Geodesic
On the asymptotic behavior of the spectrum of a~sixth-order differential operator, whose potential is~the~delta function
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 3, pp. 280-305.

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In this paper we propose a new method for studying differential operators with discontinuous coefficients. We consider a sequence of sixth-order differential operators with piecewise-smooth coefficients. The limit of the sequence of these operators’ potentials is the Dirac delta function. The boundary conditions are separated. To correctly determine solutions of differential equations with discontinuous coefficients at the points of discontinuity, “gluing” conditions are required. Asymptotic solutions were written out for large values of the spectral parameter, with the help of them the “gluing” conditions were studied and the boundary conditions were investigated. As a result, we derive an eigenvalues equation for the operator under study, which is an entire function. The indicator diagram of the eigenvalues equation, which is a regular hexagon, is investigated. In various sectors of the indicator diagram, the method of successive approximations has been used to find the eigenvalues asymptotics of the studied differential operators. The limit of the asymptotic of the spectrum determines the spectrum of the sixth-order operator, whose potential is the delta function.
Keywords: differential operator with discontinuous coefficients, asymptotic behavior of solutions, piecewise-smooth potential, Dirac delta function, asymptotic behavior of eigenvalues, spectrum of an operator. 
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S. I. Mitrokhin. On the asymptotic behavior of the spectrum of a~sixth-order differential operator, whose potential is~the~delta function. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 3, pp. 280-305. http://geodesic.mathdoc.fr/item/SVMO_2020_22_3_a2/

[1] F. A. Berezin, L. D. Faddeev, “Remarks on the Schrodinger equation with a singular potential”, Doklady Akademii nauk SSSR, 137:5 (1961), 1011–1014 (In Russ.) | MR | Zbl

[2] O. Z. Peng, X. Wang, J. Y. Zeng, “Analytic solution to the Schrodinger equation with a harmonic oscillator potential plus $\delta$-potential”, Sci. China, Ser. A., 34:10 (1991), 1215–1221 | MR | Zbl

[3] S. Fassari, G. Inglese, “On the spectrum of the harmonic oscillator with a $\delta$-type pertubation”, Helv. Phys. Acta., 67:6 (1994), 650–659 | MR | Zbl

[4] S. Fassari, G. Inglese, “On the spectrum of the harmonic oscillator with a $\delta$-type pertubation”, Annales Henri Poincare, 70:6 (1997), 858–865 | MR | Zbl

[5] V. D. Krevchik, R. V. Zaitsev, “Impurity absorption of light in structures with quantum dots”, Physics of the Solid State, 43:3 (2001), 504–507 (In Russ.) | DOI

[6] V. A. Il’in, “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, 22:5 (1977), 670–872

[7] S. I. Mitrokhin, “Regularized trace formulas for second-order differential operators with discontinuous coefficients”, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 22:6 (1986), 3–6 (In Russ.) | MR

[8] S. I. Mitrokhin, “On trace formulas for a boundary value problem with a functional differential equation with a discontinuous coefficient”, Differential Equations, 22:6 (1986), 927–931 (In Russ.) | MR | Zbl

[9] V. A. Il’in, “Necessary and sufficient conditions for being a Riesz basis of root vectors of second-order discontinuous operators”, Differential Equations, 22:12 (1986), 2059–2071 (In Russ.) | MR | MR | Zbl

[10] S. I. Mitrokhin, “On some spectral properties of second-order differential operators with discontinuous weight function”, Doklady Akademii nauk, 356:1 (1997), 13–15 (In Russ.) | MR | Zbl

[11] V. A. Vinokurov, V. A. Sadovnichy, “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 | MR | Zbl

[12] S. I. Mitrokhin, “Spectral properties of a fourth-order differential operator with integrable coefficients”, Proceedings of the Steklov Institute of Mathematics, 270 (2010), 184–193 | MR | Zbl

[13] S. I. Mitrokhin, “Spectral properties of boundary value problems for a functional differential equation with summable coefficients”, Differential Equations, 46:8 (2010), 1085–1093 (In Russ.) | DOI | MR | Zbl

[14] S. I. Mitrokhin, “Spectral properties of even-order differential operators with summable coefficients”, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 17:4, 3–15 (In Russ.) | MR

[15] S. I. Mitrokhin, “Asymptotics of the spectrum of a periodic boundary value problem for a differential operator with a summable potential”, Trudy instituta matematiki i mekhaniki URO RAN, 25:1 (2019), 136–149 (In Russ.) | DOI | MR

[16] A. M. Savchuk, A. A. Shkalikov, “Sturm-Liouville operators with singular potentials”, Mathematical Notes, 66:6 (1999), 741–753 | Zbl

[17] A. M. Savchuk, “First-order regularised trace of the Sturm-Liouville operator with $\delta$-potential”, Russian Mathematical Surveys, 55:6 (2000), 1168–1169 | MR | Zbl

[18] V. A. Vinokurov, V. A. Sadovnichy, “The asymptotics of eigenvalues and eigenfunctions and a trace formula for a potential with delta functions”, Differential Equations, 38:6 (2002), 772–789 | MR | Zbl

[19] A. M. Savchuk, A. A. Shkalikov, “Trace formula for Sturm-Liouville Operators with singular potentials”, Mathematical Notes, 69:3 (2001), 387–400 | MR | Zbl

[20] V. A. Geiler, V. A. Margulis, I. I. Chuchaev, “Potentials of zero radius and Carleman operators”, Siberian Mathematical Journal, 36:4 (1995), 714–726 | DOI | MR

[21] D. I. Borisov, “Gaps in the spectrum of the Laplacian in a strip with periodic delta interaction”, Proceedings of the Steklov Institute of Mathematics (Supplementary Issues), 305:suppl. 1 (2019), S16–S23

[22] M. A. Naimark, Linear differential operators, Nauka Publ., Moscow, 1969, 528 pp. (In Russ.) | MR | Zbl

[23] S. I. Mitrokhin, “Asymptotics of eigenvalues of differential operator with alternating weight function”, Russian Mathematics, 62:6 (2018), 27–42 | MR | Zbl

[24] R. Bellman, K. L. Cook, Differential-difference equations, Academic Press, London, 1963, 462 pp.

[25] V. A. Sadovnichyi, V. B. Lyubishkin, “On some new results in the theory of regularized traces of differential operators”, Differential Equations, 18:1 (1982), 109–116 (In Russ.) | MR