On the Eigenvalues and Eigenfunctions of the Sturm-Liouville Operator with the Barrier Potential

SARWAR Qanitah, HUANG Zhen-you, ZAHID Abdul Hannan, XU Xin-Jian

doi:10.13371/j.cnki.chin.q.j.m.2023.01.004

Chinese Quarterly Journal of Mathematics >

2023 , Vol. 38 >Issue 1: 50 - 61

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2023.01.004

On the Eigenvalues and Eigenfunctions of the Sturm-Liouville Operator with the Barrier Potential

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1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing
210094, China; 2. Department of Chemical Engineering, University of Gujrat, Gujrat 50700, Pakistan;
3. Key Laboratory for Soft Chemistry and Functional Materials, Ministry of Education, Nanjing
University of Science and Technology, Nanjing 210094, China

SARWAR Qanitah (1986-), female, native of Bahawalpur, Pakistan, engages in functional analysis and operator theory; HUANG Zhen-you (1962-), male, native of Huainan, Anhui, professor of Nanjing University of Science and Technology, engages in functional analysis and operator theory; ZAHID Abdul Hannan (1986-), male, native of Bahawalpur, Pakistan, lecturer of University of Gujrat, engages in semiconductor synthesis and processing; XU Xin-Jian (1990-), male, native of Nanjing, Jiangsu, engages in functional analysis and operator theory.

Received date: 2021-12-02

Online published: 2023-03-20

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Abstract

We aim to find the eigenvalues and eigenfunctions of the barrier potential
case for Strum-Liouville operator on the finite interval [0 ,π ] when λ> 0. Generally, the
eigenvalue problem for the Sturm-Liouville operator is often solved by using integral
equations, which are sometimes complex to solve, and difficulties may arise in computing
the boundary values. Considering the said complexity, we have successfully developed
a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for
Sturm-Liouville operator with barrier potential. The results are of significant interest in
the field of quantum mechanics and atomic systems to observe discrete energy levels.

Key words： Sturm-Liouville operator; Eigenvalue; Eigenfunction; Barrier potential;
Asymptotic estimate

Cite this article

SARWAR Qanitah, HUANG Zhen-you, ZAHID Abdul Hannan, XU Xin-Jian . On the Eigenvalues and Eigenfunctions of the Sturm-Liouville Operator with the Barrier Potential[J]. Chinese Quarterly Journal of Mathematics, 2023 , 38(1) : 50 -61 . DOI: 10.13371/j.cnki.chin.q.j.m.2023.01.004

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