# zbMATH — the first resource for mathematics

Magnetic Schrödinger operators with compact resolvent. (English) Zbl 0637.35026
The work deals with the magnetic Schrödinger operator $L_ V(\bar a)=-\sum^{n}_{j=1}(\partial /\partial x_ j-ia_ j)^ 2+V,$ where $$a_ j(x)\in L^ 2_{loc}(R^ n)$$, $$V(x)\in L^ 1_{loc}(R^ n)$$ and V(x)$$\geq 0$$. Let us denote $$\Pi_ j(\bar a)=(1/i)\partial /\partial x_ j-a_ j$$; $h_{\bar a,V}(\phi,\psi)=\sum^{n}_{j=1}(\Pi_ j(\bar a)\phi,\Pi_ j(\bar a)\psi)+(V\phi,\psi);$
$\bar h_{\bar a,V}(u,w)=\sum^{n}_{j=1}(\Pi_ j(\bar a)u,\Pi_ j(\bar a)w)+(V^{1/2}u,V^{1/2}w);$
$\ell_{\bar a,V}(\Omega)=\inf \{h_{\bar a,V}(\phi,\phi)/(\phi,\phi);\quad \phi \in C^{\infty}_ 0(\Omega),\quad \phi \not\equiv 0\}.$ Denote the self-adjoint operator in $$L^ 2(R^ n)$$ associated with $$\bar h{}_{a,V}$$ by $$H_ V(\bar a)$$, $$\bar a=(a_ 1,...,a_ n).$$
The main result is the following Theorem. The following four conditions are equivalent to each other. (a) $$H_ V(\bar a)$$ has compact resolvent. (b) $$\ell_{\bar a,V}(\Omega_ R)\to \infty$$ as $$R\to \infty$$, where $$\Omega_ R=\{x|| x| >R\}$$. (c) $$\ell_{\bar a,V}(Q_ x)\to \infty$$ as $$| x| \to \infty$$, where $$Q_ x=\{y|| x-y| <1\}$$. (d) there exists a real-valued continuous function $$\lambda$$ (x) on $$R^ n$$ such that $$\lambda$$ (x)$$\to \infty$$ as $$| x| \to \infty$$, $$h_{\bar a,V}(\phi,\phi)\geq \int \lambda (x)| \phi (x)|^ 2 dx$$ for all $$\phi \in C^{\infty}_ 0(R^ n)$$.
Reviewer: G.Derfel

##### MSC:
 35J10 Schrödinger operator, Schrödinger equation 47A10 Spectrum, resolvent 35P05 General topics in linear spectral theory for PDEs 47F05 General theory of partial differential operators
Full Text: