On the mathematical work of Mikhail Zakharovich Solomyak
HTML articles powered by AMS MathViewer
- by
G. V. Rozenblyum
Translated by: S. V. Kislyakov - St. Petersburg Math. J. 30 (2019), 391-410
- DOI: https://doi.org/10.1090/spmj/1548
- Published electronically: April 12, 2019
- PDF | Request permission
References
- M. Z. Solomyak, On characteristic values and characteristic vectors of a perturbed operator, Doklady Akad. Nauk SSSR (N.S.) 90 (1953), 29â32 (Russian). MR 0056843
- M. Z. Solomjak, On orthogonal bases in Banach space, Vestnik Leningrad. Univ. 12 (1957), no. 1, 27â36, 208 (Russian, with English summary). MR 0085466
- M. Z. Solomjak, Application of semigroup theory to the study of differential equations in Banach spaces, Dokl. Akad. Nauk SSSR 122 (1958), 766â769 (Russian). MR 0105029
- M. Z. Solomjak, Analyticity of a semigroup generated by an elliptic operator in $L_{p}$ spaces, Dokl. Akad. Nauk SSSR 127 (1959), 37â39 (Russian). MR 0121664
- M. Z. Solomjak, On differential equations in Banach spaces, Izv. VysĆĄ. UÄebn. Zaved. Matematika 1960 (1960), no. 1 (14), 198â209 (Russian). MR 0130451
- M. Z. Solomjak, Evaluation of norm of the resolvent of elliptic operators in $L_{p}$-spaces, Uspehi Mat. Nauk 15 (1960), no. 6 (96), 141â148 (Russian). MR 0131064
- M. Z. Solomjak, Elliptic operators on two-dimensional manifolds, Dokl. Akad. Nauk SSSR 139 (1961), 37â39 (Russian). MR 0140821
- M. Z. Solomjak, On the spaces conjugate to the $W_{p}{}^{l}$ spaces of S. L. Sobolev, Dokl. Akad. Nauk SSSR 143 (1962), 1289â1292 (Russian). MR 0137994
- M. Z. Solomjak, On the solvability of certain integral identities, Leningrad. Gos. Ped. Inst. UÄen. Zap. 238 (1962), 141â148 (Russian). MR 0170098
- M. Z. Solomjak, On linear elliptic systems of first order, Dokl. Akad. Nauk SSSR 150 (1963), 48â51 (Russian). MR 0176216
- M. Z. Solomyak, On local properties of factional powers if elliptic operators, Uspekhi Mat. Nauk 18 (1963), no. 2, 184â185. (Russian)
- M. Ć . Birman and M. Z. Solomjak, On double Stieltjes operator integrals, Dokl. Akad. Nauk SSSR 165 (1965), 1223â1226 (Russian). MR 0188802
- M. Z. Solomyak, On LopatinskiÄ condition for the solvability of boundary problems, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1965 vyp. 1, 143â149. (Russian)
- M. Ć . Birman and M. Z. Solomjak, Approximation of functions of the $W_{p}{}^{\alpha }$-classes by piece-wise-polynomial functions, Dokl. Akad. Nauk SSSR 171 (1966), 1015â1018 (Russian). MR 0209733
- M. Ć . Birman and M. Z. Solomjak, Double Stieltjes operator integrals and problems on multipliers, Dokl. Akad. Nauk SSSR 171 (1966), 1251â1254 (Russian). MR 0209871
- M. Ć . Birman and M. Z. Solomjak, Double Stieltjes operator integrals and problems on multipliers, Dokl. Akad. Nauk SSSR 171 (1966), 1251â1254 (Russian). MR 0209871
- M. Ć . Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. I, Vestnik Leningrad. Univ. 22 (1967), no. 7, 43â53 (Russian, with English summary). MR 0216339
- M. Ć . Birman and M. Z. Solomjak, Piecewise polynomial approximations of functions of classes $W_{p}{}^{\alpha }$, Mat. Sb. (N.S.) 73 (115) (1967), 331â355 (Russian). MR 0217487
- M. Ć . Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. II, Vestnik Leningrad. Univ. 22 (1967), no. 13, 21â28 (Russian, with English summary). MR 0217650
- M. Z. Solomjak and V. M. Tihomirov, Geometric characteristics of the imbedding of the classes $W_{p}{}^{\alpha }$ in $C$, Izv. VysĆĄ. UÄebn. Zaved. Matematika 1967 (1967), no. 10 (65), 76â82 (Russian). MR 0218888
- C. E. Rickart, Holomorphic convexity for general function algebras, Canadian J. Math. 20 (1968), 272â290. MR 227778, DOI 10.4153/CJM-1968-027-2
- M. Ć . 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. 26â60 (Russian). MR 0234304
- M. Z. Solomjak, The approximation of double Stieltjes operator integrals by intergral sums, Vestnik Leningrad. Univ. 23 (1968), no. 7, 148â149 (Russian, with English summary). MR 0232240
- M. Z. Solomjak, Double operator integrals in the ring $\hat R$, Problems of mathematical physics, No. 3: Spectral theory (Russian), Izdat. Leningrad. Univ., Leningrad, 1968, pp. 88â102 (Russian). MR 0350484
- M. Ć . Birman and M. Z. Solomjak, Estimates of singular numbers of integral operators. III. Operators in unbounded domains, Vestnik Leningrad. Univ. 24 (1969), no. 1, 35â48 (Russian, with English summary). MR 0246163
- M. Ć . Birman and M. Z. Solomjak, Remarks on the nuclearity of integral operators and the boundedness of pseudodifferential operators, Izv. VysĆĄ. UÄebn. Zaved. Matematika 1969 (1969), no. 9 (88), 11â17 (Russian). MR 0254670
- M. Z. Solomjak and V. V. StenâČkin, A certain class of multiple operator Stieltjes integrals, Problems of Math. Anal., No. 2: Linear Operators and Operator Equations (Russian), Izdat. Leningrad. Univ., Leningrad, 1969, pp. 122â134 (Russian). MR 0438161
- M. Z. Solomjak, Estimates of the singular numbers of integral operators. IV, Vestnik Leningrad. Univ. 25 (1970), no. 1, 76â87 (Russian, with English summary). MR 0264473
- M. Ć . Birman and M. Z. Solomjak, The principal term of the spectral asymptotics for ânon-smoothâ elliptic problems, Funkcional. Anal. i PriloĆŸen. 4 (1970), no. 4, 1â13 (Russian). MR 0278126
- M. Ć . Birman and M. Z. Solomjak, Asymptotics of the spectrum of weakly polar integral operators, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1142â1158 (Russian). MR 0279638
- M. Ć . 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), 16â25 (Russian). MR 0284851
- M. Ć . Birman and M. Z. Solomjak, The asymptotics of the spectrum of ânonsmoothâ elliptic equations. , Funkcional. Anal. i PriloĆŸen. 5 (1971), no. 1, 69â70 (Russian). MR 0283428
- G. P. Kostometov and M. Z. Solomjak, Extimates of the singular values of integral operators with a weak singularity, Vestnik Leningrad. Univ. 1971 (1971), no. 1, 28â39 (Russian, with English summary). MR 0288632
- A. A. Kruglov and M. Z. Solomjak, Interpolation of operators in the spaces $V_{p}$, Vestnik Leningrad. Univ. 13 (1971), 54â60 (Russian, with English summary). MR 0293387
- V. M. BabiÄ, M. Ć . Birman, V. I. Smirnov, M. Z. Solomjak, and N. N. UralâČceva, OlâČga Aleksandrovna LadyĆŸenskaja (on her fiftieth birthday), Vestnik Leningrad. Univ. 7 (1972), 159â160 (Russian). MR 0297506
- M. Ć . Birman and M. Z. Solomjak, Spectral asymptotic behavior of nonsmooth elliptic operators, Dokl. Akad. Nauk SSSR 205 (1972), 267â270 (Russian). MR 0305148
- Pierre-Philippe Dechant, CĂ©line BĆhm, and Reidun Twarock, Affine extensions of non-crystallographic Coxeter groups induced by projection, J. Math. Phys. 54 (2013), no. 9, 093508, 21. MR 3136651, DOI 10.1063/1.4820441
- M. Ć . Birman and M. Z. Solomjak, Remarks on the spectral shift function, Zap. NauÄn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 33â46 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR 0315482
- M. Ć . Birman and M. Z. Solomjak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Trudy Moskov. Mat. ObĆĄÄ. 27 (1972), 3â52; ibid. 28 (1973), 3â34 (Russian). MR 0364898
- V. I. Macaev and M. Z. Solomjak, Existence conditions for the Stieltjes integral, Mat. Sb. (N.S.) 88(130) (1972), 522â535 (Russian). MR 0507422
- I. L. Vulis and M. Z. Solomjak, Spectral asymptotic analysis of the degenerate Steklov problem, Vestnik Leningrad. Univ. 19 Mat. Meh. Astronom Vyp. 4 (1973), 148â150, 156 (Russian, with English summary). MR 0330795
- M. Ć . Birman and M. Z. Solomjak, Equivalent renormings of $W^{r}_{p}$ classes in the case of partitioning of the domain, Izv. VysĆĄ. UÄebn. Zaved. Matematika 3(130) (1973), 19â27 (Russian). MR 0344875
- M. Ć . Birman and M. Z. Solomjak, Double Stieltjes operator integrals. III, Problems of mathematical physics, No. 6 (Russian), Izdat. Leningrad. Univ., Leningrad, 1973, pp. 27â53 (Russian). MR 0348494
- I. L. Vulis and M. Z. Solomjak, Spectral asymptotic analysis for degenerate second order elliptic operators, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1362â1392 (Russian). MR 0358081
- M. Ć . Birman and M. Z. Solomjak, Quantitative analysis in Sobolevâs 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. 5â189 (Russian). MR 0482138
- M. Ć . Birman and M. Z. Solomjak, A certain âmodelâ nonelliptic spectral problem, Vestnik Leningrad. Univ. 1 Mat. Meh. Astronom. vyp. 1 (1975), 39â45, 145 (Russian, with English summary). Collection of articles dedicated to the memory of Academician V. I. Smirnov. MR 0380148
- M. Ć . Birman, L. S. Koplienko, and M. Z. Solomjak, Estimates of the spectrum of a difference of fractional powers of selfadjoint operators, Izv. VysĆĄ. UÄebn. Zaved. Matematika 3(154) (1975), 3â10 (Russian). MR 0385597
- M. Ć . Birman and M. Z. Solomjak, Estimates for the singular numbers of integral operators, Uspehi Mat. Nauk 32 (1977), no. 1(193), 17â84, 271 (Russian). MR 0438186
- M. Ć . 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), 13â21, 169 (Russian, with English summary). MR 0463738
- M. Ć . Birman and M. Z. Solomjak, Asymptotic properties of the spectrum of differential equations, Mathematical analysis, Vol. 14 (Russian), Akad. Nauk SSSR Vsesojuz. Inst. NauÄn. i Tehn. Informacii, Moscow, 1977, pp. 5â58, i. (loose errata) (Russian). MR 0467022
- William Wernick, An enumeration of logical functions, Bull. Amer. Math. Soc. 45 (1939), 885â887. MR 806, DOI 10.1090/S0002-9904-1939-07109-7
- M. Sh. Birman and M. Z. Solomyak, On asymptotic behavior of the spectrum of elliptic variational problems with constraints, Proc. AllâUnion Conf. on Partial Differential Equations Dedicated to Acad. Petrovskiy 75th Birthday, Moskov. Univ., Moscow, 1978, pp. 51â52. (Russian)
- M. Ć . Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Sibirsk. Mat. Zh. 20 (1979), no. 1, 3â22, 204 (Russian). MR 523133
- M. Ć . Birman, A. M. VerĆĄik, and M. Z. Solomjak, The product of commuting spectral measures may fail to be countably additive, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 61â62 (Russian). MR 527523
- M. Ć . Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols. II, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1979), 5â10, 121 (Russian, with English summary). MR 555971
- M. Ć . Birman and M. Z. Solomjak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, American Mathematical Society Translations, Series 2, vol. 114, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by F. A. Cezus. MR 562305
- SaÄon Ulare and M. Z. Solomjak, Spaces dual to the spaces $\Lambda _{\Omega }$, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1980), 113â115, 122 (Russian, with English summary). MR 573411
- M. Ć . Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations in unbounded domains, Funktsional. Anal. i Prilozhen. 14 (1980), no. 4, 27â35, 95 (Russian). MR 595726
- M. Ć . Birman and M. Z. Solomjak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Partial differential equations (Proc. Conf., Novosibirsk, 1978) âNaukaâ Sibirsk. Otdel., Novosibirsk, 1980, pp. 221â223, 255 (Russian). MR 601449
- M. Ć . Birman and M. Z. Solomjak, SpektralâČnaya teoriya samosopryazhennykh operatorov v gilâČbertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
- N. A. Karazeeva and M. Z. Solomyak, The asymptotic behavior of the spectrum of a contact problem for second-order elliptic equations, Partial differential equations. Spectral theory, Probl. Mat. Anal., vol. 8, Leningrad. Univ., Leningrad, 1981, pp. 36â48, 221â222 (Russian). MR 658151
- M. Sh. Birman and M. Z. Solomyak, On subspaces that admit a pseudodifferential projector, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1982), 18â25, 133 (Russian, with English summary). MR 652058
- 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), 23â39, 305 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 14. MR 660069
- A. I. KarolâČ and M. Z. Solomyak, A class of differential operators with nonstandard behavior of the spectrum of the Dirichlet problem, Mat. Zametki 31 (1982), no. 5, 747â751, 815 (Russian). MR 660580
- M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential variational problems with contraints, Spectral theory. Wave processes, Probl. Mat. Fiz., vol. 10, Leningrad. Univ., Leningrad, 1982, pp. 20â36, 296 (Russian). MR 691142
- M. Sh. Birman and M. Z. Solomyak, Compact operators with power asymptotic behavior of the singular numbers, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 21â30 (Russian, with English summary). Investigations on linear operators and the theory of functions, XII. MR 697420
- M. Z. Solomyak, The CalderĂłn projection, Operator theory and function theory, No. 1, Leningrad. Univ., Leningrad, 1983, pp. 47â55 (Russian). MR 768776
- M. Z. Solomyak, Asymptotic behavior of the spectrum of a Schrödinger operator with nonregular homogeneous potential, Dokl. Akad. Nauk SSSR 278 (1984), no. 2, 291â295 (Russian). MR 762882
- V. P. Havin, S. V. HruĆĄÄĂ«v, and N. K. NikolâČskiÄ (eds.), Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR 734178, DOI 10.1007/BFb0072183
- M. Z. Solomyak, Asymptotic behavior of the spectrum of the Schrödinger operator with nonregular homogeneous potential, Mat. Sb. (N.S.) 127(169) (1985), no. 1, 21â39, 142 (Russian). MR 791315
- M. Sh. Birman and M. Z. Solomyak, The Maxwell operator in domains with a nonsmooth boundary, Sibirsk. Mat. Zh. 28 (1987), no. 1, i, 23â36 (Russian). MR 886850
- M. Sh. Birman and M. Z. Solomyak, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1987), 23â28, 127 (Russian, with English summary). MR 928156
- M. Sh. Birman and M. Z. Solomyak, $L_2$-theory of the Maxwell operator in arbitrary domains, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 61â76, 247 (Russian). MR 933995
- M. Sh. Birman and M. Z. Solomyak, 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), no. Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. FunktsiÄ 19, 17â28, 186 (Russian, with English summary); English transl., J. Soviet Math. 49 (1990), no. 5, 1128â1136. MR 918938, DOI 10.1007/BF02208708
- M. Sh. Birman and M. Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782
- V. S. Buslaev, I. M. GelâČfand, O. A. Ladyzhenskaya, B. S. Pavlov, M. Z. Solomyak, L. D. Faddeev, and D. R. Yafaev, Mikhail ShlĂ«movich Birman (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 43 (1988), no. 3(261), 201â202 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 3, 233â235. MR 955795, DOI 10.1070/RM1988v043n03ABEH001768
- Yu. K. DemâČyanovich, V. P. IlâČin, A. I. Koshelev, M. Z. Solomyak, and N. N. UralâČtseva, Solomon GrigorâČevich Mikhlin (on the occasion of his eightieth birthday), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 2 (1988), 128â129 (Russian). MR 965115
- S. A. Vinogradov, D. A. Vladimirov, M. K. Gavurin, S. M. Ermakov, B. M. Makarov, G. I. Natanson, M. Z. Solomyak, D. K. Faddeev, and V. P. Khavin, GrigoriÄ MikhaÄlovich FikhtengolâČts (on the 100th anniversary of his birth), Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1988), 3â6 (Russian). MR 974784
- R. V. Gamkrelidze (ed.), Sovremennye problemy matematiki. FundamentalâČnye napravleniya. Tom 64, Itogi Nauki i Tekhniki. [Progress in Science and Technology], Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 (Russian). DifferentsialâČnye uravneniya s chastnymi proizvodnymi. 7. [Partial differential equations. 7]. MR 1033499
- 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. 2â18. MR 1091987
- M. Sh. Birman and M. Z. Solomjak, The $L_2$-theory of the Maxwell operator in domains with nonsmooth boundary, Symposium âPartial Differential Equationsâ (Holzhau, 1988) Teubner-Texte Math., vol. 112, Teubner, Leipzig, 1989, pp. 33â41. MR 1105797
- M. Sh. Birman and M. Z. Solomyak, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96â110 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 99â115. MR 1015335
- 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. LineÄn. Oper. Teorii FunktsiÄ. 17, 176â183, 323 (Russian, with English summary); English transl., J. Soviet Math. 63 (1993), no. 2, 212â216. MR 1039579, DOI 10.1007/BF01099312
- 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. LineÄn. Oper. Teorii FunktsiÄ. 18, 120â145, 185 (Russian, with English summary); English transl., J. Soviet Math. 61 (1992), no. 2, 2018â2035. MR 1037767, DOI 10.1007/BF01095665
- 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, BirkhĂ€user, Basel, 1990, pp. 3â16. MR 1124648
- A. D. Aleksandrov, S. S. Kutateladze, Yu. G. Reshetnyak, G. Sh. RubinshteÄn, and M. Z. Solomyak, Viktor Abramovich Zalgaller (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 46 (1991), no. 1(277), 215â216 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 1, 257â259. MR 1109046, DOI 10.1070/RM1991v046n01ABEH002729
- 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. 49â57. MR 1117952
- 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. 25â36. MR 1243027, DOI 10.1117/12.24776
- 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, 1989â90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1â55. MR 1306507
- 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, 1989â90) Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85â106. MR 1306510
- 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. 1â54. MR 1157648, DOI 10.1090/trans2/150/01
- 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, 143â159 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 125â139. MR 1220492
- Michael Solomyak, A remark on the Hardy inequalities, Integral Equations Operator Theory 19 (1994), no. 1, 120â124. MR 1271241, DOI 10.1007/BF01202293
- M. Solomyak, Piecewise-polynomial approximation of functions from $H^l((0,1)^d)$, $2l=d$, and applications to the spectral theory of the Schrödinger operator, Israel J. Math. 86 (1994), no. 1-3, 253â275. MR 1276138, DOI 10.1007/BF02773681
- V. Olevskii and M. Solomyak, An estimate for Schur multipliers in $S_p$-classes, Linear Algebra Appl. 208/209 (1994), 57â64. MR 1287339, DOI 10.1016/0024-3795(94)90430-8
- Michael Solomyak, Eigenvalue estimates for a class of operators related to self-similar measures, JournĂ©es âĂquations aux DĂ©rivĂ©es Partiellesâ (Saint-Jean-de-Monts, 1994) Ăcole Polytech., Palaiseau, 1994, pp. Exp. No. XI, 6. MR 1298682
- Joshua Newman and Michael Solomyak, Two-sided estimates on singular values for a class of integral operators on the semi-axis, Integral Equations Operator Theory 20 (1994), no. 3, 335â349. MR 1299892, DOI 10.1007/BF01205286
- Katherine Naimark and Michael Solomyak, On the eigenvalue behaviour for a class of operators related to self-similar measures on $\textbf {R}^d$, C. R. Acad. Sci. Paris SĂ©r. I Math. 319 (1994), no. 8, 837â842 (English, with English and French summaries). MR 1300953
- Michael Solomyak, Piecewise-polynomial approximation of functions from Sobolev spaces, revisited, A tribute to Ilya Bakelman (College Station, TX, 1993) Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 7â27. MR 1423362
- Michael Solomyak, My recollections of Ilya Bakelman, A tribute to Ilya Bakelman (College Station, TX, 1993) Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 28â30. MR 1423363
- V. I. ArnolâČd, M. Sh. Birman, I. M. GelâČfand et al., AnatoliÄ Moiseevich Vershik (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 49 (1994), no. 3(297), 195â204 (Russian); English transl., Russian Math. Surveys 49 (1994), no. 3, 207â221. MR 1289402, DOI 10.1070/RM1994v049n03ABEH002272
- M. Solomyak, Spectral problems related to the critical exponent in the Sobolev embedding theorem, Proc. London Math. Soc. (3) 71 (1995), no. 1, 53â75. MR 1327933, DOI 10.1112/plms/s3-71.1.53
- M. Solomyak and E. Verbitsky, On a spectral problem related to self-similar measures, Bull. London Math. Soc. 27 (1995), no. 3, 242â248. MR 1328700, DOI 10.1112/blms/27.3.242
- K. Naimark and M. Solomyak, The eigenvalue behaviour for the boundary value problems related to self-similar measures on $\textbf {R}^d$, Math. Res. Lett. 2 (1995), no. 3, 279â298. MR 1338787, DOI 10.4310/MRL.1995.v2.n3.a5
- 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, 179â187. MR 1371945, DOI 10.1007/BF01193459
- Mikhail Sh. Birman, Ari Laptev, and Michael Solomyak, The negative discrete spectrum of the operator $(-\Delta )^l-\alpha V$ in $L_2(\textbf {R}^d)$ for $d$ even and $2l\geq d$, Ark. Mat. 35 (1997), no. 1, 87â126. MR 1443037, DOI 10.1007/BF02559594
- Daniel Levin and Michael Solomyak, The Rozenblum-Lieb-Cwikel inequality for Markov generators, J. Anal. Math. 71 (1997), 173â193. MR 1454250, DOI 10.1007/BF02788029
- Michael Solomyak, On the negative discrete spectrum of the operator $-\Delta _{\scr N}-\alpha V$ for a class of unbounded domains in $\textbf {R}^d$, Partial differential equations and their applications (Toronto, ON, 1995) CRM Proc. Lecture Notes, vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 283â296. MR 1479254, DOI 10.1090/crmp/012/22
- G. Rozenblum and M. Solomyak, CLR-estimate revisited: Liebâs approach with no path integrals, JournĂ©es âĂquations aux DĂ©rivĂ©es Partiellesâ (Saint-Jean-de-Monts, 1997) Ăcole Polytech., Palaiseau, 1997, pp. Exp. No. XVI, 10. MR 1482282
- K. Naimark and M. Solomyak, Regular and pathological eigenvalue behavior for the equation $-\lambda u''=Vu$ on the semiaxis, J. Funct. Anal. 151 (1997), no. 2, 504â530. MR 1491550, DOI 10.1006/jfan.1997.3149
- G. Rozenblyum and M. Solomyak, The Cwikel-Lieb-Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, Algebra i Analiz 9 (1997), no. 6, 214â236 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1195â1211. MR 1610184
- Grigori Rozenblum and Michael Solomyak, On principal eigenvalues for indefinite problems in Euclidean space, Math. Nachr. 192 (1998), 205â223. MR 1626340, DOI 10.1002/mana.19981920112
- M. Sh. Birman, A. Laptev, and M. Solomyak, On the eigenvalue behaviour for a class of differential operators on semiaxis, Math. Nachr. 195 (1998), 17â46. MR 1654744, DOI 10.1002/mana.19981950103
- Michael Solomyak, On the discrete spectrum of a class of problems involving the Neumann Laplacian in unbounded domains, Voronezh Winter Mathematical Schools, Amer. Math. Soc. Transl. Ser. 2, vol. 184, Amer. Math. Soc., Providence, RI, 1998, pp. 233â251. MR 1729937, DOI 10.1090/trans2/184/16
- V. Buslaev, M. Solomyak, and D. Yafaev, On the scientific work of Mikhail ShlĂ«movich Birman, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 1â15. MR 1730499, DOI 10.1090/trans2/189/01
- G. Rozenblum and M. Solomyak, On the number of negative eigenvalues for the two-dimensional magnetic Schrödinger operator, Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Amer. Math. Soc., Providence, RI, 1999, pp. 205â217. MR 1730514, DOI 10.1090/trans2/189/16
- K. Naimark and M. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3) 80 (2000), no. 3, 690â724. MR 1744781, DOI 10.1112/S0024611500012272
- Michael Solomyak, Estimates for the approximation numbers of the weighted Riemann-Liouville operator in the spaces $L_p$, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., vol. 113, BirkhĂ€user, Basel, 2000, pp. 371â383. MR 1771775
- V. S. Buslaev, I. M. GelâČfand, O. A. Ladyzhenskaya, B. S. Pavlov, M. Z. Solomyak, L. D. Faddeev, and D. R. Yafaev, Mikhail ShlĂ«movich Birman (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 43 (1988), no. 3(261), 201â202 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 3, 233â235. MR 955795, DOI 10.1070/RM1988v043n03ABEH001768
- M. Z. Solomyak, A few words about Viktor Abramovich Zalgaller, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), no. Geom. i Topol. 7, 22â27, 299 (Russian); English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 138â140. MR 1879255, DOI 10.1023/B:JOTH.0000008753.25951.2c
- 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), 337â391. MR 1828497, DOI 10.1007/BF02790267
- K. Naimark and M. Solomyak, Eigenvalue distribution of some fractal semi-elliptic differential operators: combinatorial approach, Integral Equations Operator Theory 40 (2001), no. 4, 495â506. MR 1839473, DOI 10.1007/BF01198143
- K. Naimark and M. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russ. J. Math. Phys. 8 (2001), no. 3, 322â335. MR 1930378
- Alexander V. Sobolev and Michael Solomyak, Schrödinger operators on homogeneous metric trees: spectrum in gaps, Rev. Math. Phys. 14 (2002), no. 5, 421â467. MR 1912093, DOI 10.1142/S0129055X02001235
- Michael Solomyak, On eigenvalue estimates for the weighted Laplacian on metric graphs, Nonlinear problems in mathematical physics and related topics, I, Int. Math. Ser. (N. Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 327â347. MR 1970620, DOI 10.1007/978-1-4615-0777-2_{2}0
- Michael Solomyak, On approximation of functions from Sobolev spaces on metric graphs, J. Approx. Theory 121 (2003), no. 2, 199â219. MR 1971768, DOI 10.1016/S0021-9045(03)00033-9
- Michael Solomyak, Laplace and Schrödinger operators on regular metric trees: the discrete spectrum case, Function spaces, differential operators and nonlinear analysis (Teistungen, 2001) BirkhĂ€user, Basel, 2003, pp. 161â181. MR 1984167
- Mikhail Sh. Birman and Michael Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), no. 2, 131â168. MR 2002663, DOI 10.1007/s00020-003-1157-8
- Michael Solomyak, On a differential operator appearing in the theory of irreversible quantum graphs, Waves Random Media 14 (2004), no. 1, S173âS185. Special section on quantum graphs. MR 2042551, DOI 10.1088/0959-7174/14/1/018
- Michael Solomyak, On the spectrum of the Laplacian on regular metric trees, Waves Random Media 14 (2004), no. 1, S155âS171. Special section on quantum graphs. MR 2046943, DOI 10.1088/0959-7174/14/1/017
- M. Z. Solomyak, The discrete spectrum of a family of differential operators, Funktsional. Anal. i Prilozhen. 38 (2004), no. 3, 70â78 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 3, 217â223. MR 2095135, DOI 10.1023/B:FAIA.0000042806.71352.1b
- V. S. Buslaev, M. Z. Solomyak, and D. R. Yafaev, Mikhail Shlemovich Birman (on the occasion of his 75th birthday), Algebra i Analiz 16 (2004), no. 1, 5â14 (Russian); English transl., St. Petersburg Math. J. 16 (2005), no. 1, 1â8. MR 2068350, DOI 10.1090/S1061-0022-04-00841-6
- V. I. ArnolâČd, M. Sh. Birman, A. M. Vershik et al., OlâČga Aleksandrovna Ladyzhenskaya, Uspekhi Mat. Nauk 59 (2004), no. 3(357), 151â152 (Russian); English transl., Russian Math. Surveys 59 (2004), no. 3, 553â555. MR 2116538, DOI 10.1070/RM2004v059n03ABEH000739
- W. D. Evans and M. Solomyak, Smilanskyâs model of irreversible quantum graphs. I. The absolutely continuous spectrum, J. Phys. A 38 (2005), no. 21, 4611â4627. MR 2147079, DOI 10.1088/0305-4470/38/21/007
- W. D. Evans and M. Solomyak, Smilanskyâs model of irreversible quantum graphs. II. The point spectrum, J. Phys. A 38 (2005), no. 35, 7661â7675. MR 2169482, DOI 10.1088/0305-4470/38/35/004
- Sergey N. Naboko and Michael Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph, Proc. London Math. Soc. (3) 92 (2006), no. 1, 251â272. MR 2192392, DOI 10.1017/S0024611505015522
- Peter R. Massopust, Multiwavelets: some approximation-theoretic properties, sampling on the interval, and translation invariance, Harmonic, wavelet and $p$-adic analysis, World Sci. Publ., Hackensack, NJ, 2007, pp. 37â57. MR 2441428, DOI 10.1142/9789812770707_{0}003
- Michael Solomyak, On the limiting behaviour of the spectra of a family of differential operators, J. Phys. A 39 (2006), no. 33, 10477â10489. MR 2256605, DOI 10.1088/0305-4470/39/33/015
- Uzy Smilansky and Michael Solomyak, The quantum graph as a limit of a network of physical wires, Quantum graphs and their applications, Contemp. Math., vol. 415, Amer. Math. Soc., Providence, RI, 2006, pp. 283â291. MR 2277623, DOI 10.1090/conm/415/07875
- J. A. Robinson, Heuristic and complete processes in the mechanization of theorem proving, Systems and Computer Science (Proc. Conf., London, Ont., 1965) Univ. Toronto Press, Toronto, Ont., 1967, pp. 116â124. MR 0234776
- L. Friedlander and M. Solomyak, On the spectrum of narrow periodic waveguides, Russ. J. Math. Phys. 15 (2008), no. 2, 238â242. MR 2410832, DOI 10.1134/S1061920808020076
- M. Solomyak and T. Suslina, On the scientific work of M. Sh. Birman in 1998â2007, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 1â15. MR 2509773, DOI 10.1090/trans2/225/01
- Leonid Friedlander and Michael Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 103â116. MR 2509778, DOI 10.1090/trans2/225/07
- Grigori Rozenblum and Michael Solomyak, Counting Schrödinger boundstates: semiclassics and beyond, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 329â353. MR 2484631, DOI 10.1007/978-0-387-85650-6_{1}4
- Michael Solomyak, Remarks on counting negative eigenvalues of the Schrödinger operator on regular metric trees, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 165â178. MR 2500518, DOI 10.1090/trans2/226/14
- Leonid Friedlander and Michael Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math. 170 (2009), 337â354. MR 2506330, DOI 10.1007/s11856-009-0032-y
- Grigori Rozenblum and Michael Solomyak, On the spectral estimates for the Schrödinger operator on $\Bbb Z^d,\ d\ge 3$, J. Math. Sci. (N.Y.) 159 (2009), no. 2, 241â263. Problems in mathematical analysis. No. 41. MR 2544038, DOI 10.1007/s10958-009-9436-9
- M. Solomyak, T. Suslina, and D. Yafaev, Obituary: Mikhail Birman, 17.01.1928â2.07.2009, Math. Model. Nat. Phenom. 5 (2010), no. 4, 1â3. MR 2662447, DOI 10.1051/mmnp/20105400
- G. V. Rozenblyum and M. Z. Solomyak, On spectral estimates for Schrödinger-type operators: the case of small local dimension, Funktsional. Anal. i Prilozhen. 44 (2010), no. 4, 21â33 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 44 (2010), no. 4, 259â269. MR 2768562, DOI 10.1007/s10688-010-0037-7
- V. M. Babich, V. S. Buslaev, A. M. Vershik et al., Mikhail Shlemovich Birman, Uspekhi Mat. Nauk 65 (2010), no. 3(393), 185â190 (Russian); English transl., Russian Math. Surveys 65 (2010), no. 3, 569â575. MR 2682722, DOI 10.1070/RM2010v065n03ABEH004682
- M. Ć . Birman and M. Z. Solomjak, SpektralâČnaya teoriya samosopryazhennykh operatorov v gilâČbertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148
- G. Rozenblum and M. Solomyak, Spectral estimates for Schrödinger operators with sparse potentials on graphs, J. Math. Sci. (N.Y.) 176 (2011), no. 3, 458â474. Problems in mathematical analysis. No. 57. MR 2839050, DOI 10.1007/s10958-011-0401-z
- M. Z. Solomyak, T. A. Suslina, and D. R. Yafaev, On the mathematical works of M. Sh. Birman, Algebra i Analiz 23 (2011), no. 1, 5â60 (Russian); English transl., St. Petersburg Math. J. 23 (2012), no. 1, 1â38. MR 2760146, DOI 10.1090/S1061-0022-2011-01184-7
- A. Laptev and M. Solomyak, On the negative spectrum of the two-dimensional Schrödinger operator with radial potential, Comm. Math. Phys. 314 (2012), no. 1, 229â241. MR 2954515, DOI 10.1007/s00220-012-1501-4
- Michael Solomyak, On a class of spectral problems on the half-line and their applications to multi-dimensional problems, J. Spectr. Theory 3 (2013), no. 2, 215â235. MR 3042765, DOI 10.4171/JST/43
- Ari Laptev and Michael Solomyak, On spectral estimates for two-dimensional Schrödinger operators, J. Spectr. Theory 3 (2013), no. 4, 505â515. MR 3122220, DOI 10.4171/JST/53
- G. Rozenblum and M. Solomyak, On spectral estimates for the Schrödinger operators in global dimension 2, Algebra i Analiz 25 (2013), no. 3, 185â199; English transl., St. Petersburg Math. J. 25 (2014), no. 3, 495â505. MR 3184603, DOI 10.1090/S1061-0022-2014-01301-5
- Vitaly Moroz, Michael Solomyak, and Aubrey Truman, Vitali Liskevich, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 573â574. MR 3200325, DOI 10.1017/S0013091514000029
Bibliographic Information
- G. V. Rozenblyum
- Affiliation: Department of Mathematics, Chalmers University of Technology and The University of Gothenburg, S-412, 96, Gothenburg, Sweden
- MR Author ID: 209425
- Email: grigori@math.chalmers.se
- Received by editor(s): March 12, 2018
- Published electronically: April 12, 2019
- © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 30 (2019), 391-410
- MSC (2010): Primary 47-02; Secondary 35-02
- DOI: https://doi.org/10.1090/spmj/1548
- MathSciNet review: 3811996