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Numerische Mathematik

, Volume 57, Issue 1, pp 51–62 | Cite as

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

  • Jürg T. Marti
Article

Summary

The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (*)−y+qyy holds on (0, π) fory(0)=y(π)=0 and a set of given eigenvalues λ. Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues λ, asymptotic corrections for the λ's serve to get better estimates forq. Let λ k (1≦kn) be the first eigenvalues of (*), letΛ k be the corresponding discrete eigenvalues obtained by the finite element method for (*) and letμ k Λ 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(kh2)(h:=π/(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+c2≦k≦(n+1)/2, wherec1 andc2 are constants depending onq which tend to 0 for vanishingq.

Subject Classifications

AMS(MOS): 65L15 CR: G1.7 

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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Jürg T. Marti
    • 1
  1. 1.Seminar für Angewandte MathematikETH ZürichZürichSwitzerland

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