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

S. N. Naboko

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 81-89

Résumé

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 $|\arg{\sqrt{\lambda}}|<\pi/2+\theta$.