Abstract
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.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 18-11-00032
Funding statement: This work was supported by the RSF grant No. 18-11-00032.
Acknowledgements
Finally, we would like to thank the referee for thoughtful comments that helped us to improve the manuscript.
References
[1] G. Borg, Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96. 10.1007/BF02421600Search in Google Scholar
[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. 10.1002/cpa.3160460203Search in Google Scholar
[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. 10.1002/cpa.3160460204Search in Google Scholar
[4] B. Dahlberg and E. Trubowitz, The inverse Sturm–Liouville problem. III, Comm. Pure Appl. Math. 37 (1984), no. 2, 255–267. 10.1002/cpa.3160370205Search in Google Scholar
[5] J. Garnett and E. Trubowitz, Gaps and bands of one dimensional periodic Schrödinger operators, Comment. Math. Helv. 59 (1984), 258–312. 10.1007/BF02566350Search in Google Scholar
[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. Search in Google Scholar
[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. 10.1002/cpa.3160370102Search in Google Scholar
[8] E. L. Isaacson and E. Trubowitz, The inverse Sturm–Liouville problem. I, Comm. Pure Appl. Math. 36 (1983), no. 6, 767–783. 10.1002/cpa.3160360604Search in Google Scholar
[9] P. Kargaev and E. Korotyaev, The inverse problem for the Hill operator, direct approach, Invent. Math. 129 (1997), 567–593. 10.1007/s002220050173Search in Google Scholar
[10] E. Korotyaev, The estimates of periodic potentials in terms of effective masses, Comm. Math. Phys. 183 (1997), no. 2, 383–400. 10.1007/BF02506412Search in Google Scholar
[11] E. Korotyaev, Estimates of periodic potentials in terms of gap lengths, Comm. Math. Phys. 197 (1998), no. 3, 521–526. 10.1007/s002200050462Search in Google Scholar
[12] E. Korotyaev, Inverse problem and the trace formula for the Hill operator. II, Math. Z. 231 (1999), no. 2, 345–368. 10.1007/PL00004733Search in Google Scholar
[13] E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions, Int. Math. Res. Not. (2003), no. 37, 2019–2031. 10.1155/S1073792803209107Search in Google Scholar
[14] E. Korotyaev, Eigenvalues of Schrödinger operators on finite and infinite intervals, preprint (2018), https://arxiv.org/abs/1809.01371. 10.1002/mana.201900511Search in Google Scholar
[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. 10.1090/S1061-0022-2010-01116-6Search in Google Scholar
[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. 10.1007/978-1-4612-5341-9_17Search in Google Scholar
[17] B. M. Levitan, Inverse Sturm–Liouville Problems, VSP, Zeist, 1987. 10.1515/9783110941937Search in Google Scholar
[18] V. Marchenko and I. Ostrovski, A characterization of the spectrum of the Hill operator, Mat. Sb. 97(139) (1975), 540–606. Search in Google Scholar
[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. 10.2140/pjm.2002.204.497Search in Google Scholar
[20] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Pure Appl. Math. 130, Academic Press, Boston, 1987. Search in Google Scholar
[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. Search in Google Scholar
[22] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), no. 3, 321–337. 10.1002/cpa.3160300305Search in Google Scholar
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