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 $H=-\mathrm{\Delta }+V(x)$ with a complex-valued potential $V(x)$ on ${\text{bR}}^n$, $n\ge 2$. If $\mathrm{\Im }V$ is sufficiently small, $\Im V \le 0$and$\Im V \neq 0$,weshowthat$N(V) = N(\Re V)+ k$,where$k$ is the multiplicity of the zero resonance of the selfadjoint operator $-\Delta + \Re V$and$N(W)$thenumberofeigenvaluesof$-\Delta + W$, counted according to their algebraic multiplicity.