On uniform convergence of spectral expansions and their derivatives corresponding to self-adjoint extensions of the Schrödinger operator. (English) Zbl 0863.34079

Lažetić, N. L.

Mat. Vesn. 47, No. 3-4, 85-92 (1995).

Let \(\{u_n(x)\}_1^{\infty}\) be the orthonormal system of eigenfunctions corresponding to an arbitrary non-negative self-adjoint extension of the one-dimensional Schrödinger operator \[ \mathcal L(u)(x)=-u''(x)+q(x)u(x) \] with the potential \(q\in L_p(G)\), allowing a discrete spectrum, where \(1<p\leq2\) and \(G\subset\mathbb{R}\) is a finite interval, and let \(\{\lambda_n\}_1^{\infty}\) be the corresponding system of non-negative eigenvalues enumerated in nondecreasing order. Let \(f\in L_1(G)\) and \(\sigma_{\mu}(x,f)=\sum_{\sqrt{\lambda_n}<\mu}(f,u_n)_{L_2(G)}u_n(x)\). If the function \(f\) belongs to the Sobolev class \(\overset{\circ}{W}_p^{(1)}(G)\) and \(f'\) is a piecewise monotone function on \(\overline{G}\), it is proved that \(f(x)=\lim_{\mu\to\infty}\sigma_{\mu}(x,f)\) holds uniformly on \(\overline{G}\). Increasing the smoothness of the functions \(q\) and \(f\), corresponding convergence theorems are derived for the derivatives of \(\sigma_{\mu}(x,f)\).

Reviewer: Zoran Kadelburg

MSC:

34L10	Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
47E05	General theory of ordinary differential operators

Keywords:

uniform convergence; spectral expansion; self-adjoint extension; Schrödinger operator; Sobolev class