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arXiv · hep-th/9605188

On the Evolution Operator Kernel for the Coulomb and Coulomb--Like Potentials

V. A. Slobodenyuk
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Abstract

With a help of the Schwinger --- DeWitt expansion analytical properties of the evolution operator kernel for the Schrödinger equation in time variable $t$ are studied for the Coulomb and Coulomb-like (which behaves themselves as $1/|\vec q|$ when $|\vec q| \to 0$) potentials. It turned out to be that the Schwinger --- DeWitt expansion for them is divergent. So, the kernels for these potentials have additional (beyond $δ$-like) singularity at $t=0$. Hence, the initial condition is fulfilled only in asymptotic sense. It is established that the potentials considered do not belong to the class of potentials, which have at $t=0$ exactly $δ$-like singularity and for which the initial condition is fulfilled in rigorous sense (such as $V(q) = -\frac{λ(λ-1)}{2} \frac {1}{\cosh^2 q}$ for integer $λ$).

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V. A. Slobodenyuk. 1996-05-25. On the Evolution Operator Kernel for the Coulomb and Coulomb--Like Potentials. https://doi.org/10.1142/s0217732396001715

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