Geodesic
Analytic continuation to the- second sheet of the Predholm determinant of the Schr\"odinger operator
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 81-89

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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 $|\arg{\sqrt{\lambda}}|\pi/2+\theta$. 
@article{ZNSL_1974_47_a5,
     author = {S. N. Naboko},
     title = {Analytic continuation to the- second sheet of the {Predholm} determinant of the {Schr\"odinger} operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {81--89},
     publisher = {mathdoc},
     volume = {47},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a5/}
}
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S. N. Naboko. Analytic continuation to the- second sheet of the Predholm determinant of the Schr\"odinger operator. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 81-89. http://geodesic.mathdoc.fr/item/ZNSL_1974_47_a5/