Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

Abstract

We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples. MSC:31A25, 34A55, 34L05, 47E05, 65D18

Solutions of Sturm-Liouville Problems

“…For a discussion of analytical methods and numerical methods for inverse SLP, see [7,8], respectively. Iterative methods [9,10], Rayleigh-Ritz method [11], finite difference approximation [12], Quasi-Newton method [13], shooting method [14], interval Newton's method [15], finite-difference method [16], boundary value methods [17][18][19][20], Numerov's method [21][22][23], least-squares functional [24], generalized Rundell-Sacks algorithm [25,26], spectral mappings [27], Lie-group estimation method [28], Broyden method [29,30], decent flow methods [31], modified Numerov's method [32], Newton-type method [33], Fourier-Legendre series [34], and Chebyshev polynomials [35] are of particular importance among the existing methods to solve inverse SLP.…”

Solving inverse Sturm-Liouville problems: theory and practice

“…Other methods, including those using asymptotic correction, can, in principle, be similarly modified [9], though such modifications have not yet been tested numerically or studied theoretically. In the important case when q 2 has only one discontinuity in (0, π) , Efremova and Freiling [12] proposed a simple and effective method for computing q 2 , provided only that q 1 is absolutely continuous in [0, π] . This adds a new challenge to those already mentioned [8,9].…”