Borel Summable Solutions to 1D Schrödinger equation

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the 1D Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be considered as the canonical one. The latter means that any Borel summable solution can be obtained by the Borel transformation of this unique canonical Borel function multiplied by some h̄-dependent and Borel summable constant. This justify the exceptional role the fundamental solutions play in 1D quantum mechanics. PACS number(s): 03.65.-W , 03.65.Sq , 02.30.Lt , 02.30.Mv