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Keywords:
symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method
symmetric Sturm-Liouville problem; inverse potential problem; special matrix eigenvalue problem; product formula; fictitious time integration method
Summary:
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.
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