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


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