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Mathematics 2017, 5(1), 2; doi:10.3390/math5010002

On Autonomy Imposition in Zero Interval Limit Perturbation Expansion for the Spectral Entities of Hilbert–Schmidt Integral Operators

Informatics Institute, İstanbul Technical University, Ayazağa Campus, İstanbul 34469, Turkey
This paper is an extended version of our paper published in “Suha Tuna, Metin Demiralp. (2016) Validity and Failure of the Autonomy Imposition on the Eigenfunctions in Zero Interval Limit Perturbation Expansion for Hilbert–Schmidt Integral Operators. International Journal of Applied Physics, 1, 42–48”.
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Author to whom correspondence should be addressed.
Academic Editor: Hari M. Srivastava
Received: 7 September 2016 / Revised: 15 December 2016 / Accepted: 20 December 2016 / Published: 6 January 2017
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Abstract

In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed. View Full-Text
Keywords: Hilbert–Schmidt integral operators; autonomy; eigenvalues; eigenfunctions; perturbation expansions; zero interval limit Hilbert–Schmidt integral operators; autonomy; eigenvalues; eigenfunctions; perturbation expansions; zero interval limit
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Tuna, S.; Demiralp, M. On Autonomy Imposition in Zero Interval Limit Perturbation Expansion for the Spectral Entities of Hilbert–Schmidt Integral Operators. Mathematics 2017, 5, 2.

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