# zbMATH — the first resource for mathematics

Spectral theory of finite-zone nonstationary Schrödinger operators. A nonstationary Peierls model. (English. Russian original) Zbl 0637.35060
Funct. Anal. Appl. 20, 203-214 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 42-54 (1986).
Theorems about expansions in terms of Baker-Akhiezer functions associated with the time-dependent Schrödinger operator $$i\partial_ t-\partial$$ $$2_ x+u(x,t)$$ are proved. The nonlinear correlation between these functions and the potential u(x,t) are obtained. These results are applied to the construction of the exact solutions of the equations of the time-dependent Peierls model.
Reviewer: M.A.Perelmuter

##### MSC:
 35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs 35C10 Series solutions to PDEs 35J10 Schrödinger operator, Schrödinger equation 35K10 Second-order parabolic equations
Full Text:
##### References:
 [1] I. M. Krichever, ”Algebraic-geometrical construction of the Zakharov?Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291-294 (1976). · Zbl 0361.35007 [2] I. M. Krichever, ”Integration of nonlinear equations by the algebraic geometry methods,” Funkts. Anal. Prilozhen.,11, No. 1, 15-31 (1977). · Zbl 0346.35028 [3] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ”Nonlinear equations of the Korteweg?de Vries type, finite-zone linear operators and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55-136 (1976). · Zbl 0326.35011 [4] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: the Inverse Problem Method [in Russian], Nauka, Moscow (1980). · Zbl 0598.35002 [5] I. M. Krichever, ”Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 183-208 (1980). · Zbl 0372.35002 [6] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles over algebraic curves and nonlinear equations,” Usp. Mat. Nauk,35, No. 6, 47-68 (1980). · Zbl 0501.35071 [7] B. A. Dubrovin, ”Theta-functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11-80 (1981). · Zbl 0478.58038 [8] B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, ”Topological and algebraic geometry methods in contemporary mathematical physics. II,” Soviet Scientific Reviews. Math. Phys. Reviews: OPA, Amsterdam, 1982, Vol. 3, pp. 1-151. · Zbl 0534.58002 [9] S. P. Novikov, ”Two-dimensional Schr?dinger operators in periodic fields,” Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 3-32. [10] B. A. Dubrovin, ”Matrix finite-zone operators,” Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 33-78. [11] I. M. Krichever, ”Nonlinear equations and elliptic curves,” Contemporary Problems in Mathematics [in Russian], Vol. 23, Itogi Nauki i Tekhniki, VINITI Akad. Nauk SSSR, Moscow (1983), pp. 79-136. [12] S. A. Brazovskii, S. A. Gordyunin, and N. N. Kirova, ”The exact solution of the Peierls model with an arbitrary number of electrons in a cell,” Pis’ma Zh. Eksp. Teor. Fiz.,31, No. 8, 486-490 (1980). [13] E. D. Belokolos, ”The Peierls?Frolich problems and finite-zone potentials. I,” Teor. Mat. Fiz.,45, No. 2, 268-280 (1980). [14] E. D. Belokolos, ”The Peierls?Frolich problems and finite-zone potentials. II,” ibid.,48, No. 1, 60-69 (1981). [15] S. A. Brazovskii, I. E. Dzyaloshinskii, and I. M. Krichever, ”Exactly solvable discrete Peierls models,” Zh. Eksp. Teor. Fiz.,83, No. 1, 389-415 (1982). [16] I. E. Dzyaloshinskii and I. M. Krichever, ”The sound and the wave of charge density in the discrete Peierls model,” Zh. Eksp. Teor. Fiz.,85, No. 11, 1771-1789 (1983). [17] I. M. Krichever, ”The Peierls model,” Funkts. Anal. Prilozhen.,16, No. 4, 10-26 (1982). · Zbl 0508.58020 · doi:10.1007/BF01081802 [18] I. E. Dzyaloshinskii and I. M. Krichever, ”The effects of co-measurability in the discrete Peierls model,” Zh. Eksp. Teor. Fiz.,83, No. 5, 1576-1581 (1982). [19] I. V. Cherednik, ”Reality conditions in the ?finite-zone? integration,” Dokl. Akad. Nauk SSSR,252, No. 5, 1104-1108 (1980). · Zbl 0491.35044 [20] V. E. Zakharov and A. B. Shabat, ”The integration scheme for nonlinear equations of mathematical physics with the method of the inverse problem of the scattering theory. I,” Funkts. Anal. Prilozhen.,8, No. 3, 43-53 (1974). · Zbl 0303.35024 [21] I. M. Krichever, ”Rational solutions of the Kadomtsev?Petviashvili equation and integrable systems of particles on the direct line,” Funkts. Anal. Prilozhen.,12, No. 1, 76-78 (1978). · Zbl 0374.70008 [22] L. A. Bordag, A. R. Its, V. B. Matveev, S. V. Manakov, and V. E. Zakharov, ”Two-dimensional solitons of Kadomtsev?Petviashvili equation,” Phys. Lett., 205-207 (1977). [23] I. M. Krichever, ”The Laplace method, algebraic curves and nonlinear equations,” Funkts. Anal. Prilozhen.,18, No. 3, 43-56 (1984). · Zbl 0583.35086 [24] J. Fay, ”Theta-function on Riemann surfaces,” Lect. Notes in Math.,352 (Springer-Verlag), Berlin (1973). · Zbl 0281.30013 [25] V. B. Matveev, ”Darboux transformations and the solutions of KP equation depending on the functional parameters,” Lett. Math. Phys.,3, 213-225 (1979). · Zbl 0418.35005 · doi:10.1007/BF00405295 [26] E. A. Kuznetsov and A. B. Mikhailov, ”Stability of stationary waves in nonlinear media with weak dispersion,” Zh. Eksp. Teor. Fiz.,67, No. 11, 1019-1027 (1974). [27] B. A. Dubrovin, ”The periodic problem for the Korteweg?de Vries equation,” Funkts. Anal. Prilozhen.,9, No. 3, 41-51 (1975). · Zbl 0316.30019 · doi:10.1007/BF01078174 [28] B. A. Dubrovin, ”The inverse problem of scattering for periodic finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 1, 65-66 (1975). · Zbl 0318.34038 · doi:10.1007/BF01078185 [29] I. V. Cherednik, ”Differential equations for the Beiker?Akhiezer functions,” Funkts. Anal. Prilozhen.,12, No. 3, 45-54 (1978). [30] N. Yajima and M. Oikawa, ”Interactions between Langmure’s and sonic waves,” Prog. Theor. Phys.,56, 17-19 (1976). · Zbl 1080.37592 · doi:10.1143/PTP.56.1719 [31] V. G. Makhankov, ”On the stationary solutions of Schr?dinger equation with self-congruent potential satisfying Bussinesqu equation,” Phys. Lett.,50, A?P, 42-44 (1974). · doi:10.1016/0375-9601(74)90344-2 [32] Y. Bogomolov, I. Kol’chugina, A. Litvak, and A. Sergeev, ”Near-sonic Langmur solitons,” Phys. Lett.,91, A?P, 9-14 (1982). · doi:10.1016/0375-9601(82)90249-3 [33] V. K. Mel’nikov, ”Some new nonlinear evolution equations integrable by the inverse probblem method,” Mat. Sb.,121, No. 4, 469-498 (1983). [34] H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York (1955). [35] E. Date, M. Jimbo, Kashivara, and T. Miva, ”Transformation groups for soliton equations. I,” Proc. Jpn. Acad.,57A, 342-347 (1981), and III, J. Jpn.,50, 3806-3812 (1981).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.