Geodesic
On the integral characteristic function of the Sturm-Liouville problem
Sbornik. Mathematics, Tome 211 (2020) no. 11, pp. 1539-1550 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We introduce a function whose zeros, and only these zeros, are eigenvalues of the corresponding Sturm-Liouville problem. The boundary conditions of the problem depend continuously on the spectral parameter. Therefore, it makes sense to call the function thus constructed a characteristic function of the Sturm-Liouville problem (however, it is not a characteristic function in the ordinary sense). An investigation of the function thus obtained enables us to prove the solvability of the problem in question, to find the asymptotic behaviour of the eigenvalues, to obtain comparison theorems, and to introduce an indexing of the eigenvalues and the zeros of eigenfunctions in a natural way. Bibliography: 31 titles.
Keywords: integral characteristic function, asymptotic behaviour of eigenvalues, comparison theorem
Mots-clés : Sturm-Liouville problem, Riccati equation. 
@article{SM_2020_211_11_a1,
     author = {D. V. Valovik},
     title = {On the integral characteristic function of the {Sturm-Liouville} problem},
     journal = {Sbornik. Mathematics},
     pages = {1539--1550},
     year = {2020},
     volume = {211},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_11_a1/}
}
TY  - JOUR
AU  - D. V. Valovik
TI  - On the integral characteristic function of the Sturm-Liouville problem
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 1539
EP  - 1550
VL  - 211
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_11_a1/
LA  - en
ID  - SM_2020_211_11_a1
ER  - 
%0 Journal Article
%A D. V. Valovik
%T On the integral characteristic function of the Sturm-Liouville problem
%J Sbornik. Mathematics
%D 2020
%P 1539-1550
%V 211
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2020_211_11_a1/
%G en
%F SM_2020_211_11_a1
D. V. Valovik. On the integral characteristic function of the Sturm-Liouville problem. Sbornik. Mathematics, Tome 211 (2020) no. 11, pp. 1539-1550. http://geodesic.mathdoc.fr/item/SM_2020_211_11_a1/

[1] M. A. Naimark, Linear differential operators, v. I, II, Frederick Ungar Publishing Co., New York, 1967, 1968, xiii+144 pp., xv+352 pp. | MR | MR | MR | Zbl | Zbl

[2] B. M. Levitan, I. S. Sargsjan, Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), 59, Kluwer Acad. Publ., Dordrecht, 1991, xii+350 pp. | DOI | MR | MR | Zbl | Zbl

[3] A. G. Kostyuchenko, I. S. Sargsyan, Raspredelenie sobstvennykh znachenii, Nauka, M., 1979, 400 pp. | MR | Zbl

[4] F. V. Atkinson, Discrete and continuous boundary problems, Math. Sci. Eng., 8, Academic Press, New York–London, 1964, xiv+570 pp. | MR | MR | Zbl | Zbl

[5] V. A. Marchenko, Sturm–Liouville operators and applications, Oper. Theory Adv. Appl., 22, Birkhäuser Verlag, Basel, 1986, xii+367 pp. | DOI | MR | MR | Zbl | Zbl

[6] G. Sansone, Equazioni differenziali nel campo reale, v. 1, 2nd ed., N. Zanichelli, Bologna, 1948, xvii+400 pp. | MR | Zbl

[7] Ph. Hartman, Ordinary differential equations, John Wiley Sons, Inc., New York–London–Sydney, 1964, xiv+612 pp. | MR | MR | Zbl | Zbl

[8] Yu. G. Smirnov, “Eigenvalue transmission problems describing the propagation of TE and TM waves in two-layered inhomogeneous anisotropic cylindrical and planar waveguides”, Comput. Math. Math. Phys., 55:3 (2015), 461–469 | DOI | DOI | MR | Zbl

[9] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Transl. Math. Monogr., 71, Amer. Math. Soc., Providence, RI, 1988, iv+250 pp. | MR | MR | Zbl | Zbl

[10] I. C. Gohberg, M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969, xv+378 pp. | MR | MR | Zbl | Zbl

[11] J. Ben Amara, A. A. Shkalikov, “A Sturm–Liouville problem with physical and spectral parameters in boundary conditions”, Math. Notes, 66:2 (1999), 127–134 | DOI | DOI | MR | Zbl

[12] R. Mennicken, H. Schmid, A. A. Shkalikov, “On the eigenvalue accumulation of Sturm–Liouville problems depending nonlinearly on the spectral parameter”, Math. Nachr., 189:1 (1998), 157–170 | DOI | MR | Zbl

[13] H. Hochstadt, “Asymptotic estimates for the Sturm–Liouville spectrum”, Comm. Pure Appl. Math., 14:4 (1961), 749–764 | DOI | MR | Zbl

[14] Ch. T. Fulton, “Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions”, Proc. Roy. Soc. Edinburgh Sect. A, 77:3-4 (1977), 293–308 | DOI | MR | Zbl

[15] P. A. Binding, P. J. Browne, K. Seddighi, “Sturm–Liouville problems with eigenparameter dependent boundary conditions”, Proc. Edinburgh Math. Soc. (2), 37:1 (1994), 57–72 | DOI | MR | Zbl

[16] J. Walter, “Regular eigenvalue problems with eigenvalue parameter in the boundary condition”, Math. Z., 133:4 (1973), 301–312 | DOI | MR | Zbl

[17] H. Coşkun, N. Bayram, “Asymptotics of eigenvalues for regular Sturm–Liouville problems with eigenvalue parameter in the boundary condition”, J. Math. Anal. Appl., 306:2 (2005), 548–566 | DOI | MR | Zbl

[18] N. Yu. Kapustin, “Oscillation properties of solutions to a nonselfadjoint spectral problem with spectral parameter in the boundary condition”, Differ. Equ., 35:8 (1999), 1031–1034 | MR | Zbl

[19] N. B. Kerimov, Kh. R. Mamedov, “On one boundary value problem with a spectral parameter in the boundary conditions”, Siberian Math. J., 40:2 (1999), 281–290 | DOI | MR | Zbl

[20] N. Yu. Kapustin, E. I. Moiseev, “The basis property in $L_p$ of the systems of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition”, Differ. Equ., 36:10 (2000), 1498–1501 | DOI | MR | Zbl

[21] Z. S. Aliev, A. A. Dun'yamalieva, “Basis properties of root functions of the Sturm–Liouville problem with a spectral parameter in the boundary conditions”, Dokl. Math., 88:1 (2013), 441–445 | DOI | MR | Zbl

[22] N. B. Kerimov, R. G. Poladov, “Basis properties of the system of eigenfunctions in the Sturm–Liouville problem with a spectral parameter in the boundary conditions”, Dokl. Math., 85:1 (2012), 8–13 | DOI | MR | Zbl

[23] D. B. Marchenkov, “Basis property in $L_p(0,1)$ of the system of eigenfunctions corresponding to a problem with a spectral parameter in the boundary condition”, Differ. Equ., 42:6 (2006), 905–908 | DOI | MR | Zbl

[24] F. G. Tricomi, Differential equations, Hafner Publishing Co., New York; Blackie Son Ltd., London, 1961, x+273 pp. | MR | MR | Zbl | Zbl

[25] R. Courant, D. Hilbert, Methoden der mathematischen Physik, v. I, Grundlehren Math. Wiss., 12, 2. verb. Aufl., Julius Springer, Berlin, 1931, xiv+469 pp. | MR | MR | Zbl | Zbl

[26] V. P. Mikhaĭlov, Partial differential equations, Mir, Moscow, 1978, 397 pp. | MR | MR

[27] S. V. Kurochkin, “Existence conditions of negative eigenvalues in the regular Sturm–Liouville boundary value problem and explicit expressions for their number”, Comput. Math. Math. Phys., 58:12 (2018), 1937–1947 | DOI | DOI | MR | Zbl

[28] D. V. Valovik, “Propagation of electromagnetic waves in an open planar dielectric waveguide filled with an nonlinear medium I: TE waves”, Comput. Math. Math. Phys., 59:6 (2019), 958–977 | DOI | DOI | MR | Zbl

[29] D. V. Valovik, “Propagation of electromagnetic waves in an open planar dielectric waveguide filled with a nonlinear medium II: TM waves”, Comput. Math. Math. Phys., 60:3 (2020), 427–447 | DOI | DOI | MR

[30] D. V. Valovik, “Study of a nonlinear eigenvalue problem by the integral characteristic equation method”, Differ. Equ., 56:2 (2020), 171–184 | DOI | DOI | Zbl

[31] D. V. Valovik, “Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem”, Nonlinear Anal. Real World Appl., 20 (2014), 52–58 | DOI | MR | Zbl