Abstract
We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg–Marchenko theorem for Schrödinger operators on the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.
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Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Hafner, New York (1965)
Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover, New York (1993)
Aktosun, T., Weder, R.: Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schrödinger equation. Inverse Problems 22, 89–114 (2006)
Aktosun, T., Weder, R.: The Borg–Marchenko theorem with a continuous spectrum. In: Recent Advances in Differential Equations and Mathematical Physics, Contemp. Math., 412, 15–30. AMS, Providence, RI (2006)
Berezans’kiĭ, J.M.: Expansions in Eigenfunctions of Selfadjoint Operators. Transl. Math. Monogr. Amer. Math. Soc. Providence, R.I. 17 (1968)
Borg, G.: Uniqueness theorems in the spectral theory of y″+(λ–q(x))y=0. In: Proceedings of the 11th Scandinavian Congress of Mathematicians. Johan Grundt Tanums Forlag, Oslo pp. 276–287 (1952)
Brown, B.M., Naboko, S., Weikard, R.: The inverse resonance problem for Jacobi operators. Bull. London Math. Soc. 37, 727–737 (2005)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer, Berlin Heidelberg New York (1987)
Donoghue, Jr., W.F.: On the perturbation of spectra. Comm. Pure Appl. Math. 18, 559–579 (1965)
Fu, L., Hochstadt, H.: Inverse theorems for Jacobi matrices. J. Math. Anal. Appl. 47, 162–168 (1974)
Gesztesy, F., Simon, B.: Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators. Trans. Amer. Math. Soc. 348, 349–373 (1996)
Gesztesy, F., Simon, B.: m-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices. J. Anal. Math. 73, 267–297 (1997)
Gesztesy, F., Simon, B.: On local Borg–Marchenko uniqueness results. Comm. Math. Phys. 211, 273–287 (2000)
Guseĭnov, G.Š.: The determination of the infinite Jacobi matrix from two spectra. Mat. Zametki 23, 709–720 (1978)
Halilova, R.Z.: An inverse problem. Izv. Akad. Nauk Azerbaĭdžan. SSR Ser. Fiz.-Tehn. Mat. Nauk 3–4, 169–175 (1967)
Kato, T.: Perturbation Theory of Linear Operators. Second Edition. Springer, Berlin Heidelberg New York (1976)
Koelink, E.: Spectral theory and special functions. In: Laredo Lectures on Orthogonal Polynomials and Special Functions. Adv. Theory Spec. Funct. Orthogonal Polynomials, pp. 45–84. Nova, Hauppauge, NY (2004)
Levin, B.Ja.: Distribution of zeros of entire functions. Transl. Math. Monogr. Am. Math. Soc. Providence 5 (1980)
Levitan, B.M., Gasymov, M.G.: Determination of a differential equation by two spectra. Uspekhi Mat. Nauk 19, 3–63 (1964)
Marchenko, V.A.: Some questions in the theory of one-dimensional linear differential operators of the second order. I, Tr. Mosk. Mat. Obš. 1, 327–420 (1952) [Amer. Math. Soc. Transl. (Ser. 2) 101, 1–104 (1973)]
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic, New York (1978)
Schechter, M.: Spectra of Partial Differential Operators. Second Edition. Applied Mathematics and Mechanics 14. North Holland, Amsterdam (1986)
Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)
Teschl, G.: Trace formulas and inverse spectral theory for Jacobi operators. Comm. Math. Phys. 196, 175–202 (1998)
Teschl, G.: Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs. Amer. Math. Soc., Providence 72, 2000
Weidmann, J.: Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics 68. Springer, Berlin Heidelberg New York (1980)
Weikard, R.: A local Borg–Marchenko theorem for difference equations with complex coefficients. In: Partial differential equations and inverse problems. Contemp. Math., vol. 362, pp. 403–410. AMS, Providence, RI (2004)
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Research partially supported by Universidad Nacional Autónoma de México under Project PAPIIT-DGAPA IN 105799, and by CONACYT under Project P42553F.
R. Weder is a fellow of Sistema Nacional de Investigadores.
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Silva, L.O., Weder, R. On the Two Spectra Inverse Problem for Semi-infinite Jacobi Matrices. Math Phys Anal Geom 9, 263–290 (2006). https://doi.org/10.1007/s11040-007-9014-7
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DOI: https://doi.org/10.1007/s11040-007-9014-7