On the Riesz summability of eigenfunction expansions. (English) Zbl 0554.35030

Let \(\Omega\) be a bounded domain in \(R^ N\) (N\(\geq 3)\) with \(C^{\infty}\) boundary. Let \(g\in L_ 2(\Omega)\) be non-negative and spherically symmetric, i.e., \(g(x)=a(| x-x_ 0|)/| x-x_ 0|,\) \(x\in \Omega\), where \(t^ k| a^{(k)}(t)| \leq C_{\tau}t^{\tau -1}\) for \(t>0\) and some \(\tau >0\) \((\tau >1/2\) if \(N=3)\). Let \(L=-\Delta +g(x)\) with \(\hat L\) a positive self-adjoint extension of L from \(C_ 0^{\infty}(\Omega)\) with discrete spectrum. Let \(0<\lambda_ 1\leq \lambda_ 2\leq..\). denote the eigenvalues of \(\hat L\) with \(\{u_ n\}^{\infty}\), the complete orthonormal sequence of eigenfunctions. For \(s\geq 0\) and \(f\in L_ 2(\Omega)\), the s-th Riesz mean is \(E^ s_{\lambda}f(x)=\sum_{\lambda_ n<\lambda}(1- \lambda_ n/\lambda)^ s(f,u_ n)u_ n(x).\) For \(f\in L_ p^{\ell}(R^ N)\) and \(\sup p f\subset {\bar \Omega},\) the author proves: Theorem 1. If \(p\geq 1\), \(s\geq 0\), \(\ell >0\), \(\ell +s\geq (N- 1)/2,\) \(p\ell >N\), then \(\lim_{\lambda \to \infty}E^ s_{\lambda}f(x)=f(x),\quad x\in \Omega.\) Theorem 2. If \(s\geq 0\), \(\ell \geq 0\), \(\ell +s\geq (N-1)/2,\) then \(\lim_{\lambda \to \infty}E^ s_{\lambda}f(x)=0,\) \(x\in \Omega \setminus \sup p f.\) The method is that of V. A. Il’in [Usp. Mat. Nauk 23, No.2(140), 61-120 (1968; Zbl 0189.357)], and involves a sequence of technical lemmas estimating various functionals.
Reviewer: J.R.Kuttler


35J10 Schrödinger operator, Schrödinger equation
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
47F05 General theory of partial differential operators


Zbl 0189.357