On uniform convergence of spectral expansions and their derivatives arising by self-adjoint extensions of an one-dimensional Schrödinger operator

Abstract

In this paper we consider the problem of global uniform convergence of spectral expansions and their derivatives, $\sum_{n=1}^{ınfty}f_nu^{(j)}_n(x)$$(j=0,1,\dots)$, generated by non-negative self-adjoint extensions of the operator $\Cal L(u)(x) = - u^{\prime\prime}(x) + q(x)u(x)$ with discrete spectrum, for functions from the class $W^{(1)}_2(G)$, where $G$ is a finite interval of the real axis. Two theorems giving conditions on functions $q(x)$, $f(x)$ which are sufficient for the absolute and uniform convergence on $\overline G$ of the mentioned series, are proved. Also, some convergence rate estimates are obtained.