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The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula
Applications of Mathematics, Tome 69 (2024) no. 3, pp. 355-372 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 ${\bf A}:=[a_{ij}]$, that is, $a_{ij}=0$ if $i+\nobreak j$ is odd.\looseness +1 \endgraf 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.
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 $i+\nobreak j$ is odd.\looseness +1 \endgraf 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.
DOI : 10.21136/AM.2024.0005-21
Classification : 34A55, 34B24
Keywords: symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method 
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     journal = {Applications of Mathematics},
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Liu, Chein-Shan; Li, Botong. The eigenvalues of symmetric Sturm-Liouville problem and inverse potential problem, based on special matrix and product formula. Applications of Mathematics, Tome 69 (2024) no. 3, pp. 355-372. doi: 10.21136/AM.2024.0005-21

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