The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems
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Abstract
We study the direct and inverse problems for the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as a smooth surface (as the values of a real analytic function of two variables), which has specific properties. We call this function ”the eigenvalues function of the family of Sturm-Liouville operators (EVF)”. From the properties of this function we select those, which are sufficient for a function of two variables be the EVF a family of Sturm-Liouville operators.
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Harutyunyan, T. (2019). The eigenvalues’ function of the family of Sturm-Liouville operators and the inverse problems: . Tamkang Journal of Mathematics, 50(3), 233–252. https://doi.org/10.5556/j.tkjm.50.2019.3352
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References
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[2] Marchenko, V.A. The Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, (in Russian), 1977.
[3] Levitan, B.M. and Sargsyan, I.S. Sturm-Liouville and Dirac operators, Nauka, Moscow, (in Russian), 1988.
[4] Harutyunyan, T.N. “The Dependence of the Eigenvalues of the Sturm-Liouville Problem on Boundary Conditions.” Matematicki Vesnik, 60, no. 4, (2008): 285–294.
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[7] Liouville, J. “M ́emoire sur le d ́eveloppement des fonctions ou parties de fonctions en s ́eries dont les divers termes sont assuj ́etis `a satisfaire `a une mˆeme ́equation diff ́erentielle du second ordre, contenant un param`etre variable.” Journal de math ́ematiques pures et appliqu ́ees, (1836): 253–265.
[8] Isaacson, E.L. and Trubowitz, E. “The inverse Sturm-Liouville problem. I.” Comm. Pure Appl. Math., 36, no. 6, (1983): 767–783.
[9] Isaacson, E.L., McKean, H.P. and Trubowitz, E. “The inverse Sturm-Liouville problem. II.” Comm. Pure Appl. Math., 37, no. 1, (1984): 1–11.
[10] Dahlberg, B.E.J. and Trubowitz, E. “The inverse Sturm-Liouville problem. III.” Comm. Pure Appl. Math., 37, no. 2, (1984): 255–267.
[11] P ̈oschel, J. and Trubowitz, E. Inverse spectral theory, Academic Press, Inc., Boston, MA, 1987.
[12] Atkinson, F.V. Discrete and continuous boundary problems, Academic Press, New York-London, 1964.
[13] Zhikov, V.V. “On inverse Sturm-Liouville problems on a finite segment.” Izv. Akad. Nauk SSSR, ser. Math., 31, no. 5, (in Russian), (1967): 965–976.
[14] Zettl, A. Sturm-liouville theory, American Mathematical Soc., 2005.
[15] Yurko, V.A. Introduction to the theory of inverse spectral problems, Fizmatlit, Moscow, (in Russian), 2007.
[16] Marchenko, V.A. “Some questions of the theory of one-dimensional linear differential operators of the second order.” Trudy Moskov. Mat. Obsh., 1, (in Russian), (1952): 327–420.
[17] Harutyunyan, T.N. and Navasardyan, H.R. “Eigenvalue function of a family of Sturm-Liouville opera- tors.” Izvestia NAN Armenii, Mathematika, 35, no. 5, (in Russian), (2000): 1–11.
[18] Harutyunyan, T.N. “The eigenvalue function of a family of Sturm-Liouville operators.” Izvestiya: Math- ematics 74:3, (2010):439–459 (Izvestiya RAN: Ser. Mat. 74:3, 3–22)
[19] Borg, G. “Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe.” Acta Mathematica, 78, no. 1, (1946): 1–78
[20] Krein, M.G. “The solution of inverse Sturm-Liouville problem.” Dokl. Akad. Nauk, LXXVI, no. 1, (in Russian), (1951): 21–24
[21] Levitan, B.M. “On the determination of Sturm-Liouville equation by two spectra.” Izvestia AN SSSR, ser. Math., 28, no. 1, (in Russian), (1964): 63–78.
[22] Gasymov, M.G. and Levitan, B.M. “Determination of a differential equation by two of its spectra.” Uspekhi Mat. Nauk, 19, no. 2, (in Russian), (1964): 3–63.
[23] Harutyunyan, T.N. “The asymptotics of eigenvalues of Sturm-Liouville problem.” Journal of Contem- porary Mathematical Analysis, 51, no. 4, (2016): 174–183.
[24] Harutyunyan, T.N. “The representation of norming constants by two spectra.” Electronic Journal of Differential Equations, Vol. 2010, No. 159, (2010):1–10
[25] Ashrafyan, Yu.A. and Harutyunyan, T.N. “Inverse Sturm-Liouville problems with fixed boundary con- ditions.” Electronic Journal of Differential Equations, Vol. 2015, no. 27, (2015): 1–8.
[26] Ashrafyan, Yu.A. and Harutyunyan, T.N. “Inverse Sturm-Liouville problems with summable potential.” Mathematical Inverse Problems, Vol. 5, no. 1, (2018): 35–45.
[27] Jodeit, Max, Jr. and Levitan B.M. “The isospectrality problem for the classical Sturm-Liouville equa- tion.” Adv. Differential Equations, 2, no. 2, (1997): 297–318.
[28] Korotyaev, E.L. and Chelkak D.S. “The inverse Sturm-Liouville problem with mixed boundary condi- tions.” Algebra i Analiz, 21, no. 5, (in Russian), (2009): 114-137.
[29] Pahlevanyan, A.A. “Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem.” Izv. NAN Armenii Mat., 52(6), (in Russian), (2017): 77–90.
[30] Harutyunyan, T.N. and Pahlevanyan, A.A. “On the norming constants of the Sturm-Liouville problem.” Bulletin of Kazan State Power Engineering University, no. 3(31), (2016): 7–26.
[2] Marchenko, V.A. The Sturm-Liouville Operators and their Applications, Naukova Dumka, Kiev, (in Russian), 1977.
[3] Levitan, B.M. and Sargsyan, I.S. Sturm-Liouville and Dirac operators, Nauka, Moscow, (in Russian), 1988.
[4] Harutyunyan, T.N. “The Dependence of the Eigenvalues of the Sturm-Liouville Problem on Boundary Conditions.” Matematicki Vesnik, 60, no. 4, (2008): 285–294.
[5] Hobson, E.W. “On a general convergence theorem and theory of the representation of a function by series of normal functions.” Proc. of the London Math. Soc. 2, no. 1, (1908): 349–395.
[6] Kneser, A. “Untersuchungen u ̈ber die Darstellung willku ̈rlicher Funktionen in der mathematischen Physik.” Mathematische Annalen, 58, no. 1, (1903): 81–147.
[7] Liouville, J. “M ́emoire sur le d ́eveloppement des fonctions ou parties de fonctions en s ́eries dont les divers termes sont assuj ́etis `a satisfaire `a une mˆeme ́equation diff ́erentielle du second ordre, contenant un param`etre variable.” Journal de math ́ematiques pures et appliqu ́ees, (1836): 253–265.
[8] Isaacson, E.L. and Trubowitz, E. “The inverse Sturm-Liouville problem. I.” Comm. Pure Appl. Math., 36, no. 6, (1983): 767–783.
[9] Isaacson, E.L., McKean, H.P. and Trubowitz, E. “The inverse Sturm-Liouville problem. II.” Comm. Pure Appl. Math., 37, no. 1, (1984): 1–11.
[10] Dahlberg, B.E.J. and Trubowitz, E. “The inverse Sturm-Liouville problem. III.” Comm. Pure Appl. Math., 37, no. 2, (1984): 255–267.
[11] P ̈oschel, J. and Trubowitz, E. Inverse spectral theory, Academic Press, Inc., Boston, MA, 1987.
[12] Atkinson, F.V. Discrete and continuous boundary problems, Academic Press, New York-London, 1964.
[13] Zhikov, V.V. “On inverse Sturm-Liouville problems on a finite segment.” Izv. Akad. Nauk SSSR, ser. Math., 31, no. 5, (in Russian), (1967): 965–976.
[14] Zettl, A. Sturm-liouville theory, American Mathematical Soc., 2005.
[15] Yurko, V.A. Introduction to the theory of inverse spectral problems, Fizmatlit, Moscow, (in Russian), 2007.
[16] Marchenko, V.A. “Some questions of the theory of one-dimensional linear differential operators of the second order.” Trudy Moskov. Mat. Obsh., 1, (in Russian), (1952): 327–420.
[17] Harutyunyan, T.N. and Navasardyan, H.R. “Eigenvalue function of a family of Sturm-Liouville opera- tors.” Izvestia NAN Armenii, Mathematika, 35, no. 5, (in Russian), (2000): 1–11.
[18] Harutyunyan, T.N. “The eigenvalue function of a family of Sturm-Liouville operators.” Izvestiya: Math- ematics 74:3, (2010):439–459 (Izvestiya RAN: Ser. Mat. 74:3, 3–22)
[19] Borg, G. “Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe.” Acta Mathematica, 78, no. 1, (1946): 1–78
[20] Krein, M.G. “The solution of inverse Sturm-Liouville problem.” Dokl. Akad. Nauk, LXXVI, no. 1, (in Russian), (1951): 21–24
[21] Levitan, B.M. “On the determination of Sturm-Liouville equation by two spectra.” Izvestia AN SSSR, ser. Math., 28, no. 1, (in Russian), (1964): 63–78.
[22] Gasymov, M.G. and Levitan, B.M. “Determination of a differential equation by two of its spectra.” Uspekhi Mat. Nauk, 19, no. 2, (in Russian), (1964): 3–63.
[23] Harutyunyan, T.N. “The asymptotics of eigenvalues of Sturm-Liouville problem.” Journal of Contem- porary Mathematical Analysis, 51, no. 4, (2016): 174–183.
[24] Harutyunyan, T.N. “The representation of norming constants by two spectra.” Electronic Journal of Differential Equations, Vol. 2010, No. 159, (2010):1–10
[25] Ashrafyan, Yu.A. and Harutyunyan, T.N. “Inverse Sturm-Liouville problems with fixed boundary con- ditions.” Electronic Journal of Differential Equations, Vol. 2015, no. 27, (2015): 1–8.
[26] Ashrafyan, Yu.A. and Harutyunyan, T.N. “Inverse Sturm-Liouville problems with summable potential.” Mathematical Inverse Problems, Vol. 5, no. 1, (2018): 35–45.
[27] Jodeit, Max, Jr. and Levitan B.M. “The isospectrality problem for the classical Sturm-Liouville equa- tion.” Adv. Differential Equations, 2, no. 2, (1997): 297–318.
[28] Korotyaev, E.L. and Chelkak D.S. “The inverse Sturm-Liouville problem with mixed boundary condi- tions.” Algebra i Analiz, 21, no. 5, (in Russian), (2009): 114-137.
[29] Pahlevanyan, A.A. “Convergence of expansions for eigenfunctions and asymptotics of the spectral data of the Sturm-Liouville problem.” Izv. NAN Armenii Mat., 52(6), (in Russian), (2017): 77–90.
[30] Harutyunyan, T.N. and Pahlevanyan, A.A. “On the norming constants of the Sturm-Liouville problem.” Bulletin of Kazan State Power Engineering University, no. 3(31), (2016): 7–26.