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}_n{u}_n^{(j)}(x)$$(j=0,1,\dots )$, generated by non-negative self-adjoint extensions of the operator $\mathcal{L}\mathcal{(}\mathcal{u}\mathcal{)}\mathcal{(}\mathcal{x}\mathcal{)}\mathcal{=}\mathcal{-}{\mathcal{u}}^{\prime \prime }\mathcal{(}\mathcal{x}\mathcal{)}\mathcal{+}\mathcal{q}\mathcal{(}\mathcal{x}\mathcal{)}\mathcal{u}\mathcal{(}\mathcal{x}\mathcal{)}$ with discrete spectrum, for functions from the class $W_2^{(1)}(G)$, 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 $\bar{G}$ of the mentioned series, are proved. Also, some convergence rate estimates are obtained.