Change of variables, fundamental solutions and Borel resummation

Borel summable solutions to one-dimensional Schrödinger equation

“…Namely, all these results show that the semiclassical theory in 1D quantum mechanics can be completely formulated on the base of the Borel method of resummation. This is certainly true in the case of the polynomial potentials and it seems to be true with some modifications for meromorphic potentials as well [32]. In the formulation of such a theory the essential role as we have shown in the present paper is played by the fundamental solutions (see also [1,12,25]).…”

“…These are, for example, the rational potentials being the next class of potentials of the modeling importance. In particular, the universality of the Borel function in the later case of potentials seems to be not satisfied [32].…”

“…a) The factors χ 1,2 given by (26) and (27) respectively can be expanded in corresponding domains D 1,2 = {x : ℜξ(x) < ℜξ(x 0 )} into the semiclassical series determined by the following formulae c) The asymptotic series (32) and (33) can be Borel summed with the following results…”