Number of eigenvalues for a class of non-selfadjoint Schr\"odinger operators

In this article, we prove the finiteness of the number of eigenvalues for a class of Schr\"odinger operators $H = -\Delta + V(x)$ with a complex-valued potential $V(x)$ on $\bR^n$, $n \ge 2$. If $\Im V$ is sufficiently small, $\Im V \le 0$ and $\Im V \neq 0$, we show that $N(V) = N(\Re V)+ k$, where $k$ is the multiplicity of the zero resonance of the selfadjoint operator $-\Delta + \Re V$ and $N(W)$ the number of eigenvalues of $-\Delta + W$, counted according to their algebraic multiplicity.