×

The Levinson-type formula for a class of Sturm-Liouville equation. (English) Zbl 1488.34483

Summary: The boundary value problem \[ -\psi^{\prime \prime}+q(x)\psi=\lambda^2 \psi, \quad 0<x<\infty, \] \[ \psi^\prime(0) - (\alpha_0+\alpha_1\lambda) \psi(0)=0 \] is considered, where \(\lambda\) is a spectral parameter, \( q(x)\) is real-valued function such that \[ \int\limits_0^{\infty}(1+x)|q(x)|\,dx<\infty \] with \(\alpha_0, \alpha_1\geq 0\) (\(\alpha_0,\alpha_1\in \mathbb{R} \)).
In this paper, for the above-mentioned boundary value problem, the scattering data is considered and the characteristics properties (such as continuity of the scattering function \(S(\lambda)\) and giving the Levinson-type formula) of this data are studied.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
34B07 Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34B24 Sturm-Liouville theory
34B09 Boundary eigenvalue problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 1.S. A. Alimov:A. N. Tihonov’s works on inverse problems for the Sturm-Liouville equation. Matematicheskikh Nauk,31(6) (1976) 84-88; English Trans:English translation in Russian Mathematical Surveys Uspekhi 31 (6) (1976) 87-92; · Zbl 0367.34016
[2] 2.D. S. Cohen:An integral transform associated with boundary conditions containing an eigenvalue parameter. SIAM Journal on Applied Mathematics,14(5) (1966) 1164- 1175. · Zbl 0237.34040
[3] 3.C. T. Fulton:Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditions. Proceedings of the Royal Society of Edinburgh. Section A,87 (1-2) (1980) 1-34. · Zbl 0458.34013
[4] 4.C. T. Fulton:Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proceedings of the Royal Society of Edinburgh Section A.77(3-4) (1977) 293-308. · Zbl 0376.34008
[5] 5.G. Freiling, V. A. Yurko:Inverse Sturm-Liouville problems and their applications. NOVA Science Publishers, New York, 2001. · Zbl 1037.34005
[6] 6.B. M. Levitan:On the solution of the inverse problem of quantum scattering theory. Mathematical Notes,17(4) (1975) 611-624. · Zbl 0322.34016
[7] 7.B. M. Levitan:Inverse Sturm-Liouville Problems. VSP, Zeist:The Netherlands, 1987. · Zbl 0749.34001
[8] 8.V. E. Lyantse:On a differential equations with spectral singularities I. Matematicheskii Sbornik,106(4) (1964) 521-561. · Zbl 0127.03904
[9] 9.Kh. R. Mamedov:On the inverse problem for Sturm-Liouville operator with a nonlinear spectral parameter in the boundary condition. Journal of Korean Mathematical Society,46(6) (2009) 1243-1254. · Zbl 1191.34013
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.