×

zbMATH — the first resource for mathematics

An evaluation of powers of the negative spectrum of Schrödinger operator equation with a singularity at zero. (English) Zbl 1378.34082
Summary: In this study, we investigate the discreteness and finiteness of the negative spectrum of the differential operator \(L\) in the Hilbert space \(L_{2} (H, [0,\infty)) \), defined by \[ Ly=- \frac{d^{2} y}{d x^{2}} + \frac{A(A+I)}{x^{2}} y-Q(x)y \] under the boundary condition \(y (0) =0\).
In the case when the negative spectrum is finite, we obtain an evaluation for the sums of powers of the absolute values of negative eigenvalues. The obtained result is applied to a class of equations of mathematical physics.
MSC:
34G10 Linear differential equations in abstract spaces
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kostyuchenko, AG; Levitan, BM, Asymptotic behavior of the eigenvalues of the Sturm-Liouville operator problem, Funct. Anal. Appl., 1, 75-83, (1967) · Zbl 0168.12302
[2] Gorbachuk, VI; Gorbachuk, ML, Classes of boundary-value problems for the Sturm- Liouville equation with an operator potential, Ukr. Math. J., 24, 241-250, (1972) · Zbl 0258.34020
[3] Gorbachuk, VI; Gorbachuk, ML, Some problems of spectral theory of elliptic type differential equations in the space of vector-functions on a finite interval, Ukr. Math. J., 28, 12-26, (1976) · Zbl 0344.34017
[4] Otelbaev, M, On titchmarsh’s method of resolvent estimation, Dokl. Akad. Nauk SSSR, 211, 787-790, (1973)
[5] Solomyak, MZ, Asymptotics of the spectrum of the Schrödinger operator with nonregular homogeneous potential, Math. USSR Sb., 55, 19-37, (1986) · Zbl 0657.35099
[6] Maksudov, FG; Bayramoglu, M; Adiguzelov, EE, On asymptotics of spectrum and trace of high order differential operator with operator coefficients, Doğa Türk Mat. Derg., 17, 113-128, (1993)
[7] Vladimirov, AA, Estimates of the number of eigenvalues of self-adjoınt operator functions, Math. Notes, 74, 794-802, (2003) · Zbl 1131.47014
[8] Aslanova, NM, Trace formula of one boundary value problem for operator Sturm-Liouville equation, Sib. Math. J., 49, 1207-1215, (2008) · Zbl 1224.34277
[9] Bayramoglu, M, Aslanova, NM: Study of the asymptotic eigenvalue distribution and trace formula of a second order operator differential equation. Boundary Value Problems 2012(8) (2012). doi:10.1186/1687-2770-2011-7
[10] Gesztesy, F; Weikard, R; Zinchenko, M, On spectral theory for Schrödinger operators with operator-valued potentials, J. Differ. Equ., 255, 1784-1827, (2013) · Zbl 1291.34139
[11] Hashimoglu, I, Asymptotics of the number of eigenvalues of one-term second-order operator equations, Adv. Differ. Equ., 2015, (2015) · Zbl 1422.34190
[12] Birman, MS, On the spectrum of singular boundary-value problems, Mat. Sb. (N.S.), 55, 125-174, (1961)
[13] Skachek, BY, The asymptotic behavior of the negative part of the spectrum of one-dimensional differential operators, 96-109, (1963), Kiev
[14] Skachek, BY, On asymptotics of the negative part of the spectrum of multidimensional differential operators, Dokl. Akad. Nauk Ukr. SSR, Ser. A. Fiz.-Mat. Teh. Nauki, 1, 14-17, (1964)
[15] Rozenblyum, GV, An asymptotic of the negative discrete spectrum of the Schrödinger operator, Math. Notes, 21, 222-227, (1977) · Zbl 0399.35083
[16] Birman, MS; Solomyak, MZ, Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations. estimates and asymptotics for discrete spectra of integral and differential equations, No. 7, 1-55, (1991), Providence
[17] Laptev, A, Asymptotics of the negative discrete spectrum of a class of Schrödinger operators with large coupling constant, Proc. Am. Math. Soc., 119, 481-488, (1993) · Zbl 0795.35071
[18] Birman, MS; Laptev, A, The negative discrete spectrum of a two-dimensional Schrödinger operator, Commun. Pure Appl. Math., XLIX, 967-997, (1996) · Zbl 0864.35080
[19] Laptev, A; Safronov, O, The negative discrete spectrum of a class of two-dimensional Schrödinger operators with magnetic fields, Asymptot. Anal., 41, 107-117, (2005) · Zbl 1083.35070
[20] Laptev, A; Solomyak, M, On the negative spectrum of the two-dimensional Schrödinger operator with radial potential, Commun. Math. Phys., 314, 229-241, (2012) · Zbl 1253.35037
[21] Gasymov, MG; Zhikov, VV; Levitan, BM, Conditions for discreteness and finiteness of the negative spectrum of schrödinger’s operator equation, Math. Notes, 2, 813-817, (1967) · Zbl 0159.19801
[22] Yafaev, DR, Negative spectrum of the Schrödinger operational equation, Math. Notes, 7, 451-457, (1970) · Zbl 0206.44603
[23] Adygezalov, AA, On asymptotics of negative part of the spectrum of Sturm-Liouville operator problem, Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Teh. Mat. Nauk, 6, 8-12, (1980) · Zbl 0459.47042
[24] Adigözelov, E; Bakşi, Ö; Bayramov, A, The asymptotic behaviour of the negative part of the spectrum of Sturm-Liouville operator with the operator coefficient which has singularity, Int. J. Differ. Equ. Appl., 6, 315-329, (2002) · Zbl 1048.34136
[25] Adigüzelov, E; Şengül, S; Akyol, M, The asymptotic formulas for the sum of squares of negative eigenvalues of the singular Sturm-Liouville operator, Int. J. Contemp. Math. Sci., 1, 341-358, (2006) · Zbl 1165.34426
[26] Aslanov, HI; Gadirli, NA, On asymptotic distribution of negative eigenvalues of second order equation with operator coefficients on a semi-axis, Trans. Azerb. Natl. Acad. Sci., Ser. Phys.-Tech. Math. Sci., 37, 44-52, (2017) · Zbl 1377.34105
[27] Bayramoglu, M, On the expansion in eigenfunctions of a non-selfadjoint Schrödinger operator in certain unbounded domains, Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Teh. Mat. Nauk, 3-4, 162-168, (1967)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.