In this paper we consider problem of the global uniform convergence of spectral expansions and their derivatives generated by arbitrary non-negative self-adjoint extensions of the Schrödinger operator $$ \Cal L(u)(x)=-u^{\prime\prime}(x)+q(x)u(x) \tag 1 $$ with discrete spectrum, for functions in the Sobolev class $\overset\circ\to{W}_p^{(k)}(G)$ ($p>1$) defined on a finite interval $G\subset R$. Assuming that the potential $q(x)$ of the operator $\Cal L$ belongs to the class $L_p(G)$ ($1