A convergent approximation scheme for the inverse Sturm-Liouville problem

Abstract

For the Sturm-Liouville operator L=Lp:y mod to -y"+py one seeks to reconstruct the coefficient p from knowledge of the sequence of eigen-frequencies ( lambda j with Lyj= lambda jyj for some yj not=0). An implementable scheme is: for some N determine pN so (approximately) pN has minimum norm with eigen-frequencies ( lambda 1,. . ., lambda N) as given. This is the method of 'generalised interpolation' and is shown to give a convergent approximation scheme: pN to p. The principal technical difficulties are the continuities of the functionals p mod to lambda j, which are shown for p topologised by weak convergence in (H1)', and the injectivity of p mod to ( lambda j:j=1,2,. . .).