On combinatorics of Schrödinger perturbations
In this paper, the author gives a tight upper bound for Schrödinger-type perturbations of integral kernels. The proofs are based on the identities involving Stirling numbers of the first kind. These results may be of independent interest in the theory of Stirling numbers. Also, the author specializes to transition densities \(p\) and explains that, in this case, the perturbed kernel \(\tilde p\) is a transition density, too, thus allowing for a construction of new semigroups and evolutions. For recent developements in Schrödinger perturbations of time-nonhomogeneous transition probabilties, the reader may refer to [\textit{K.\,Bogdan, W.\,Hansen} and the author, Stud.\ Math.\ 189, No.\,3, 235--254 (2008; Zbl 1161.47009)] and [\textit{A.\,Gulisashvili}, Trans.\ Am.\ Math.\ Soc.\ 360, No.\,8, 4063--4098 (2008; Zbl 1141.47031)].
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