×

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line. (English) Zbl 0970.34072

An inverse scattering theory is constructed for the selfadjoint system of differential equations \[ -iJ_{2n}Y'(x,\lambda)-V(x)Y(x,\lambda)=\lambda Y(x,\lambda),\quad -\infty<x<\infty, \] on the line, with \[ J_{2n}=\left[\begin{matrix} I_n & 0 \\ 0 & -I_n\end{matrix}\right],\qquad V(x)=\left[\begin{matrix} 0 & k(x) \\ k^*(x) & 0 \end{matrix}\right], \] \(I_n\) is the identity matrix of odd order \(n\), \(k(x)\in L(-\infty,\infty)\) is the \(n\times n\)-matrix function, and \(*\) denotes the matrix conjugate transpose.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U40 Inverse scattering problems in quantum theory
34A55 Inverse problems involving ordinary differential equations
74J25 Inverse problems for waves in solid mechanics
47A40 Scattering theory of linear operators
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adamyan, V. M. and Arov, D. Z.: On scattering operators and contraction semigroups in Hilbert space.Soviet Math. Dokl. 6, 1377-1380 (1965); also:Dokl. Akad. Nauk SSSR 165, 9-12 (1965) [Russian]. · Zbl 0149.10204
[2] Ablowitz, M. J. and Clarkson, P. A.:Solitons, nonlinear evolution equations and inverse scattering. London Math. Soc. Lecture Note Series149, Cambridge Univ. Press, Cambridge, 1991. · Zbl 0762.35001
[3] Alpay, D. and Gohberg, I.: Inverse spectral problems for differential operators with rational scattering matrix functions.J. Differential Equations 118, 1-19 (1995). · Zbl 0819.47008
[4] Alpay, D. and Gohberg, I.: Inverse scattering problem for differential operators with rational scattering matrix functions. In: Böttcher, A. and Gohberg, I. (Eds.):Singular integral operators and related topics (Tel Aviv, 1995), 1-18. Birkhäuser OT90, Basel, 1996. · Zbl 0819.47008
[5] Alpay, D. and Gohberg, I.: Potential associated with rational weights. In: Gohberg, I. and Lyubich, Yu. (Eds.):New results in operator theory and its applications, 23-40. Birkhäuser OT98, Basel, 1997. · Zbl 0888.34068
[6] Alpay, D. and Gohberg, I.: State space method for inverse spectral problems.Progr. Systems Control Theory 22, 1-16 (1997). · Zbl 0865.93014
[7] Alpay, D., Gohberg, I., and Sakhnovich, L.: Inverse scattering problem for continuous transmission lines with rational reflection coefficient function. In: Gohberg, I., Lancaster, P., and Shivakumar, P. N. (Eds.):Recent developments in operator theory and its applications (Winnipeg, MB, 1994), 1-16. Birkhäuser OT87, Basel, 1996. · Zbl 0859.34063
[8] Arov, D. Z.: Darlington realization of matrix-valued functions.Math. USSR Izv. 7, 1295-1326 (1973); also:Izv. Akad. Nauk SSSR Ser. Mat. 37, 1299-1331 (1973) [Russian]. · Zbl 0316.30037
[9] Bava, G. P., Ghione, G., and Maio, I.: Fast exact inversion of the generalized Zakharov-Shabat problem for rational scattering data: Application to the synthesis of optical couplers.SIAM J. Appl. Math. 48, 689-702 (1988).
[10] Beals, R. and Coifman, R. R.: Scattering and inverse scattering for first order systems.Comm. Pure Appl. Math. 37, 39-90 (1984). · Zbl 0523.34020
[11] Beals, R. and Coifman, R. R.: Scattering and inverse scattering for first-order systems: II.Inverse Problems 3, 577-593 (1987). · Zbl 0663.35054
[12] Beals, R., Deift, P., and Tomei, C.:Direct and inverse scattering on the line. Math. Surveys and Monographs,28, Amer. Math. Soc., Providence, 1988. · Zbl 0679.34018
[13] Bart, H., Gohberg, I., and Kaashoek, M. A.:Minimal factorization of matrix and operator functions. Birkhäuser OT1, Basel, 1979. · Zbl 0424.47001
[14] Bart, H., Gohberg, I., and Kaashoek, M. A.:Exponentially dichotomous operators and inverse Fourier transforms. Report 8511/M, Econometric Institute, Erasmus University of Rotterdam, The Netherlands, 1985. · Zbl 0606.47021
[15] Bart, H., Gohberg, I., and Kaashoek, M. A.: Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators.J. Funct. Anal. 68, 1-42 (1986). · Zbl 0606.47021
[16] Beals, R., Deift, P., and Zhou, X.: The inverse scattering transform on the line. In: Fokas, A. S. and Zakharov, V. E. (Eds.):Important developments in soliton theory, 7-32. Springer, Berlin, 1993. · Zbl 0926.35129
[17] Bhatia, R.:Matrix analysis. Graduate Texts in Mathematics169, Springer, New York, 1997.
[18] Böttcher, A. and Silbermann, B.:Analysis of Toeplitz operators. Springer, New York, 1990. · Zbl 0732.47029
[19] Clancey, K. and Gohberg, I.:Factorization of matrix functions and singular integral operators. Birkhäuser OT3, Basel, 1981. · Zbl 0474.47023
[20] Deift, P. and Trubowitz, E.: Inverse scattering on the line.Comm. Pure Appl. Math. 32, 121-251 (1979). · Zbl 0395.34019
[21] Faddeev, L. D.: Properties of theS-matrix of the one-dimensional Schrödinger equation.Amer. Math. Soc. Transl. (Ser. 2),65, 139-166 (1967); also:Trudy Mat. Inst. Steklova 73, 314-336 (1964) [Russian]. · Zbl 0181.56704
[22] Gasymov, M. G.: The inverse scattering problem for a system of Dirac equations of order 2n.Trans. Moscow Math. Soc. 19, 41-119 (1968); also:Trudy Moscov. Mat. Ob??. 19, 41-112 (1968) [Russian]. · Zbl 0197.26102
[23] Gohberg, I., Goldberg, S., and Kaashoek, M. A.:Classes of linear operators. Vol. I. Birkhäuser OT49, Basel, 1990. · Zbl 0745.47002
[24] Gohberg, I., Kaashoek, M. A., and Sakhnovich, A. L.: Canonical systems with rational spectral densities: Explicit formulas and applications.Math. Nachr. 149, 93-125 (1998). · Zbl 0917.34066
[25] Gohberg, I., Kaashoek, M. A., and Sakhnovich, A. L.: Pseudo-canonical systems with rational Weyl functions: Explicit formulas and applications.J. Differential Equations 146, 375-398 (1998). · Zbl 0917.34074
[26] Gohberg, I., Kaashoek, M. A., and Sakhnovich, A. L.: Canonical systems on the line with rational spectral densities: Explicit formulas. To appear in: Proceedings of the Mark Krein International Conference on Operator Theory and Applications, Odessa, August 18-22, Birkhäuser OT series. · Zbl 0979.34055
[27] Gohberg, I. and Rubinstein, S.: Proper contractions and their unitary minimal completions. In: Gohberg, I. (Ed.):Topics in interpolation theory of rational matrix-valued functions, 223-247. Birkhäuser OT33, Basel, 1988.
[28] Grébert, B.: Inverse scattering for the Dirac operator on the real line,Inverse Problems 8, 787-807 (1992). · Zbl 0764.34010
[29] Hinton, D. B., Jordan, A. K., Klaus, M., and Shaw, J. K.: Inverse scattering on the line for a Dirac system.J. Math. Phys. 32, 3015-3030 (1991).
[30] Hinton, D. B. and Shaw, J. K.: Hamiltonian systems of limit point or limit circle type with both endpoints singular.J. Differential Equations 50, 444-464 (1983). · Zbl 0515.34022
[31] Lancaster, P. and Rodman, L.:Algebraic Riccati equations. Oxford University Press, New York, 1995. · Zbl 0836.15005
[32] Melik-Adamyan, F. È.: On the properties of theS-matrix of canonical differential equations on the whole axis.Akad. Nauk Armjan. SSR Dokl. 58, 199-205 (1974) [Russian]. · Zbl 0313.34012
[33] Melik-Adamyan, F. È.: Canonical differential operators in a Hilbert space.Izv. Akad. Nauk Armjan. SSR Ser. Mat. 12, 10-31, 85 (1977) [Russian]. · Zbl 0364.34010
[34] Melik-Adamyan, F. È.: On a class of canonical differential operators.Soviet J. Contemporary Math. Anal. 24, 48-69 (1989); also:Izv. Akad. Nauk Armyan. SSR Ser. Mat. 24, 570-592, 620 (1989) [Russian]. · Zbl 0697.34023
[35] Petrovski, I. G.:Ordinary differential equations. Prentice-Hall, Englewood Cliffs, 1966.
[36] Ran, A. C. M.: Minimal factorization of selfadjoint rational matrix functions.Integral Equations Operator Theory 5, 850-869 (1982). · Zbl 0492.47013
[37] Reed, M. and Simon, B.:Methods of modern mathematical physics. III. Scattering theory. Academic Press, New York, 1979. · Zbl 0405.47007
[38] Sakhnovich, A. L.: Nonlinear Schrödinger equation on a semi-axis and an inverse problem associated with it.Ukr. Math. J. 42, 316-323 (1990); also:Ukrain. Mat. Zh. 42, 356-363 (1990) [Russian]. · Zbl 0711.35132
[39] Sakhnovich, L. A.: Factorization problems and operator identities.Russ. Math. Surv. 41, 1-64 (1986); also:Usp. Mat. Nauk 41, 4-55, 240 (1986) [Russian]. · Zbl 0613.47017
[40] Sakhnovich, L. A.: The method of operator identities and problems in analysis.St. Petersburg Math. J. 5, 1-69 (1994); also:Algebra i Analiz 5, 3-80 (1993) [Russian]. · Zbl 0823.47017
[41] Sakhnovich, L. A.: Spectral problems for canonical systems of equations on the axis.Russian J. Math. Phys. 2, 517-526 (1995). · Zbl 0913.34068
[42] Shabat, A. B.: An inverse scattering problem.Differential Equations 15, 1299-1307 (1980); also:Differ. Urav. 15, 1824-1834, 1918 (1979) [Russian]. · Zbl 0451.34022
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.