Open problems: Lieb-Thirring type inequalities for Schrödinger operators with a complex-valued potential
The Lieb-Thirring inequality states that, if \(H=-\Delta +V\), \(V \in L^{p}(\mathbb{R}^{d};\mathbb{R})\), \(p\geq 1\) if \(d=1\), \(p>1\) if \(d=2\) and \(p \geq d/2\) if \(d \geq3\), then \[ \sum_{\lambda \in \sigma_{d}(H)} | \lambda | ^{p-d/2} \leq C(p,d) \| V \| ^{p}_{L^{p}}, \] where \(C(p,d)\) depends only on \(p\) and on \(d\). The authors raise the following question: is it true that, if \(V\) is complex valued, then the inequality \[ \sum_{\lambda \in \sigma_{d}(H)} \frac{\operatorname{dist} (\lambda, [0, \infty))^{p}} { | \lambda | ^{d/2}} \leq C(p,d) \| V \| ^{p}_{L^{p}}\tag{1} \] holds? They review several results which may be considered weakened versions of (1).
scientific article; zbMATH DE number 6153954