New problems of the spectral theory of one-dimensional Schrodinger operators are formulated and solved: It is proved that the Banach algerbras generated by the generalized shift are isomorphic to the algebras with ordinary convolution. An asymptotic formula for spectral functions is obtained. It is proved that the spectral functions uniquely define the operators. The stability of the inverse problems of the spectral analysis is investigated. The inverse scattering problem is solved. The inverse problem for the operators with periodic potentials is solved (with I. V. Ostrovsky). Characteristic properties of the Weyl solutions are found. An asymptotic theory of the boundary problems with fine-grained boundary is constructed (with E. Ya. Khruslov). The theory gives the possibility to find the limits to which the solutions are converged under the unbounded fragmentation of the boundary and to evaluate the rate of convergence. The limits of the integral density of distribution of the eigenvalues of the random matrix ensembles in the limit of infinite matrix order, are obtained (with L. A. Pastur). A new method of solving the non-linear evolution equations, which allows one to solve the non-linear Cauchy problems with initial data which are not stabilizing at infinity, is developed.
Biography
Graduated from the Faculty of Physics and Mathematics of Kharkov State University in 1945. Ph.D. thesis was defended in 1948. D.Sci. thesis was defended in 1952. The list of my works contains near 100 titles.
Academician of the Academy of Sciences of Ukraine. In 1962 I was awarded the Lenin prize for a series of papers on the inverse problems of spectral analysis of differential operators (with B. M. Levitan). In 1983 I was awarded the N. M. Krylov prize of the Presidium of the Academy of Science of Ukraine for the monograph "Stourm–Liouville operators and their applications". In 1989 I was awarded the State prize of Ukraine for a series of papers "Boundary problems of mathematical physics with fine-grained boundary" (with E. Ya. Khruslov and I. V. Skrypnik). In 1996 I was awarded the N. N. Bogolyubov prize of the Presidium of the Academy of Science of Ukraine for a series of papers "Functional algebraic methods of mathematical physics" (with O. S. Parasyuk and A. N. Tavkhelidze).
Main publications:
Z. S. Agranovich, V. A. Marchenko. The Inverse Problem of Scattering Theory. Gordon and Breach, 1963.
I. E. Egorova, E. A. Kopylova, V. A. Marchenko, G. Teschl, “Dispersion estimates for one-dimensional Schrödinger and Klein–Gordon equations revisited”, Uspekhi Mat. Nauk, 71:3(429) (2016), 3–26; Russian Math. Surveys, 71:3 (2016), 391–415
V. A. Marchenko, “Hill discriminants and spectra of periodic Jacobi matrices”, Algebra i Analiz, 25:2 (2013), 155–161; St. Petersburg Math. J., 25:2 (2014), 265–269
Yu. I. Lyubarskii, V. A. Marchenko, “Direct and Inverse Multichannel Scattering Problems”, Funktsional. Anal. i Prilozhen., 41:2 (2007), 58–77; Funct. Anal. Appl., 41:2 (2007), 126–142
Anne Boutet de Monvel, Vladimir Marchenko, “The Cauchy problem for nonlinear Schrödinger equation with bounded initial data”, Mat. Fiz. Anal. Geom., 4:1/2 (1997), 3–45
D. Sh. Lundina, V. A. Marchenko, “Compactness of the set of multisoliton solutions of the nonlinear Schrödinger equation”, Mat. Sb., 183:4 (1992), 3–19; Russian Acad. Sci. Sb. Math., 75:2 (1993), 429–443
A. F. Barannik, V. A. Marchenko, W. I. Fushchych, “Reductions and exact solutions of nonlinear multidimensional Schrödinger equations”, TMF, 87:2 (1991), 220–234; Theoret. and Math. Phys., 87:2 (1991), 488–498
V. A. Marchenko, “Spectral parameter asymptotics of the Weil solutions of Sturm-Liouville equations”, Zap. Nauchn. Sem. LOMI, 170 (1989), 184–206; J. Soviet Math., 63:2 (1993), 217–232
1975
9.
V. A. Marchenko, I. V. Ostrovskii, “A characterization of the spectrum of Hill’s operator”, Dokl. Akad. Nauk SSSR, 222:6 (1975), 1283–1286
10.
V. A. Marchenko, I. V. Ostrovskii, “A characterization of the spectrum of Hill's operator”, Mat. Sb. (N.S.), 97(139):4(8) (1975), 540–606; Math. USSR-Sb., 26:4 (1975), 493–554
V. A. Marchenko, K. V. Maslov, “Stability of the problem of recovering the Sturm–Liouville operator from the spectral function”, Mat. Sb. (N.S.), 81(123):4 (1970), 525–551; Math. USSR-Sb., 10:4 (1970), 475–502
D. Sh. Lundina, V. A. Marchenko, “Refinement of inequalities characterizing stability of the inverse problem of scattering theory”, Mat. Sb. (N.S.), 78(120):4 (1969), 475–484; Math. USSR-Sb., 7:4 (1969), 467–476
V. A. Marchenko, “Stability of the inverse problem in scattering theory”, Mat. Sb. (N.S.), 77(119):2 (1968), 139–162; Math. USSR-Sb., 6:2 (1968), 125–148
V. A. Marchenko, L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices”, Mat. Sb. (N.S.), 72(114):4 (1967), 507–536; Math. USSR-Sb., 1:4 (1967), 457–483
V. A. Marchenko, “Expansion in eigenfunctions of non-self-adjoint singular second-order differential operators”, Mat. Sb. (N.S.), 52(94):2 (1960), 739–788
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
Z. S. Agranovich, V. A. Marchenko, “Construction of tensor forces from scattering data”, Dokl. Akad. Nauk SSSR, 118:6 (1958), 1055–1058
1957
22.
Z. S. Agranovich, V. A. Marchenko, “Re-establishment of the potential from the scattering matrix for a system of differential equations”, Dokl. Akad. Nauk SSSR, 113:5 (1957), 951–954
23.
Z. S. Agranovich, V. A. Marchenko, “Reconstruction of the potential energy from the scattering matrix”, Uspekhi Mat. Nauk, 12:1(73) (1957), 143–145
V. A. Marchenko, “Some questions of the theory of one-dimensional linear differential operators of the second order. II”, Tr. Mosk. Mat. Obs., 2 (1953), 3–83
V. A. Marchenko, “Some questions of the theory of one-dimensional linear differential operators of the second order. I”, Tr. Mosk. Mat. Obs., 1 (1952), 327–420
V. M. Babich, A. R. Its, V. A. Marchenko, L. A. Pastur, B. A. Plamenevskii, T. A. Suslina, L. D. Faddeev, A. A. Fedotov, N. N. Ural'tseva, “Vladimir Savel'evich Buslaev (obituary)”, Uspekhi Mat. Nauk, 69:1(415) (2014), 163–168; Russian Math. Surveys, 69:1 (2014), 153–158
2011
29.
Yu. I. Lyubich, V. A. Marchenko, S. P. Novikov, M. I. Ostrovskii, L. A. Pastur, A. N. Plichko, M. M. Popov, E. M. Semenov, S. L. Troyanskii, V. P. Fonf, E. Ya. Khruslov, “Mikhail Iosifovich Kadets (obituary)”, Uspekhi Mat. Nauk, 66:4(400) (2011), 179–180; Russian Math. Surveys, 66:4 (2011), 809–811
V. M. Babich, V. S. Buslaev, A. M. Vershik, S. G. Gindikin, S. V. Kislyakov, A. A. Laptev, V. A. Marchenko, N. K. Nikol'skii, L. A. Pastur, B. A. Plamenevskii, M. Z. Solomyak, T. A. Suslina, N. N. Ural'tseva, L. D. Faddeev, V. P. Khavin, D. R. Yafaev, “Mikhail Shlemovich Birman (obituary)”, Uspekhi Mat. Nauk, 65:3(393) (2010), 185–190; Russian Math. Surveys, 65:3 (2010), 569–575
D. V. Anosov, V. V. Kozlov, V. A. Marchenko, Yu. A. Mitropol'skii, E. F. Mishchenko, S. M. Nikol'skii, B. E. Paton, V. A. Sadovnichii, E. Ya. Khruslov, F. L. Chernous'ko, “Anatolii Mikhailovich Samoilenko (on his 70th birthday)”, Uspekhi Mat. Nauk, 64:1(385) (2009), 169–174; Russian Math. Surveys, 64:1 (2009), 175–181
2008
32.
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
33.
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
2007
34.
S. I. Bezuglyi, L. L. Vaksman, S. L. Gefter, A. I. Danilenko, G. N. Zholtkevich, V. M. Kulagin, V. A. Marchenko, N. I. Nessonov, L. A. Pastur, S. D. Sinel'shchikov, E. Ya. Khruslov, “Valentin Yakovlevich Golodets (on his seventieth birthday)”, Zh. Mat. Fiz. Anal. Geom., 3:2 (2007), 277–279
35.
D. V. Anosov, S. I. Bezuglyi, A. M. Vershik, S. L. Gefter, A. I. Danilenko, R. S. Ismagilov, V. A. Marchenko, N. I. Nessonov, S. P. Novikov, Ya. G. Sinai, S. D. Sinel'shchikov, A. M. Stepin, “Valentin Yakovlevich Golodets (on his 70th birthday)”, Uspekhi Mat. Nauk, 62:6(378) (2007), 191–192; Russian Math. Surveys, 62:6 (2007), 1231–1232
2005
36.
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
37.
V. S. Azarin, A. A. Gol'dberg, A. I. Il'inskii, V. A. Marchenko, L. A. Pastur, I. V. Skrypnik, M. L. Sodin, A. M. Ulanovskii, G. M. Feldman, E. Ya. Khruslov, G. P. Chistyakov, “Iosif Vladimirovich Ostrovskii (on his 70th birthday)”, Uspekhi Mat. Nauk, 60:1(361) (2005), 186–188; Russian Math. Surveys, 60:1 (2005), 189–192
2004
38.
V. A. Marchenko, S. P. Novikov, I. V. Ostrovskii, M. I. Ostrovskii, L. A. Pastur, A. N. Plichko, M. M. Popov, C. L. Troyanscii, V. P. Fonf, E. Ya. Khruslov, “Mikhail Iosifovich Kadets (on his 80th birthday)”, Uspekhi Mat. Nauk, 59:5(359) (2004), 183–185; Russian Math. Surveys, 59:5 (2004), 1001–1004
2003
39.
Yu. M. Berezanskii, A. G. Brusentsev, V. A. Marchenko, K. V. Maslov, L. A. Pastur, I. V. Ostrovskii, V. I. Khrabustovskii, E. Ya. Khruslov, “Fedor Semenovich Rofe-Beketov (on his 70th birthday)”, Uspekhi Mat. Nauk, 58:4(352) (2003), 173–176; Russian Math. Surveys, 58:4 (2003), 817–820
40.
V. A. Aleksandrov, V. I. Arnol'd, A. A. Borisenko, Yu. F. Borisov, V. A. Zalgaller, S. S. Kutateladze, V. A. Marchenko, A. D. Milka, Yu. G. Reshetnyak, I. Kh. Sabitov, “Aleksei Vasil'evich Pogorelov (obituary)”, Uspekhi Mat. Nauk, 58:3(351) (2003), 173–175; Russian Math. Surveys, 58:3 (2003), 593–596
A. D. Aleksandrov, A. A. Borisenko, V. A. Zalgaller, V. A. Marchenko, K. V. Maslov, A. D. Milka, S. P. Novikov, Yu. G. Reshetnyak, I. V. Skrypnik, E. Ya. Khruslov, “Aleksei Vasil'evich Pogorelov (on his 80th birthday)”, Uspekhi Mat. Nauk, 54:4(328) (1999), 188–190; Russian Math. Surveys, 54:4 (1999), 869–872
42.
P. Z. Agranovich, A. A. Gol'dberg, V. P. Gurarii, V. A. Marchenko, S. P. Novikov, I. V. Ostrovskii, L. A. Pastur, A. Yu. Rashkovskii, S. Yu. Favorov, E. Ya. Khruslov, “Lev Isaakovich Ronkin (obituary)”, Uspekhi Mat. Nauk, 54:3(327) (1999), 147–148; Russian Math. Surveys, 54:3 (1999), 619–621
1998
43.
V. P. Gurarii, V. E. Zakharov, V. A. Marchenko, V. P. Maslov, A. V. Pogorelov, Ya. G. Sinai, E. Ya. Khruslov, “Leonid Andreevich Pastur (on his 60th birthday)”, Uspekhi Mat. Nauk, 53:2(320) (1998), 181–182; Russian Math. Surveys, 53:2 (1998), 427–429
1997
44.
O. A. Ladyzhenskaya, V. A. Marchenko, Yu. A. Mitropol'skii, S. P. Novikov, A. V. Pogorelov, “Evgenii Yakovlevich Khruslov (on his 60th birthday)”, Uspekhi Mat. Nauk, 52:6(318) (1997), 205–206; Russian Math. Surveys, 52:6 (1997), 1351–1353
1995
45.
V. A. Azarin, A. A. Gol'dberg, A. I. Il'inskii, V. A. Marchenko, L. A. Pastur, L. I. Ronkin, M. L. Sodin, G. M. Feldman, A. E. Fryntov, G. P. Chistyakov, “Iosif Vladimirovich Ostrovskii (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 50:2(302) (1995), 232–235; Russian Math. Surveys, 50:2 (1995), 465–468
1994
46.
Yu. M. Berezanskii, V. A. Marchenko, S. P. Novikov, I. V. Ostrovskii, M. I. Ostrovskii, L. A. Pastur, A. V. Pogorelov, L. I. Ronkin, “Mikhail Iosifovich Kadets (on his seventieth birthday)”, Uspekhi Mat. Nauk, 49:3(297) (1994), 205–206; Russian Math. Surveys, 49:3 (1994), 223–225
V. S. Azarin, A. A. Gol'dberg, E. A. Gorin, V. A. Marchenko, I. V. Ostrovskii, A. V. Pogorelov, L. I. Ronkin, M. L. Sodin, V. A. Tkachenko, “Boris Yakovlevich Levin (obituary)”, Uspekhi Mat. Nauk, 49:1(295) (1994), 201–202; Russian Math. Surveys, 49:1 (1994), 219–221
1991
48.
L. A. Aizenberg, A. A. Gol'dberg, B. Ya. Levin, V. N. Logvinenko, V. A. Marchenko, V. V. Napalkov, I. V. Ostrovskii, S. Yu. Favorov, “Lev Isaakovich Ronkin (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 46:5(281) (1991), 181–183; Russian Math. Surveys, 46:5 (1991), 207–209
1990
49.
I. I. Vorovich, A. Yu. Ishlinskii, M. A. Krasnosel'skii, N. N. Krasovskii, A. B. Kurzhanskii, V. A. Marchenko, V. P. Maslov, Yu. A. Mitropol'skii, L. V. Ovsyannikov, Yu. S. Osipov, A. N. Sharkovskii, “Anatolii Dmitrievich Myshkis (on his seventieth birthday)”, Uspekhi Mat. Nauk, 45:4(274) (1990), 179–181; Russian Math. Surveys, 45:4 (1990), 207–210
1989
50.
A. D. Aleksandrov, V. A. Marchenko, S. P. Novikov, Yu. G. Reshetnyak, “Aleksei Vasil'evich Pogorelov (on his seventieth birthday)”, Uspekhi Mat. Nauk, 44:4(268) (1989), 245–249; Russian Math. Surveys, 44:4 (1989), 217–223
I. M. Gel'fand, M. G. Krein, V. A. Marchenko, N. K. Nikol'skii, I. V. Ostrovskii, “Boris Yakovlevich Levin (on his eightieth birthday)”, Uspekhi Mat. Nauk, 42:4(256) (1987), 207–210; Russian Math. Surveys, 42:4 (1987), 175–178
1986
52.
M. L. Gorbachuk, A. Yu. Ishlinskii, V. A. Marchenko, Yu. A. Mitropol'skii, L. P. Nizhnik, S. L. Sobolev, “Yurii Makarovich Berezanskii (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 41:1(247) (1986), 213–214; Russian Math. Surveys, 41:1 (1986), 253–255
1985
53.
M. G. Gasymov, M. G. Krein, B. Ya. Levin, V. A. Marchenko, I. S. Sargsyan, S. L. Sobolev, “Boris Moiseevich Levitan (on his seventieth birthday)”, Uspekhi Mat. Nauk, 40:2(242) (1985), 209–210; Russian Math. Surveys, 40:2 (1985), 247–249
1984
54.
I. M. Gel'fand, B. Ya. Levin, V. A. Marchenko, A. V. Pogorelov, S. L. Sobolev, “Mikhail Iosifovich Kadets (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 39:6(240) (1984), 249–250; Russian Math. Surveys, 39:6 (1984), 231–232
1981
55.
A. N. Kolmogorov, M. G. Krein, B. Ya. Levin, B. M. Levitan, Yu. I. Lyubich, V. A. Marchenko, I. V. Ostrovskii, A. Ya. Povzner, A. V. Pogorelov, “Naum Il'ich Akhiezer (obituary)”, Uspekhi Mat. Nauk, 36:4(220) (1981), 183–184; Russian Math. Surveys, 36:4 (1981), 207–208
A. D. Aleksandrov, Ya. P. Blank, N. V. Efimov, V. A. Marchenko, “Aleksei Vasil'evich Pogorelov (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 34:4(208) (1979), 221–226; Russian Math. Surveys, 34:4 (1979), 199–207
1977
57.
N. I. Akhiezer, N. V. Efimov, M. G. Krein, M. A. Lavrent'ev, V. A. Marchenko, I. V. Ostrovskii, B. V. Shabat, “Boris Yakovlevich Levin (on his seventieth birthday)”, Uspekhi Mat. Nauk, 32:5(197) (1977), 211–213; Russian Math. Surveys, 32:5 (1977), 181–184
1975
58.
S. Yu. Dobrokhotov, V. P. Maslov, P. P. Mosolov, B. M. Gurevich, Yu. M. Sukhov, V. A. Zlatoustov, V. B. Matveev, A. P. Markeev, Ya. B. Zel'dovich, V. A. Marchenko, E. Ya. Khruslov, “Sessions of the I. G. Petrovskii Seminar on Differential Equations and Mathematical Problems of Physics”, Uspekhi Mat. Nauk, 30:6(186) (1975), 197–206
Yu. M. Berezanskii, I. M. Gel'fand, B. Ya. Levin, V. A. Marchenko, K. V. Maslov, “Aleksandr Yakovlevich Povzner (on his sixtieth birthday)”, Uspekhi Mat. Nauk, 30:5(185) (1975), 221–226; Russian Math. Surveys, 30:5 (1975), 177–183
1971
60.
Yu. M. Berezanskii, A. N. Kolmogorov, M. G. Krein, B. Ya. Levin, B. M. Levitan, V. A. Marchenko, “Naum Il'ich Akhiezer (on his seventieth birthday)”, Uspekhi Mat. Nauk, 26:6(162) (1971), 257–261; Russian Math. Surveys, 26:6 (1971), 233–237
N. I. Akhiezer, E. M. Zhmud', Yu. I. Lyubich, V. A. Marchenko, “Izrail' Markovich Glazman (obituary)”, Uspekhi Mat. Nauk, 24:5(149) (1969), 215–219; Russian Math. Surveys, 24:5 (1969), 185–189
1965
62.
N. I. Akhiezer, V. A. Marchenko, “Boris Moiseevich Levitan (on his fiftieth birthday)”, Uspekhi Mat. Nauk, 20:3(123) (1965), 227–234; Russian Math. Surveys, 20:3 (1965), 175–182