On uniform convergence of spectral expansions and their derivatives corresponding to self-adjoint extensions of Schrödinger operator

Abstract

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 $L_p(G)$ ($1<p\le 2$), we establish conditions ensuring the absolute and uniform convergence on the entire closed interval $\overline G$ of the series$$\sum_{n=1}^{ınfty}(f,u_n)_{L_2(G)}u_n(x),\qquad \sum_{n=1}^{ınfty}(f,u_n)_{L_2(G)}u_n^{\prime}(x)$$if $fın \overset\circ\to{W}_p^{(1)}(G)$ or $fın \overset\circ\to{W}_p^{(2)}(G)$ respectively, where $\{u_n(x)\}_1^{ınfty}$ is the orthonormal system of eigenfunctions corresponding to one of the mentioned extensions of operator~(1). Also, increasing the smoothness of the functions $f(x)$ and $q(x)$ correspondingly, we prove a theorem concerning the absolute and uniform convergence on the entire closed interval $\overline G$ of the series$$\sum_{n=1}^{ınfty}(f,u_n)_{L_2(G)}u_n^{(2k)}(x),\qquad \sum_{n=1}^{ınfty}(f,u_n)_{L_2(G)}u_n^{(2k+1)}(x),\qquad k\ge 1.$$