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ON THE EVOLUTION OPERATOR KERNEL FOR THE COULOMB AND COULOMB-LIKE POTENTIALS
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With the 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 potentials (which becomes 1/|q| as |q|→0). It turned out that the Schwinger-DeWitt expansion for them is divergent. So, the kernels for these potentials have additional (beyond δ-like) singularities 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 exactly δ-like singularity at t=0 and for which the initial condition is fulfilled in rigorous sense (such as [Formula: see text] for integer λ).
Title: ON THE EVOLUTION OPERATOR KERNEL FOR THE COULOMB AND COULOMB-LIKE POTENTIALS
Description:
With the 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 potentials (which becomes 1/|q| as |q|→0).
It turned out that the Schwinger-DeWitt expansion for them is divergent.
So, the kernels for these potentials have additional (beyond δ-like) singularities 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 exactly δ-like singularity at t=0 and for which the initial condition is fulfilled in rigorous sense (such as [Formula: see text] for integer λ).
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