The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula

Chein-Shan Liu; Botong Li

Applications of Mathematics (2024)

  • Volume: 69, Issue: 3, page 355-372
  • ISSN: 0862-7940

Abstract

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The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an n -dimensional matrix eigenvalue problem is derived with a special matrix 𝐀 : = [ a i j ] , that is, a i j = 0 if i + j is odd.Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function q ( x ) in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.

How to cite

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Liu, Chein-Shan, and Li, Botong. "The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula." Applications of Mathematics 69.3 (2024): 355-372. <http://eudml.org/doc/299382>.

@article{Liu2024,
abstract = {The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$-dimensional matrix eigenvalue problem is derived with a special matrix $\{\bf A\}:=[a_\{ij\}]$, that is, $a_\{ij\}=0$ if $i+j$ is odd.Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function $q(x)$ in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.},
author = {Liu, Chein-Shan, Li, Botong},
journal = {Applications of Mathematics},
keywords = {symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method},
language = {eng},
number = {3},
pages = {355-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula},
url = {http://eudml.org/doc/299382},
volume = {69},
year = {2024},
}

TY - JOUR
AU - Liu, Chein-Shan
AU - Li, Botong
TI - The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula
JO - Applications of Mathematics
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 3
SP - 355
EP - 372
AB - The Sturm-Liouville eigenvalue problem is symmetric if the coefficients are even functions and the boundary conditions are symmetric. The eigenfunction is expressed in terms of orthonormal bases, which are constructed in a linear space of trial functions by using the Gram-Schmidt orthonormalization technique. Then an $n$-dimensional matrix eigenvalue problem is derived with a special matrix ${\bf A}:=[a_{ij}]$, that is, $a_{ij}=0$ if $i+j$ is odd.Based on the product formula, an integration method with a fictitious time, namely the fictitious time integration method (FTIM), is developed to obtain the higher-index eigenvalues. Also, we recover the symmetric potential function $q(x)$ in the Sturm-Liouville operator by specifying a few lower-index eigenvalues, based on the product formula and the Newton iterative method.
LA - eng
KW - symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method
UR - http://eudml.org/doc/299382
ER -

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