Geodesic
On the spectrum of the multipoint boundary value problem for an odd order differential operator with summable potential
Matematičeskie zametki SVFU, Tome 24 (2017) no. 1, pp. 26-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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A boundary value problem for an odd order differential operator with multipoint boundary conditions is studied. The interior points in which boundary conditions are set can divide the segment on which the operator is considered into the incommensurable parts. The potential of the differential operator is a function Lebesgue integrable at the segment on which the operator is considered. We study the asymptotic behaviour of the solutions to the corresponding differential equation for large values of the spectral parameter. The equation on the eigenvalues of the operator is received. We obtain the indicator diagram of that equation. The asymptotic behavior of the eigenvalues in all sectors of the indicator diagram is studied.
Keywords: differential operator, summable potential, boundary value problem, multipoint boundary conditions, indicator diagram, the asymptotics of the eigenvalues. 
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S. I. Mitrokhin. On the spectrum of the multipoint boundary value problem for an odd order differential operator with summable potential. Matematičeskie zametki SVFU, Tome 24 (2017) no. 1, pp. 26-42. http://geodesic.mathdoc.fr/item/SVFU_2017_24_1_a3/

[1] Konyaev Yu. A., “Initial-value and multipoint boundary-value problems for nonautonomous systems with polynomial matrix and their applications”, J. Math. Sci., 170:3 (2010), 332-339 | DOI | MR | Zbl

[2] Krall A. M., “Boundary value problems with interior point boundary conditions”, Pac. J. Math., 29 (1969), 161–166 | DOI | MR | Zbl

[3] Mustafokulov R. and Soliev S. K, “About one multipoint boundary value problem”, Dokl. Akad. Nauk Resp. Tadzh., 57:10 (2014), 725–731

[4] Pokorny Yu. V. and Lazarev K. P, “Some oscillation theorems for many-point problems”, Differ. Equations, 23:4 (1987), 658–670 | MR | Zbl

[5] Klimov V. S., “A multipoint boundary value problem for a system of differential equations”, Differ. Uravn., 5:8 (1969), 1532–1534 | MR | Zbl

[6] Levin A. Yu., “Differential properties of the Green’s function in a many point boundary value problem”, Sov. Math. Dokl., 2 (1961), 154–157 | MR | Zbl

[7] Kodlyuk T. I. and Mikhailets V. A., “Multipoint boundary value problems with parameter in Sobolev space”, Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, 2012, no. 11, 15–19 | MR | Zbl

[8] Haseinov K. A., Multipoint and conjugate boundary value problems, their applications, Nauka, Moscow, 2006

[9] Belabbasi Yu., “Regularized trace of a several point problem”, Vestn. Mosk. Univ., Ser. I, 1981, no. 2, 35–41

[10] Belabbasi Yu., ““Regularized traces of many point problems for high-order ordinary differential operators”, Differ. Equations, 19:6 (1983), 938-944 | MR

[11] Mitrokhin S. I, “Formulas for the regularized traces of the second order differential operators with discontinuous coefficients”, Mosc. Univ. Math. Bull., 41:6 (1986), 1–5 | MR | Zbl

[12] Mitrokhin S. I., “Spectral properties of second-order differential operators with a discontinuous positive weight function”, Dokl. Math., 56:2 (1997), 652–654 | MR | Zbl

[13] Yurko V. A., “About differential operators of the highest orders with a singularity inside the interval”, Math. Notes, 71:1 (2002), 152–156 | DOI | MR | Zbl

[14] Mitrokhin S. I., “On the ’splitting’ in multiplicity in principal eigenvalues of multipoint boundary value problems”, Russ. Math., 41:3 (1997), 37–42 | MR | Zbl

[15] Vinokurov V. A. and Sadovnichii V. A., “Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary value problem on a segment with a summable potential”, Izv. Math., 64:4 (2000), 695–754 | DOI | MR | Zbl

[16] Mitrokhin S. I., “The asymptotics of the eigenvalues of a fourth-order differential operator with summable coefficients”, Mosc. Univ. Math. Bull., 64:3 (2009), 102–104 | DOI | MR | Zbl

[17] Mitrokhin S. I., “On spectral properties of a differential operator with summable coefficients and retarded argument”, Ufim. Math. J., 3:4 (2011), 95–115 | MR | Zbl

[18] Mitrokhin S. I., ““On the study of the spectrum of the boundary value problem for a fifth-order differential operator with summable potential”, Mat. Zamet. SVFU, 23:2 (2016), 78–89

[19] Bellman R. and Cooke K. L., Differential-Difference Equations, Mir, Moscow, 1967 | MR

[20] Mitrokhin S. I., “On the ’splitting’ in multiplicity in principal eigenvalues of multipoint boundary value problems”, Mathematical Modelling and Boundary Problems, v. 3, 2008, 130–133

[21] Lidskii V. B. and Sadovnichii V. A., “Asymptotic formulas for roots of one class of the entire functions”, Mat. Sb., N. Ser., 75:117 (1968), 558–566 | MR | Zbl

[22] Sadovnichii V. A., Lyubishkin V. A., and Belabbasi Yu., “On regularized sums of roots of an entire function of a certain class”, Sov. Math., Dokl., 22 (1980), 613–616 | MR