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 $$ \Cal L(u)(x)=-\bigl(p(x)\,u'(x)\bigr)'+q(x)\,u(x) $$ defined on a finite or infinite interval $G\subseteq R$. We suppose that the complex-valued potential $q=q(x)$ belongs to the class $L_1^{loc}(G)$ 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 $(k\geq 2$) of the eigenfunctions and associated functions $\{\,\overset{i}\to{u}_{\lambda}(x)\,|\,i=0,1,\dots\,\}$ of the operator $\Cal L$ in terms of their norms in metric $L_2$ on compact subsets of $G$ (on the entire interval $G$). Also, order-sharp upper estimates are established for the integrals (over closed intervals $[y_1,y_2]\subseteq \overline G$) $$ \int_{y_1}^{y_2}\biggl(\int_a^y\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy, \qquad \int_{y_1}^{y_2}\biggl(\int_y^b\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy $$ in terms of $L_2$-norms of the mentioned functions when $G$ is finite. The corresponding estimates for derivatives $\overset{i}\to{u}_{\lambda}'(x)$ and integrals $\int_{y_1}^{y_2}\overset{i}\to{u}_{\lambda}(y)\,dy$ were proved in [5]--[6].

Keywords: Sturm-Liouville operator, estimation of eigenfunctions.

MSC: 34L20, 47E05

Pages:  29$-$46     

Volume  48 ,  Issue  1$-$2 ,  1996