On the studying the spectrum of differential operators’ family whose potentials converge to the Dirac delta function 

Sergey I. Mitrokhin, Candidate of physical and mathematical sciences, associate professor, senior staff scientist of Research Computing Center, Lomonosov Moscow State University (building 6, 1 Leninskiye Gory, Moscow, Russia), E-mail: mitrokhin-sergey@yandex.ru 

517.9 

10.21685/2072-3040-2021-1-3 

Background. The paper proposes a new method for studying differential operators with discontinuous coefficients. We study a sequence of differential operators of high even order whose potentials converge to the Dirac delta function. It is assumed that the operator’s potential is a piecewise-summable function on the segment of the operator task. At the points of discontinuity of the potential, the fulfillment of the “gluing” conditions is required to correctly determine the solutions of the corresponding differential equations. The spectral properties of differential operators defined on a finite segment with one of the types of separated boundary conditions are investigated. For large values of the spectral parameter, the Naimark method is used to obtain the asymptotics of the fundamental system of solutions of the corresponding differential equations. With the help of this asymptotics, the conditions for “gluing” the considered differential operator are studied. Then the boundary conditions of the operator under study are studied. As a result, we derive an eigenvalue equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular hexadecimal, is investigated. In various sectors of the indicator diagram, the method of successive Picard approximations has been used to find the eigenvalue asymptotics of the studied differential operators. In the limiting case, the found asymptotics of the eigenvalues tends to the asymptotics of the eigenvalues of an operator whose potential is the Dirac delta function.
Materials and methods. The asymptotics of the solutions’ fundamental system of differential equations with summable potentials for large values of the spectral parameter is obtained by the generalized Naimark method. To find the roots of the equation for the eigenvalues of the operator under study, the Bellman-cook method is used to study the indicator diagram, which is a regular hexadecagon. The asymptotics of the eigenvalues of the studied differential operators in various sectors of the indicator diagram are found by the method of successive Picard approximations.
Results. The spectrum of a previously unexplored family of differential operators of high even order whose potentials converge to the Dirac Delta function is studied. Taking into account the “gluing” condition at the points of discontinuity of the potentials, it is proved that the eigenvalue equation is a quasi-polynomial, the roots of which can be found by the Bellman-Cook method. Similar results can be obtained for other types of separated boundary conditions.
Conclusions. The new results obtained on the asymptotics of the spectrum of a family of differential operators can be applied to the study of the basis property of the eigenfunctions of similar operators, the study of the Green’s function, and the calculation of formulas for regularized traces of operators whose sequence of potentials converges to the Dirac delta function. The delta potential method is used in physics to study short-range impurities and defects in various systems. In atomic and nuclear physics, the model of point potentials is very popular; this confirms the need to study operators with delta potentials. 

differential operator with discontinuous coefficients, asymptotics of differential equation solutions, piecewise-summable potential, Dirac delta function, asymptotics of eigenvalues, spectrum of the operator 

1. Il'in V.A. On the convergence of the expansion in terms of eigenfunctions at points of discontinuity of the coefficients of a differential operator. Matematicheskie zametki = Mathematical notes. 1977;22(5):698–723. (In Russ.)
2. Mitrokhin S.I. Regularized trace formulas for second-order differential operators with discontinuous coefficients. Vestnik Moskovskogo universiteta. Ser. 1, Matematika. Mekhanika = Bulletin of Moscow University. Series 1, Mathematics, Mechanics. 1986;6:3–6. (In Russ.)
3. 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;356(1):13–15. (In Russ.)
4. 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;22(6):927–931. (In Russ.)
5. Mitrokhin S.I. Spectral properties of differential operators with discontinuous coefficients. Differentsial'nye uravneniya = Differential equations. 1992;28(3):530–532. (In Russ.)
6. Tikhonov A.N., Samarskiy A.A. Uravneniya matematicheskoy fiziki = Equations of mathematical physics. Moscow: Nauka, 1977:736. (In Russ.)
7. Savchuk A.M., Shkalikov A.A. Sturm – Liouville operators with singular potentials. Matematicheskie zametki = Mathematical notes.1999;66(6):897–912. (In Russ.)
8. Vinokurov V.A., Sadovnichiy V.A. Asymptotics of any order of eigenvalues and eigenfunctions of the Sturm – Liouville boundary value problem on an interval with summable potential. Differentsial'nye uravneniya = Differential equations. 1998;34(10):1423–1426. (In Russ.)
9. Mitrokhin S.I. Spectral properties of a fourth-order differential operator with summable coefficients. Trudy Matematicheskogo instituta im. V. A. Steklova = Proceedings of Steklov Mathematical Institute. 2010;270:188–197. (In Russ.)
10. Mitrokhin S.I. Spectral properties of a certain differential operator with summable coefficients with lagging argument. Ufimskiy matematicheskiy zhurnal = Ufa mathematical journal. 2011;3(4):95–115. (In Russ.)
11. Mitrokhin S.I. Spectral properties of a differential operator with an integrable potential and a smooth weight function. Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya = Bulletin of Samara University. Natural sciences’ series. 2008;8:172–187. (In Russ.)
12. Mitrokhin S.I. Spectral properties of boundary value problems for a functional differential equation with summable coefficients. Differentsial'nye uravneniya = Differential equations. 2010;46(8):1085–1093. (In Russ.)
13. Mitrokhin S.I. Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with summable potential. Trudy Instituta matematiki i mekhaniki UrO RAN = Proceedings of Institute of mathematics and mechanics of Ural department of the Russian Academy of Sciences. 2019;25(1):136–149. (In Russ.)
14. Vinokurov V.A., Sadovnichiy V.A. Asymptotics of eigenvalues and eigenfunctions and the trace formula for potential containing δ -functions. Differentsial'nye uravneniya = Differential equations. 2002;38(6):735–751. (In Russ.)
15. Savchuk A.M. Regularized trace of the first order of the Sturm – Liouville operator with δ -potential. Uspekhi matematicheskikh nauk = Advances in mathematical sciences. 2000;55(6):155–156. (In Russ.)
16. Savchuk A.M., Shkalikov A.A. Trace formula for Sturm-Liouville operators with singular potentials. Matematicheskie zametki = Mathematical notes. 2001;69(3):427–442. (In Russ.)
17. Berezin F.A., Fadeev L.D. Remark on the Schrödinger equation with singular potential. Doklady Akademii nauk = Reports of the Academy of Sciences. 1961;137(5):1011–1014. (In Russ.)
18. Borisov D.I. On gaps in the Laplacian spectrum in a band with periodic delta interaction. Trudy Instituta matematiki i mekhaniki UrO RAN = Proceedings of Institute of mathematics and mechanics of Ural department of the Russian Academy of Sciences. 2018;24(2):46–53. (In Russ.)
19. Mitrokhin S.I. Asymptotics of the eigenvalues of a differential operator with an alternating weight function. Izvestiya vuzov. Ser.: Matematika = University proceedings. Series: Mathematics. 2018;6:31–47. (In Russ.)
20. Mitrokhin S.I. Spectral properties of differential operators of even order with summable coefficients. Vestnik Moskovskogo universiteta. Ser. 1, Matematika. Mekhanika = Bulletin Moscow University. Series 1, Mathematics. Mechanics. 2017;4:3–15. (In Russ.)
21. Naymark M.A. Lineynye differentsial'nye operatory = Linear differential operators. Moscow: Nauka, 1969:528. (In Russ.)
22. Bellman R., Kuk K.L. Differentsial'no-raznostnye uravneniya = Differential-difference equations. Moscow: Mir, 1967:548. (In Russ.)
23. Sadovnichiy V.A., Lyubishkin V.A. On some new results in the theory of regularized traces of differential operators. Differentsial'nye uravneniya = Differential equations. 1982;18(1):109–116. (In Russ.)