Schrödinger operators and de Branges spaces. (English) Zbl 1054.34019

The author gives a new view on the inverse spectral theory of one-dimensional Schrödinger operators by recognizing it as a part of the de Branges theory of Hilbert spaces of entire functions. The de Branges theorem on the connection between a regular de Branges space and canonical systems is considered as the mother of many inverse theorems.


34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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[1] Arov, D. Z.; Dym, H., \(J\)-inner matrix functions, interpolation and inverse problems for canonical systems, Ifoundations, Integral Equations Operator Theory, 29, 373-454 (1997) · Zbl 0902.30026
[2] Atkinson, F. V., On the location of Weyl circles, Proc. Roy. Soc. Edinburgh, 88A, 345-356 (1981) · Zbl 0485.34012
[3] Chadan, K.; Sabatier, P. C., Inverse Problems in Quantum Scattering Theory (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0681.35088
[4] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602
[5] de Branges, L., Some Hilbert spaces of entire functions I, Trans. Amer. Math. Soc., 96, 259-295 (1960) · Zbl 0094.04705
[6] de Branges, L., Some Hilbert spaces of entire functions II, Trans. Amer. Math. Soc., 99, 118-152 (1961) · Zbl 0100.06901
[7] de Branges, L., Some Hilbert spaces of entire functions III, Trans. Amer. Math. Soc., 100, 73-115 (1961) · Zbl 0112.30101
[8] de Branges, L., Some Hilbert spaces of entire functions IV, Trans. Amer. Math. Soc., 105, 43-83 (1962) · Zbl 0109.04703
[9] de Branges, L., Hilbert Spaces of Entire Functions (1968), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0157.43301
[10] Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005
[11] Dym, H., An introduction to de Branges spaces of entire functions with applications to differential equations of the Sturm-Liouville type, Adv. Math., 5, 395-471 (1970) · Zbl 0213.39503
[12] Dym, H.; McKean, H. P., Gaussian Processes, Function Theory, and the Inverse Spectral Problem (1976), Academic Press: Academic Press New York · Zbl 0327.60029
[13] Garnett, J. B., Bounded Analytic Functions (1981), Academic Press: Academic Press New York · Zbl 0469.30024
[14] Gelfand, I. M.; Levitan, B. M., On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2), 1, 253-304 (1955) · Zbl 0066.33603
[15] Gesztesy, F.; Simon, B., A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure, Ann. Math., 152, 593-643 (2000) · Zbl 0983.34013
[16] Gohberg, I. C.; Krein, M. G., Theory and Applications of Volterra Operators in Hilbert Space, Translations of Mathematical Monographs, Vol. 24 (1970), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0194.43804
[17] Harris, B. J., The asymptotic form of the Titchmarsh-Weyl \(m\)-function associated with a second order differential equation with locally integrable coefficient, Proc. Roy. Soc. Edinburgh, 102A, 243-251 (1986) · Zbl 0615.34028
[18] Hassi, S.; de Snoo, H.; Winkler, H., Boundary-value problems for two-dimensional canonical systems, Integral Equations Operator Theory, 36, 445-479 (2000) · Zbl 0966.47012
[19] Hinton, D.; Klaus, M.; Shaw, J. K., Series representation and asymptotics for Titchmarsh-Weyl \(m\)-functions, Differential Integral Equations, 2, 419-429 (1989) · Zbl 0715.34044
[20] Horvath, M., On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc., 353, 4155-4171 (2001) · Zbl 0977.34018
[21] Levitan, B. M., Inverse Sturm-Liouville Problems (1987), VNU Science Press: VNU Science Press Utrecht · Zbl 0749.34001
[22] Levitan, B. M.; Gasymov, M. G., Determination of a differential equation by two of its spectra, Russ. Math. Surveys, 19, 1-63 (1964) · Zbl 0145.10903
[23] Marchenko, V. A., Sturm-Liouville Operators and Applications (1986), Birkhäuser: Birkhäuser Basel
[24] Pöschel, J.; Trubowitz, E., Inverse Spectral Theory (1987), Academic Press: Academic Press Orlando
[26] Sakhnovich, L. A., Spectral Theory of Canonical Systems. Method of Operator Identities (1999), Birkhäuser: Birkhäuser Basel · Zbl 0987.35152
[27] Simon, B., A new approach to inverse spectral theory, I. Fundamental formalism, Ann. Math., 150, 1029-1057 (1999) · Zbl 0945.34013
[28] Symes, W., Inverse boundary value problems and a theorem of Gelfand and Levitan, J. Math. Anal. Appl., 71, 379-402 (1979) · Zbl 0425.35092
[29] Thurlow, C., A generalisation of the inverse spectral theorem of Levitan and Gasymov, Proc. Roy. Soc. Edinburgh, 84A, 185-196 (1979) · Zbl 0422.34026
[30] Weidmann, J., Spectral Theory of Ordinary Differential Operators, Lecture Notes in Mathematics, Vol. 1258 (1987), Springer: Springer New York · Zbl 0647.47052
[31] Woracek, H., De Branges spaces of entire functions closed under forming difference quotients, Integral Equations Operator Theory, 37, 238-249 (2000) · Zbl 0957.46019
[32] Yuditskii, P., A special case of de Branges’ theorem on the inverse monodromy problem, Integral Equations Operator Theory, 39, 229-252 (2001) · Zbl 0973.46019
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