Reconstruction techniques for classical inverse Sturm-Liouville problems

Abstract

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

The inverse spectral problem for exterior transmission eigenvalues

“…From (4.9) we now have that ∂ ∂ξ K 1 (γ , t ) and ∂ ∂ξ K 2 (γ , t ) are uniquely determined. Hence, following Rundell and Sacks [13], we can conclude that p ∈ C 1 [0, γ ] is uniquely determined and from this it is easily seen that n(r) is uniquely determined (cf [8] or section 9.4 of [3]).…”

Reconstruction for a class of the inverse transmission eigenvalue problem

“…We also note the results on the missing eigenvalue problem for differential operators in previous studies and other works. On the other hand, the results on numerical solutions for differential operators can be found in previous works . To the best of our knowledge, the numerical solution for the inverse transmission eigenvalue problem has been not completely solved.…”

“…Figure A is the corresponding approximate solution q ( x ) with the given coefficient and σ ( L ( q ,1)). The numerical method for the regular inverse Sturm‐Liouville problem may be used in Rundell and Sacks, Sacks, and Uhlmann . Let η i be the discrete form of η ( r ( x )) at x = x i .…”

A Qualitative Approach to Inverse Scattering Theory

“…For subsequent use will also need the following result due to Rundell and Sacks [147] (see also [98], p. 162).…”