Estimates for derivatives and integrals of eigenfunctions and associated functions of nonself-adjoint Sturm-Liouville operator with discontinuous coefficients (III)

Authors

  • Nebojša L. Lažetić Faculty of Mathematics, University of Belgrade, Studentski trg 16/IV, 11000 Beograd, Yugoslavia Author

Keywords:

Sturm-Liouville operator, estimation of eigenfunctions

Subjects:

34L20, 47E05

Abstract

In this paper we consider derivatives of higher order and certain "double'' integrals of the eigenfunctions and associated functions of the formal Sturm-Liouville operator L(u)(x)=(p(x)u(x))+q(x)u(x)defined on a finite or infinite interval GR. We suppose that the complex-valued potential q=q(x) belongs to the class L1loc(G) and that piecewise continuously differentiable coefficient p=p(x) has a finite number of the discontinuity points in G. Order-sharp upper estimates are obtained for the suprema of the moduli of the k-th order derivatives (k2) of the eigenfunctions and associated functions {iuλ(x)|i=0,1,} of the operator L in terms of their norms in metric L2 on compact subsets of G (on the entire interval G). Also, order-sharp upper estimates are established for the integrals (over closed intervals [y1,y2]G)ınty1y2(ıntayiuλ(ξ)dξ)dy,ınty1y2(ıntybiuλ(ξ)dξ)dyin terms of L2-norms of the mentioned functions when G is finite. The corresponding estimates for derivatives iuλ(x) and integrals ınty1y2iuλ(y)dy were proved in [5]–[6].

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Published

1996-04-15