Abstract
The inverse problem for canonical differential equations is investigated for Hamiltonians with singularities. The usual notion of a spectral function is not adequate in this generality, and it is replaced by a more general notion of spectral data. The method of operator identities is used to describe a solution of the inverse problem in this setting. The solution is explicitly computable in many cases, and a number of examples are constructed.
Keywords
- Inverse problem
- spectral function
- singularity
- operator identity
- interpolation
- generalized Nevanlinna function
Mathematics Subject Classification (2000)
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References
D.Z. Arov and H. Dym, J-inner matrix functions, interpolation and inverse problems for canonical systems. I. Foundations, Integral Equations Operator Theory, 29 no. 4 (1997), 373–454; II. The inverse monodromy problem, ibid. 36 no. 1 (2000), 11–70; III. More on the inverse monodromy problem, ibid. 36 no. 2 (2000), 127–181; IV. Direct and inverse bitangential input scattering problems, ibid. 43 no. 1 (2002), 1–67; V. The inverse input scattering problem for Wiener class and rational p x q input scattering matrices, ibid. 43 no. 1 (2002), 68–129.
-, The bitangential inverse input impedance problem for canonical systems I. Weyl-Titchmarsh classification, existence and uniqueness, Integral Equations Operator Theory 47 no. 1 (2003), 3–49; II. Formulas and examples, ibid. 51 no. 2 (2005), 155–213.
K. Daho and H. Langer, Matrix functions of the class N k , Math. Nachr. 120 (1985), 275–294.
L. de Branges, Hilbert spaces of entire functions, Prentice-Hall Inc., Englewood Cliffs, N.J., 1968.
H. Dym and L.A. Sakhnovich, On dual canonical systems and dual matrix string equations, Operator theory, system theory and related topics (Beer-Sheva/Rehovot, 1997), Oper. Theory Adv. Appl., vol. 123, Birkhäuser, Basel, 2001, pp. 207–228.
I.C. Gohberg and M.G. Kreîn, Theory and applications of Volterra operators in Hilbert space, American Mathematical Society, Providence, R.I., 1970.
M. Kaltenbäck and H. Woracek, Pontryagin spaces of entire functions. I, Integral Equations Operator Theory, 33 no. 1 (1999), 34–97; II, ibid. 33 no. 3 (1999), 305–380; III, Acta Sci. Math. (Szeged), 69 no. 1–2 (2003), 241–310; IV, ibid. 72 no. 3–4 (2006), 709–835.
T. Kato, Perturbation theory for linear operators, second ed., Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, Band 132.
M.G. Krem and H. Langer, On some extension problems which are closely connected with the theory of Hermitian operators in a space IIk. III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Part I, Beiträge Anal. 14 (1979), 25–40; Part II, ibid. 15, (1981), 27–45.
-, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume IIk zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1997), 187–236.
H. Langer and H. Winkler, Direct and inverse spectral problems for generalized strings, Integral Equations Operator Theory 30 no. 4 (1998), 409–431, Dedicated to the memory of Mark Grigorievich Krein (1907–1989).
J. Rovnyak and L.A. Sakhnovich, On indefinite cases of operator identities which arise in interpolation theory, The extended field of operator theory (Newcastle, 2004), Oper. Theory Adv. Appl., vol. 171, Birkhäuser, Basel, 2006, pp. 281–322.
-, Some indefinite cases of spectral problems for canonical systems of difference equations, Linear Algebra Appl. 343/344 (2002), 267–289.
-, On the Kreîn-Langer integral representation of generalized Nevanlinna functions, Electron. J. Linear Algebra 11 (2004), 1–15 (electronic).
-, Spectral problems for some indefinite cases of canonical differential equations, J. Operator Theory 51 (2004), 115–139.
A.L. Sakhnovich, Spectral functions of a second-order canonical system, Mat. Sb. 181 no. 11 (1990), 1510–1524, Engl. transi., USSR-Sb. 71 no. 2 (1992), 355–369.
-, Modification of V. P. Potapov’s scheme in the indefinite case, Matrix and operator valued functions, Oper. Theory Adv. Appl., vol. 72, Birkhäuser, Basel, 1994, pp. 185–201.
L.A. Sakhnovich, Problems of factorization and operator identities, Uspekhi Mat. Nauk 41 no. 1 (1986), (247), 4–55, Engl. transi., Russian Math. Surveys 41:1 (1986), 1–64.
-, Integral equations with difference kernels on finite intervals, Oper. Theory Adv. Appl., vol. 84, Birkhäuser Verlag, Basel, 1996.
-, Interpolation theory and its applications, Kluwer, Dordrecht, 1997.
-, Spectral theory of canonical differential systems. Method of operator identities, Oper. Theory Adv. Appl., vol. 107, Birkhäuser Verlag, Basel, 1999.
-, On reducing the canonical system to two dual differential systems, J. Math. Anal. Appl. 255 no. 2 (2001), 499–509.
G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, Cambridge, England, 1944.
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Rovnyak, J., Sakhnovich, L.A. (2007). Inverse Problems for Canonical Differential Equations with Singularities. In: Ball, J.A., Eidelman, Y., Helton, J.W., Olshevsky, V., Rovnyak, J. (eds) Recent Advances in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 179. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8539-2_16
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