Abstract
We consider the Sturm-Liouville operator L(y) = −d 2 y/dx 2 + q(x)y in the space L 2[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L 2[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y [1](0) = 0, y(π) = 0] and Neumann conditions [y [1](0) = 0, y [1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form.
Similar content being viewed by others
References
Savchuk, A.M. and Shkalikov, A.A., Sturm-Liouville Operators with Singular Potentials, Mat. Zametki, 1999, vol. 66, no. 6, pp. 897–912.
Savchuk, A.M. and Shkalikov, A.A., Sturm-Liouville Operators with Distribution Potentials, Tr. Mosk. Mat. Obs., 2003, vol. 64, pp. 159–219.
Savchuk, A.M., On Eigenfunctions of the Sturm-Liouville Operator with Potentials from Sobolev Spaces, arXiv 1003.3172, 2010.
Sadovnichaya, I.V., On the Rate of Equiconvergence of Expansions in Series in a Trigonometric System and in Eigenfunctions of the Sturm-Liouville Operator with a Distribution Potential, Differ. Uravn., 2008, vol. 44, no. 5, pp. 656–664.
Sadovnichaya, I.V., Equiconvergence of Expansions in Series in Eigenfunctions of Sturm-Liouville Operators with Distribution Potentials, Mat. Sb., 2010, vol. 201, no. 9, pp. 61–76.
Hartman, Ph., Ordinary Differential Equations, New York, 1969. Translated under the title Obyknovennye differentsial’nye uravneniya, Moscow, 1970.
Shveikina, O.A., On Asymptotic Expansions of Eigenfunctions of Sturm-Liouville Operators with Distribution Potentials, Differ. Uravn., 2013, vol. 49, no. 8, pp. 985–992.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © O.A. Shveikina, 2014, published in Differentsial’nye Uravneniya, 2014, Vol. 50, No. 5, pp. 626–635.
Rights and permissions
About this article
Cite this article
Shveikina, O.A. Theorems on asymptotics for singular Sturm-Liouville operators with various boundary conditions. Diff Equat 50, 623–632 (2014). https://doi.org/10.1134/S001226611405005X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S001226611405005X