Analytic continuation to the- second sheet of the Predholm determinant of the Schrödinger operator

The paper deals with the non-selfadjoint Schrödinger operator in $L_2(R^3)$ with the complexvalued potential $q(x)$. The main subject under consideration is the Fredholm determinant $D(\lambda)$, $\lambda\notin[0,\infty)$ of the equation for resolvent kernel of the operator playing the same role as the determinant of the characteristic function. It is proved that if $q(x)$ can be analytically continued in the sector $|\arg{z}|<\theta$ as the function of argument $r$ ($r=|x|$), $D(\lambda)$ can be analytically continued to the second sheet in the sector $|\arg{\sqrt{\lambda}}|<\pi/2+\theta$.