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

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_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.