Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Abstract

The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (*)?y ?+qy=?y holds on (0, ?) fory(0)=y(?)=0 and a set of given eigenvalues ?. Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues ?, asymptotic corrections for the ?'s serve to get better estimates forq. Let ? k (1?k?n) be the first eigenvalues of (*), let? k be the corresponding discrete eigenvalues obtained by the finite element method for (*) and letμ k ?? k for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors $$\bar \Lambda _k - \Lambda _k $$ of the corrected discrete eigenvalues $$\bar \Lambda _k : = \lambda _k + \mu _k - k^2 $$ are obtained and confirm and improve the knownO(kh 2)(h:=?/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that $$|\bar \Lambda _k - \Lambda _k | \leqq c_1 kh^2 $$ for 4+c 2 ?k?(n+1)/2, wherec 1 andc 2 are constants depending onq which tend to 0 for vanishingq.