On the study of the spectrum of a functional-differential operator with a summable potential

S. I. Mitrokhin

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 2, pp. 38-57

Résumé

The paper deals with a functional-differential operator of the eighth order with a summable potential. The boundary conditions are separated. Functional-differential operators of this kind arise in the study of vibrations of beams and bridges made up of materials of different density. To solve the functional-differential equation that defines a differential operator, the method of variation of constants is applied. The solution of the initial functional-differential equation is reduced to the solution of the Volterra integral equation. The resulting Volterra integral equation is solved by Picard's method of successive approximations. As a result of the investigation of the integral equation, asymptotic formulas and estimates for the solutions of the functional-differential equation that defines the differential operator are obtained. For large values of the spectral parameter, the asymptotics of the solutions of the differential equation defining the differential operator is derived. Similar to the asymptotic estimates of solutions of the differential operator of the second order with smooth and piecewise smooth coefficients, asymptotic estimates of solutions of the initial functional differential equation are established. The obtained asymptotic formulas are used to study the boundary conditions. As a result, we come to the study of the roots of a function represented as a determinant of the eighth order. To find the roots of this function, it is necessary to study the indicator diagram. The roots of the eigenvalue equation are in eight sectors of an infinitesimal solution, defined by the indicator diagram. The behavior of the roots of this equation in each of the sectors of the indicator diagram and the asymptotics of the eigenvalues of the differential operator under study are studied.

Bibliographie

[1] Naimark M. A., Linear Differential Operators, Nauka, M., 1969, 528 pp. (in Russian)

[2] Lidskyi V. B., Sadovnichiy V. A., “Asymptotic Formulas for the Zeros of a Class of Entire Functions”, Mathematics of the USSR-Sbornik, 4:4 (1968), 519–527 | DOI | MR | Zbl

[3] Sadovnichiy V. A., “About Traces of Ordinary Differential Operators of the Highest Orders”, Mathematics of the USSR-Sbornik, 1:2 (1967), 263–288 | DOI | MR | Zbl

[4] Lidskyi V. B., Sadovnichiy V. A., “The Trace of Ordinary Differential Operators of High Order”, Mathematics of the USSR-Sbornik, 1:2 (1967), 263–288 | DOI | MR | MR | Zbl

[5] Mitrokhin S. I., “About Formulas of Regularized Traces for Differential Operators of the Second Order with Discontinuous Coefficients”, Vestnik MGU. Ser. Mathematics, Mechanics, 1986, no. 6, 3–6 (in Russian) | MR

[6] Mitrokhin S. I., “About Spectral Properties of Differential Operators with Discontinuous Coefficients”, Differential Equations, 28:3 (1992), 530–532 (in Russian) | MR | Zbl

[7] Mitrokhin S. I., “About Some Spectral Properties of Differential Operators of the Second Order with Discontinuous Weight Function”, Reports of the Russian Academy of Sciences, 356:1 (1997), 13–15 (in Russian) | MR | Zbl

[8] Martinovich M., “On a Boundary Value Problem for a Functional-Differential Equation”, Differential Equations, 18:2 (1982), 239–245 (in Russian) | MR | Zbl

[9] Martinovich M., “The Zeta-Function and Trace Formulas for one Boundary-Value Problem with a Functional-Differential Equation”, Differential Equations, 18:3 (1982), 537–540 (in Russian) | MR

[10] Mitrokhin S. I., “On the Trace Formulas for a Boundary Value Problem with a Functional-Differential Equation with a Discontinuous Coefficient”, Differential Equations, 22:6 (1986), 927–931 (in Russian) | MR | Zbl

[11] Vinokurov V. A., 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”, Izvestiya: Mathematics, 64:4 (2000), 695–754 | DOI | DOI | MR | Zbl

[12] Savchuk A. M., Shkalikov A. A., “Sturm–Liouville Operators with Singular Potentials”, Mathematical Notes, 66:6 (1999), 741–753 | DOI | DOI | MR | Zbl

[13] Savchuk A. M., “First-Order Regularised Trace of the Sturm–Liouville Operator with $\delta$-Potential”, Russian Mathematical Surveys, 55:6 (2000), 1168–1169 | DOI | DOI | MR | Zbl

[14] Mitrokhin S. I., “The Asymptotics of the Eigenvalues of a Fourth Order Differential Operator with Summable Coefficients”, Moscow University Mathematics Bulletin, 64:3 (2009), 102–104 | DOI | MR | Zbl

[15] Mitrokhin S. I., “On the Spectral Properties of Odd-Order Differential Operators with Integrable Potential”, Differential Equations, 47:12 (2011), 1833–1836 | DOI | MR | Zbl

[16] Mitrokhin S. I., “Study of Differential Operator with Summable Potential with Discontinuous Weight Function”, Ufa Mathematical J., 9:4 (2017), 72–84 | DOI | MR

[17] Mitrokhin S. I., “A Periodic Boundary Value Problem for a Fourth Order Differential Operator with a Summable Potential”, Vladikavkaz Math. J., 19:1 (2017), 35–49 | DOI | MR

[18] Bellman R., Cooke K. L., Differential-Difference Equations, Academic Press, New York, 1963, 482 pp. | MR | Zbl

[19] Mitrokhin S. I., “Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function”, Russian Mathematics (Izvestiya VUZ. Matematika), 62:6 (2018), 27–42 | DOI | MR | Zbl

[20] Sadovnichii V. A., Lubishkin V. A., Belabassi Yu., “On Regularized Sums of Roots of an Entire Function of a Certain Class”, Dokl. Akad. Nauk SSSR, 254:6 (1980), 1346–1348 (in Russian) | MR

[21] Sadovnichii V. A., Lubishkin V. A., “Some New Results of the Theory of Regularized Traces of Differential Operators”, Differential Equations, 18:1 (1982), 109–116 (in Russian) | MR