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Inverse Sturm-Liouville problems for non-Borg conditions. (English) Zbl 1418.81032

Summary: We consider Sturm-Liouville problems on the finite interval with non-Borg conditions. Using eigenvalues of four Sturm-Liouville problems, we construct the spectral data and show that the mapping from potential to spectral data is a bijection. Moreover, we obtain estimates of spectral data in terms of potentials.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
47N50 Applications of operator theory in the physical sciences
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[1] G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1-96. · Zbl 0063.00523
[2] C. F. Coleman and J. R. McLaughlin, Solution of the inverse spectral problem for an impedance with integrable derivative. I, Comm. Pure Appl. Math. 46 (1993), no. 2, 145-184. · Zbl 0793.34014
[3] C. F. Coleman and J. R. McLaughlin, Solution of the inverse spectral problem for an impedance with integrable II, Comm. Pure Appl. Math. 46 (1993), no. 2, 185-212. · Zbl 0793.34015
[4] B. Dahlberg and E. Trubowitz, The inverse Sturm-Liouville problem. III, Comm. Pure Appl. Math. 37 (1984), no. 2, 255-267. · Zbl 0601.34017
[5] J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 (1984), 258-312. · Zbl 0554.34013
[6] R. Hryniv and Y. Mykytyuk, Inverse spectral problems for Sturm-Liouville operators with singular potentials. II. Reconstruction by two spectra, Functional Analysis and its Applications, North-Holland Math. Stud. 197, Elsevier, Amsterdam (2004), 97-114. · Zbl 1095.34008
[7] E. L. Isaacson, H. P. McKean and E. Trubowitz, The inverse Sturm-Liouville problem. II, Comm. Pure Appl. Math. 37 (1984), no. 1, 1-11. · Zbl 0552.58024
[8] E. L. Isaacson and E. Trubowitz, The inverse Sturm-Liouville problem. I, Comm. Pure Appl. Math. 36 (1983), no. 6, 767-783. · Zbl 0507.58037
[9] P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, direct approach, Invent. Math. 129 (1997), 567-593. · Zbl 0878.34011
[10] E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Comm. Math. Phys. 183 (1997), no. 2, 383-400. · Zbl 0870.34080
[11] E. Korotyaev, Estimates of periodic potentials in terms of gap lengths, Comm. Math. Phys. 197 (1998), no. 3, 521-526. · Zbl 0924.34073
[12] E. Korotyaev, Inverse problem and the trace formula for the Hill operator. II, Math. Z. 231 (1999), no. 2, 345-368. · Zbl 0929.34016
[13] E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions, Int. Math. Res. Not. (2003), no. 37, 2019-2031. · Zbl 1104.34059
[14] E. Korotyaev, Eigenvalues of Schrödinger operators on finite and infinite intervals, preprint (2018), https://arxiv.org/abs/1809.01371.
[15] E. Korotyaev and D. Chelkak, The inverse Sturm-Liouville problem with mixed boundary conditions, St. Petersburg Math. J. 21 (2009), no. 5, 114-137.
[16] N. Levinson, On the uniqueness of the potential in a Schrödinger equation for a given asymptotic phase, Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1949), no. 9, 1-29. · Zbl 0032.20702
[17] B. M. Levitan, Inverse Sturm-Liouville Problems, VSP, Zeist, 1987. · Zbl 0749.34001
[18] V. Marchenko and I. Ostrovski, A characterization of the spectrum of the Hill operator, Mat. Sb. 97(139) (1975), 540-606. · Zbl 0327.34021
[19] V. Pierce, Determining the potential of a Sturm-Liouville operator from its Dirichlet and Neumann spectra, Pacific J. Math. 204 (2002), no. 2, 497-509. · Zbl 1075.34009
[20] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure Appl. Math. 130, Academic Press, Boston, 1987.
[21] A. M. Savchuk and A. A. Shkalikov, Inverse problem for Sturm-Liouville operators with distribution potentials: Reconstruction from two spectra, Russ. J. Math. Phys. 12 (2005), no. 4, 507-514. · Zbl 1396.34008
[22] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321-337. · Zbl 0403.34022
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