Magnetic Schrödinger operators with discrete spectra on non-compact Kähler manifolds (Q2859476)

Let \((M;g)\) be a complete non-compact oriented Riemannian (\(C^\infty\)) manifold of dimension \(n\geq 2\), equipped with the metric \(g\). Let \(a\) be a 1-form defined on \((M;g)\) and \(H_a=(d^a)^*d^a\), where \(d^a(\varphi)=d\varphi+i\varphi a\) for complex-valued functions \(\varphi\). \(H_a\) is called the (scalar) magnetic Schrödinger operator generated by the potential \(a\). \(H_a\) is an unbounded operator in \(L^2(M;{\mathbb C})\).NEWLINENEWLINE In the case when the manifold is complete, non-compact and Kähler, with Kähler form \(\omega\), the author proves that the following condition NEWLINE\[NEWLINE\lim\limits_{x\to\infty} \langle da(x), \omega(x)\rangle =-\inftyNEWLINE\]NEWLINE implies that the spectrum of \(H_a\) is discrete.