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Characterization of the Weyl solutions. (English) Zbl 0807.34032

This letter describes a new characterization of Weyl solutions (i.e. loosely speaking proper spectral data for the inverse problem associated with nonlinear partial differential equations) in terms of their asymptotic behaviour when the spectral parameters tend to infinity whilst the coordinate itself remains fixed. The study is focused on the Schrödinger and Dirac equations. As a consequence, one has a criterion to select Weyl solutions according to their behaviour for large value of \(z\).

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
35L40 First-order hyperbolic systems
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35J20 Variational methods for second-order elliptic equations
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] Marchenko, V. A., The Cauchy problem for the KdV equation with non-decreasing initial data, in V. E. Zaharov (ed.),What is Integrability?, Springer-Verlag, New York, 1990, pp. 273-318. · Zbl 0810.34090
[2] Levitan, B. M. and Sargsyan, I. S.,Introduction to the Spectral Theory, Nauka, Moscow, 1970, p. 671 (in Russian). · Zbl 0225.47019
[3] Martinov, V. V., Conditions of discreteness and continuity of spectrum in the case of a self-adjoint system of first-order differential equations,Dokl. Akad. Nauk SSSR 165, 996-999 (1965).
[4] Marchenko, V. A., Asymptotics of the Weyl solutions of the Sturm-Liouville equations with respect to the spectral parameter,Proc. LOMI Seminars, v. 170, Nauka, Leningrad, pp. 184-206. · Zbl 0703.34031
[5] Misyura, T. V., Asymptotic formula for the Weyl solutions of the Dirac equations,Dokl. Akad. Nauk Ukrain. SSR 5, 26-28 (1991).
[6] Marchenko, V. A.,The Sturm-Liouville Operators and Applications, Birkhäuser Verlag, Basel, 1986, p. 367. · Zbl 0592.34011
[7] Glazman, I. M.,Direct Method of Qualitative Analysis of Singular Differential Operators, Fizmatgiz, Moscow, 1963, p. 340 (in Russian). · Zbl 0143.36504
[8] Marchenko, V. A., Characterization of the Weyl solutions, Preprint BIBOS No 578/93, Universität Bielefeld. · Zbl 0807.34032
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