Geodesic
On the asymptotic behavior of solutions of the time-dependent Schrödinger equation
Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 169-188 Cet article a éte moissonné depuis la source Math-Net.Ru

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The asymptotics for large time of solutions of an evolution equation with a selfadjoint Hamiltonian $H(t)$ having discrete spectrum are studied. Conditions are found under which the evolution equation has a solution which behaves asymptotically like an eigenfunction of the operator $H(t)$. In application to the Schrödinger differential equation it is shown that “bound states” may exist for an interaction which decays (or grows) sufficiently slowly in time. Bibliography: 20 titles. 
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     author = {D. R. Yafaev},
     title = {On the asymptotic behavior of solutions of the time-dependent {Schr\"odinger} equation},
     journal = {Sbornik. Mathematics},
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     year = {1981},
     volume = {39},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_39_2_a1/}
}
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D. R. Yafaev. On the asymptotic behavior of solutions of the time-dependent Schrödinger equation. Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 169-188. http://geodesic.mathdoc.fr/item/SM_1981_39_2_a1/

[1] J. Howland, “Stationary scattering theory for time-dependent Hamiltonians”, Math. Ann., 207 (1974), 315–335 | DOI | MR | Zbl

[2] V. G. Deich, E. L. Korotyaev, D. R. Yafaev, “Potentsialnoe rasseyanie pri uchete prostranstvennoi anizotropii”, DAN SSSR, 235:4 (1977), 749–752 | MR | Zbl

[3] V. G. Deich, E. D. Korotyaev, D. R. Yafaev, “Teoriya potentsialnogo rasseyaniya pri uchete prostranstvennoi anizotropii”, Zapiski nauchnykh seminarov LOMI, 73 (1977), 35–51 | MR | Zbl

[4] A. I. Baz, Ya. B. Zeldovich, A. M. Perelomov, Rasseyanie, reaktsii i raspady v nerelyativistskoi kvantovoi mekhanike, izd-vo “Nauka”, Moskva, 1971

[5] S. T. Kuroda, H. Morita, “An estimate for solutions of Schrödinger equations with timedependent potentials and associated scattering theory”, J. Fac. Sci. Univ. Tokyo, s. 1A, 24:2 (1977), 459–475 | MR | Zbl

[6] D. R. Yafaev, “O narushenii unitarnosti pri rasseyanii na nestatsionarnom potentsiale”, DAN SSSR, 243:5 (1978), 1150–1153 | MR | Zbl

[7] T. Kato, “Integration of the equation of evolution in a Banach space”, J. Math. Soc. Japan, 5 (1953), 208–234 | MR | Zbl

[8] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, t. 2, izd-vo “Mir”, Moskva, 1978 | MR

[9] L. Hormander, “The existence of wave operators in scattering theory”, Math. Z., 146 (1976), 69–91 | DOI | MR

[10] K. Yajima, “Scattering theory for Schrödinger equations with potentials periodic in time”, J. Math. Soc. Japan, 29:4 (1977), 729–743 | DOI | MR | Zbl

[11] H. Sohr, “Über die Existenz von Wellenoperatoren für zeitabhängige Störungen”, Monatsch. Math., 86:1 (1978), 63–81 | DOI | MR | Zbl

[12] B. M. Levitan, I. S. Sargsyan, Vvedenie v spektralnuyu teoriyu, izd-vo “Nauka”, Moskva, 1970 | MR

[13] M. Sh. Birman, M. G. Krein, “K teorii volnovykh operatorov i operatorov rasseyaniya”, DAN SSSR, 144:3 (1972), 475–478 | MR

[14] G. Beitman, A. Erdeii, Vysshie transtsendentnye funktsii, t. 1, izd-vo “Nauka”, Moskva, 1973

[15] L. D. Faddeev, “Svoistva S-matritsy odnomernogo uravneniya Shredingera”, Trudy matem. in-ta im. V. A. Steklova, 73 (1964), 314–336 | MR | Zbl

[16] M. Klaus, “On the bound state of Schrödinger operators in one dimension”, Ann. Phys., 108:2 (1977), 288–300 | DOI | MR | Zbl

[17] B. Simon, “The bound state of weakly conpled Schrödinger operators in one and two dimensions”, Ann. Phys., 97 (1976), 279–288 | DOI | MR | Zbl

[18] V. I. Smirnov, Kurs vysshei matematiki, t. 3, ch. 2, izd-vo “Nauka”, Moskva, 1974 | MR

[19] M. M. Popov, “Primery tochno reshaemykh zadach rasseyaniya dlya parabolicheskogo uravneniya teorii diffraktsii”, Zapiski nauchnykh seminarov LOMI, 78 (1978), 184–202 | Zbl

[20] V. S. Buslaev, V. B. Matveev, “Volnovye operatory dlya uravneniya Shredingera s medlenno ubyvayuschim potentsialom”, TMF, 2:3 (1970), 367–376 | MR