On the convergence of eigenfunction expansion in the norm of Sobolev spaces
Let \(S=\cup^{\ell}_{k=1}S_ k\), where each \(S_ k\) is a manifold of dimension \(m_ k\leq n-3\) given by \[ S_ k=\{(\xi,y)\in R^ n: y_ j=\phi^ k_ j(\xi),| \nabla \phi^ k_ j(\xi)| \leq C^ k_ j\} \] for some coordinate system \((\xi,y)=(\xi_ 1,...,\xi_{m_ k}\); \(y_ 1,...,y_{n-m_ k})\) and functions \(\phi^ k_ j\in C^ 1(R^{m_ k}\to R^{n-m_ k})\). The author considers the Schrödinger operator \(L_ 0=(-\Delta +q)\upharpoonright C_ 0^{\infty}(R^ n)\) under the assumption that the potential \(q\in C^{\infty}(R^ n\setminus S)\), is real-valued and \(| D^{\alpha}q(x)| \leq C[dist(x,S)]^{-\tau -| \alpha |}\) \((0\leq | \alpha | \leq 1)\) for some \(\tau\geq 0\). In this case \(L_ 0\) is self-adjoint and bounded below in \(L^ 2(R^ n)\). Let L be the Friedrichs extension of \(L_ 0\) and denote its spectral family by \(\{E_{\lambda}\}\). The main result proved is that if either \(\tau\in [0,3/2]\) and \(s\in [0,2]\), or \(\tau\in [0,)\) and \(0\leq s<7/2-\tau\), then \(\| E_{\lambda}f-f\|_{H^ s(R^ n)}\to 0\) as \(\lambda\to \infty\) for every \(f\in H^ s(R^ n)\), the Sobolev space of functions in \(L^ 2(R^ n)\) with norm \(\| f\|_{H^ s(R^ n)}:=\| (1- \Delta)^{s/2}f\|_{L^ 2(R^ n)}.\)
scientific article; zbMATH DE number 4065545