On the number of eigenvalues of Schrödinger operators with complex potentials (Q2835325)

Let \(V\) be a complex-valued potential on \({\mathbb R}^d\). It is proved that the number \(N\) of eigenvalues of \(-\Delta+V\) in \(L^2({\mathbb R}^d)\) (\(d\) is odd) counting algebraic multiplicities, satisfies, for any \(\varepsilon > 0\), NEWLINE\[NEWLINEN\leq C_d \frac{1}{\varepsilon^2} \left(\;\int\limits_{{\mathbb R}^d}e^{\varepsilon|x|}|V(x)| dx\right)^2 .NEWLINE\]NEWLINE Similar results are proved for the operator \(\frac{d^2}{dx^2}+V\) in \(L^2({\mathbb R}_+)\) with a Dirichlet boundary condition.NEWLINENEWLINEThe proofs are based on a ``trace formula approach'', which consists of identifying eigenvalues with zeroes of a certain analytic function and in using bounds on the zeroes of analytic functions. The authors use also trace ideal bounds for the Birman-Schwinger operator and complex interpolation.