Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
DOI: 10.46698/s6842-7321-6945-l
Regularized Trace of a Multipoint Boundary Value Problem with a Discontinuous Weight Function
Mitrokhin, S. I.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 1.
Abstract: The article proposes a method for calculating the regularized trace for a differential operator with a piecewise smooth potential and multipoint boundary conditions. The weight function of the differential operator is discontinuous. Using the Naimark method on the sections of the continuity of the potential and the weight function for large values of the spectral parameter, the asymptotics of solutions of differential equations defining the operator under study is obtained. The asymptotics of the solutions enables us to study the conditions of "conjugation" at the point of discontinuity of the coefficients. The necessity of the conditions of "conjugation" follows from physical considerations. The studied boundary value problems arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The multipoint boundary conditions defining the operator are studied. The technically difficult part of the study was successfully completed - the indicator diagram of the equation whose roots are the eigenvalues of the operator was studied. The asymptotics of the eigenvalues of the operator is calculated. Using the asymptotics of the eigenvalues by the Lidsky-Sadovnichy method, the first regularized trace of the differential operator is calculated.
Keywords: differential operator, spectral parameter, multipoint boundary conditions, eigenvalues, indicator diagram, regularized trace of the operator
For citation: Mitrokhin, S. I. Regularized Trace of a Multipoint Boundary Value Problem with a Discontinuous Weight Function, Vladikavkaz Math. J., 2022, vol. 24, no. 1, pp. 65-86 (in Russian). DOI 10.46698/s6842-7321-6945-l
1. Il'in, V. A. Convergence of Eigenfunction Expansions at Points of Discontinuity
of the Coefficients of a Differential Operator, Mathematical Notes of the Academy
of Sciences of the USSR, 1977, vol. 22, no. 5, pp. 679-698. DOI: 10.1007/BF01098352.
2. Budak, A. B. Expansion in Eigenfunctions of a Fourth-Order Differential Operator with a Piecewise
Constant Leading Coefficient, Differentsial'nye Uravneniya [Differential Equations],
1980, vol. 16, no. 9, pp. 1545-1558 (in Russian).
3. Gottlieb, H. P. W. Iso-Spectral Operators: Some Model Examples
with Discontinuous Coefficients, Journal of Mathematical Analysis and Applications,
1988, vol. 132, pp. 123-137. DOI: 10.1016/0022-247X(88)90048-0.
4. Belabassi, Yu. Regularized Trace of a Multipoint Problem, Bulletin of the Moscow
University. Ser. Mathematics, Mechanics, 1981, no. 2, pp. 35-41 (in Russian).
5. Gel'fand, I. M. and Levitan, B. M. On a Simple Identity for the Eigenvalues
of a Second-Order Differential Operator, Doklady Akademii Nauk SSSR,
1953, vol. 88, no. 4, pp. 593-596 (in Russian).
6. Sadovnichy, V. A. The Trace of Ordinary Differential Operators of High Order,
Mathematics of the USSR-Sbornik, 1967, vol. 1(2), no. 2, pp. 263-288.
DOI: 10.1070/SM1967v001n02ABEH001979.
7. Lidskii, V. B. and Sadovnichii, V. A. Regularized Sums of Zeros of a Class
of Entire Functions, Functional Analysis and Its Applications, 1967, vol. 1,
no. 2, pp. 133-139. DOI: 10.1007/BF01076085.
8. Mitrokhin, S. I. Regularized Trace Formulas for Second-Order Differential
Operators with Discontinuous Coefficients, Vestnik Moskovskogo Universiteta.
Seriya 1. Matematika. Mekhanika, 1986, no. 6, pp. 3-6 (in Russian).
9. Mitrokhin, S. I. Trace Formulas for a Boundary Value Problem with
a Functional-Differential Equation with a Discontinuous Coefficient,
Differentsial'nye Uravneniya [Differential Equations], 1986, vol. 22, no. 6, pp. 927-931 (in Russian).
10. Mitrokhin, S. I. Spectral Properties of Differential Operators with Discontinuous
Coefficients, Differentsial'nye Uravneniya [Differential Equations], 1992, vol. 28, no. 3, pp. 530-532 (in Russian).
11. Mitrokhin, S. I. On Some Spectral Properties of Second-Order Differential
Operators with a Discontinuous Weight Function, Doklady Akademii Nauk [Reports of the Academy of
Sciences], 1997, vol. 356, no. 1, pp. 13-15 (in Russian).
12. Mitrokhin, S. I.
Multipoint Differential Operators: "Splitting" of Multiples in the main Eigenvalues,
Izvestiya Saratov University. New Series. Series: Mathematics. Mechanics.
Computer Science, 2017, vol. 17, no. 1, pp. 5-8 (in Russian).
DOI: 10.18500/1816-9791-2017-17-1-5-18.
13. Naimark, M. A. Lineynye differentsial'nye operatory
[Linear Differential Operators], Moscow, Nauka, 1969, 528 p. (in Russian).
14. Mitrokhin, S. I. Asymptotics of the Eigenvalues of a Differential
Operator with an Alternating Weight Function,
Russian Mathematics, 2018, vol. 62, no. 6, pp. 27-42.
DOI: 10.3103/s1066369x1806004x.
15. Bellman, R. and Cook, K. L. Differentsial'no-raznostnye uravneniya
[Differential-Difference Equations], Moscow, Mir, 1967, 548 p. (in Russian).
16. Mitrokhin, S. I. Asymptotic Behavior of the Spectrum
of a Multipoint Differential Operator with an Integrable Potential,
Journal of Mathematical Sciences, 2018, vol. 231, no. 2, pp. 243-254.
DOI: 10.1007/s10958-018-3819-8.
17. Sadovnichiy, V. A. Teoriya operatorov [Theory of Operators],
Moscow, Drofa, 2001, 384 p. (in Russian).
