## 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

### MSC:

 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