petersburg Math. J. 18 (2007), no. 5, 681–697. MR 2301038, https://doi.org/10.1090/S1061-0022-07-00968-5
  • 158. M. Sh. Birman and T. A. Suslina, Averaging of periodic differential operators taking a corrector into account. Approximation of solutions in the Sobolev class 𝐻²(ℝ^{𝕕}), Algebra i Analiz 18 (2006), no. 6, 1–130 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 6, 857–955. MR 2307356, https://doi.org/10.1090/S1061-0022-07-00977-6
  • 159. M. Sh. Birman and T. A. Suslina, Averaging of a stationary periodic Maxwell system in the case of constant magnetic permeability, Funktsional. Anal. i Prilozhen. 41 (2007), no. 2, 3–23, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 41 (2007), no. 2, 81–98. MR 2345036, https://doi.org/10.1007/s10688-007-0009-8
  • 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, https://doi.org/10.1090/trans2/220/02
  • 161. M. Sh. Birman and T. A. Suslina, Operator error estimates for the averaging of nonstationary periodic equations, Algebra i Analiz 20 (2008), no. 6, 30–107 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 6, 873–928. MR 2530894, https://doi.org/10.1090/S1061-0022-09-01077-2
  • 162. M. Sh. Birman and T. A. Suslina, The limiting absorption principle and the homogenization procedure for periodic elliptic operators, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 105–108 (Russian); English transl., Funct. Anal. Appl. 42 (2008), no. 4, 336–339. MR 2492431, https://doi.org/10.1007/s10688-008-0047-x
  • 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 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 43 (2009), no. 3, 184–189. MR 2583637, https://doi.org/10.1007/s10688-009-0026-x
  • 165. M. Sh. Birman and V. A. Sloushch, 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, https://doi.org/10.1051/mmnp/20105402
  • 166. M. Solomyak and T. Suslina, On the scientific work of M. Sh. Birman in 1998–2007, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 1–15. MR 2509773, https://doi.org/10.1090/trans2/225/01

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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