01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
24.12.1932
Keywords:
linear differential operators; spectral theory; scattering; direct and inverse problems; finite and infinite systems of differential equations; eigenfunction expansions; spectral analysis; theory of extensions; deficiency indices; essential self-adjointness; oscillatory theory; the perturbations of spectra of the periodic or almost periodic operators.
Subject:
Direct and inverse (by a generalized V. A. Marchenko type spectral matrix) problems of spectral analysis for non-selfadjoint operator differnetial equations of second order were solved. An effective method of binary relations has been worked out and introduced to the theory of extensions of operators. The inverse Sturm–Liouville problem on the whole axis was solved. Perturbations of spectra for periodic and almost periodic operaotrs were investigated; Kneser type constants (critical constants) for each spectral gap were introduced. An expansion in eigenfunctions and a description of selfadjoint extensions for infinite systems of differential equations of arbitrary order were obtained. Also, for such systems an oscillation theory was produced (jointly with A. M. Kholkin). A broad class of conditions for essential selfadjointness for elliptic operators was obtained (jointly with A. G. Brusentsev and H. Kalf). A scattering theory on high singular potentials was worked out (jointly with E. H. Hristov). Inverse scattering problems for some classes of non-selfadjoint systems were solved (jointly with E. I. Bondarenko). There are some works on resolvent convergence and a generalization of the phase functions method.
Biography
Graduated from Faculty of of Physics and Mathematics of Kharkov State University in 1955 (department of Mathematical Physics), and then the prost-graduate course there in 1958. Ph.D. thesis was defended in 1962. D.Sci. thesis was defended in 1987. The list of my works contains more than 80 titles. 5 Ph.D. thesis were defended under my supervision. One of my former post-graduate students became a doctor of sciences. Since 1966 I periodically work (part time) at Kharkov State Univbersity.
Main publications:
Rofe–Beketov F. S. Deficiency indices and properties of spectrum of some classes of differential operators // Lecture Notes in Math., (Spectral theory and differential equations), 1975, 448, 273–293.
Rofe–Beketov F. S. The inverse Sturm–Liouville problem for the spectral matrix on the whole axis and associated problems // Pitman Research Notes in Math. Series 235. Longman Sci. and Techn. (Integral Equations and Inverse Problems, Ed. V. Petkov and R. Lazarov). 1991, 234–238.
A. M. Kholkin, F. S. Rofe-Beketov, “On Spectrum of Differential Operator with Block-Triangular Matrix Coefficients”, Zh. Mat. Fiz. Anal. Geom., 10:1 (2014), 44–63
E. I. Zubkova, F. S. Rofe-Beketov, “Necessary and sufficient conditions in inverse scattering problem on the axis for the triangular $2\times 2$ matrix potential”, Zh. Mat. Fiz. Anal. Geom., 5:3 (2009), 296–309
E. I. Zubkova, F. S. Rofe-Beketov, “Inverse scattering problem on the axis for the Schrödinger operator with triangular $2\times 2$ matrix potential. II. Addition of the discrete spectrum”, Zh. Mat. Fiz. Anal. Geom., 3:2 (2007), 176–195
E. I. Zubkova, F. S. Rofe-Beketov, “Inverse scattering problem on the axis for the Schrödinger operator with triangular $2\times 2$ matrix potential. I. Main theorem”, Zh. Mat. Fiz. Anal. Geom., 3:1 (2007), 47–60
E. I. Bondarenko, F. S. Rofe-Beketov, “Inverse scattering problem on the semiaxis for the system with the triangle matrix potential”, Mat. Fiz. Anal. Geom., 10:3 (2003), 412–424
F. S. Rofe-Beketov, A. M. Khol'kin, “A fundamental system of solutions for an operator differential equation with a boundary condition at infinity”, Mat. Zametki, 36:5 (1984), 697–709; Math. Notes, 36:5 (1984), 846–853
F. S. Rofe-Beketov, A. M. Khol'kin, “The connection between spectral and oscillatory properties of differential systems of arbitrary order”, Dokl. Akad. Nauk SSSR, 261:3 (1981), 551–555
F. S. Rofe-Beketov, “Perturbations and Friedrichs extensions of semibounded operators on variable domains”, Dokl. Akad. Nauk SSSR, 255:5 (1980), 1054–1058
F. S. Rofe-Beketov, A. M. Khol'kin, “On the connection between spectral and oscillation properties of the Sturm–Liouville matrix problem”, Mat. Sb. (N.S.), 102(144):3 (1977), 410–424; Math. USSR-Sb., 31:3 (1977), 365–378
A. G. Brusentsev, F. S. Rofe-Beketov, “Selfadjointness conditions for strongly elliptic systems of arbitrary order”, Mat. Sb. (N.S.), 95(137):1(9) (1974), 108–129; Math. USSR-Sb., 24:1 (1974), 103–126
A. G. Brusentsev, F. S. Rofe-Beketov, “Self-adjointness of higher-order elliptic integrals”, Funktsional. Anal. i Prilozhen., 7:4 (1973), 78–79; Funct. Anal. Appl., 7:4 (1973), 319–321
L. B. Zelenko, F. S. Rofe-Beketov, “The limit spectrum of systems of first order differential equations with slowly changing coefficients”, Differ. Uravn., 7:11 (1971), 1982–1991
F. S. Rofe-Beketov, “Conditions for the self-adjointness of the Schrodinger operator”, Mat. Zametki, 8:6 (1970), 741–751; Math. Notes, 8:6 (1970), 888–894
F. S. Rofe-Beketov, E. Kh. Khristov, “Certain analytical questions and the inverse Sturm–Liouville problem for an equation with highly singular potential”, Dokl. Akad. Nauk SSSR, 185:4 (1969), 768–771
F. S. Rofe-Beketov, “Selfadjoint extensions of differential operators in a space of vector-valued functions”, Dokl. Akad. Nauk SSSR, 184:5 (1969), 1034–1037
F. S. Rofe-Beketov, E. Kh. Khristov, “Transformation operators and the scattering function in the case of a highly singular potential”, Dokl. Akad. Nauk SSSR, 168:6 (1966), 1265–1268
1964
19.
F. S. Rofe-Beketov, “Expansion in eigenfunctions of systems with summable potential”, Dokl. Akad. Nauk SSSR, 156:5 (1964), 1029–1032
20.
F. S. Rofe-Beketov, “A finiteness test for the number of discrete levels which can be introduced into the gaps of the continuous
spectrum by perturbations of a periodic potential”, Dokl. Akad. Nauk SSSR, 156:3 (1964), 515–518
F. S. Rofe-Beketov, “On the spectrum of non-selfadjoint differential operators with periodic coefficients”, Dokl. Akad. Nauk SSSR, 152:6 (1963), 1312–1315
F. S. Rofe-Beketov, “Expansion in eigenfunctions of infinite systems of differential equations in the non-self-adjoint and self-adjoint cases”, Mat. Sb. (N.S.), 51(93):3 (1960), 293–342
V. A. Marchenko, F. S. Rofe-Beketov, “Expansion in characteristic functions of no self-adjoined singular differential operators”, Dokl. Akad. Nauk SSSR, 120:5 (1958), 963–966
F. S. Rofe-Beketov, “О краевой задаче для нелинейного дифференциального уравнения”, Mat. Pros., Ser. 2, 1 (1957), 149–154
2012
25.
Yu. M. Berezanskii, V. E. Zakharov, V. P. Kotlyarov, K. V. Maslov, S. P. Novikov, L. A. Pastur, F. S. Rofe-Beketov, A. M. Samoilenko, E. Ya. Khruslov, “Vladimir Aleksandrovich Marchenko (on his 90th birthday)”, Uspekhi Mat. Nauk, 67:5(407) (2012), 186–191; Russian Math. Surveys, 67:5 (2012), 967–972
2008
26.
Yu. M. Berezanskii, V. A. Marchenko, K. V. Maslov, S. P. Novikov, L. A. Pastur, F. S. Rofe-Beketov, Ya. G. Sinai, L. D. Faddeev, E. Ya. Khruslov, “Aleksandr Yakovlevich Povzner (obituary)”, Uspekhi Mat. Nauk, 63:4(382) (2008), 175–176; Russian Math. Surveys, 63:4 (2008), 771–772
27.
Yu. M. Berezanskii, V. P. Kotlyarov, V. A. Marchenko, K. V. Maslov, S. P. Novikov, F. S. Rofe-Beketov, A. M. Samoilenko, Ya. G. Sinai, E. Ya. Khruslov, M. V. Shcherbina, “Leonid Andreevich Pastur (to his 70th brithday)”, Uspekhi Mat. Nauk, 63:1(379) (2008), 190–191; Russian Math. Surveys, 63:1 (2008), 197–199
2005
28.
V. P. Gurarii, M. I. Kadets, V. A. Marchenko, V. I. Matsaev, I. V. Ostrovskii, M. I. Ostrovskii, L. A. Pastur, F. S. Rofe-Beketov, E. Ya. Khruslov, “Vladimir Ilyich Gurarii (1935–2005)”, Zh. Mat. Fiz. Anal. Geom., 1:2 (2005), 245–247
2003
29.
Yu. M. Berezanskii, V. E. Zakharov, Yu. A. Mitropol'skii, I. V. Ostrovskii, L. A. Pastur, A. V. Pogorelov, F. S. Rofe-Beketov, I. V. Skrypnik, E. Ya. Khruslov, “Vladimir Aleksandrovich Marchenko (on his 80th birthday)”, Uspekhi Mat. Nauk, 58:4(352) (2003), 169–172; Russian Math. Surveys, 58:4 (2003), 813–816
1998
30.
Yu. M. Berezanskii, V. E. Zakharov, V. P. Maslov, Yu. A. Mitropol'skii, I. V. Ostrovskii, L. A. Pastur, A. V. Pogorelov, F. S. Rofe-Beketov, E. Ya. Khruslov, “Vladimir Aleksandrovich Marchenko (on his 75th birthday)”, Uspekhi Mat. Nauk, 53:2(320) (1998), 177–180; Russian Math. Surveys, 53:2 (1998), 423–426
1965
31.
F. S. Rofe-Beketov, “Поправки к статье “О разложении по собственным функциям систем с суммируемым потенциалом” (ДАН, т. 156, № 5, 1964 г.)”, Dokl. Akad. Nauk SSSR, 164:2 (1965), 246
32.
F. S. Rofe-Beketov, “Поправки к статье “Признак конечности числа дискретных уровней, вносимых в лакуны непрерывного
спектра возмущениями периодического потенциала” (ДАН, т. 156, № 3, 1964 г.)”, Dokl. Akad. Nauk SSSR, 164:2 (1965), 246
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