Borel summable solutions to one-dimensional Schrödinger equation

Abstract

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.

Change of variables as Borel resummation of semiclassical series

“…A standard way to introduce FS's is a construction of a Stokes graph (SG) [7,8,9] for a given (meromorphic) potential V (x). SG consists of Stokes lines (SL) emerging from roots (turning points) of the equation:…”

“…Nevertheless, even in such cases the choice of x 0 can be arbitrary. Only the constants C k accompanied to the choice are definite depending on the choice [7]. The representation (3.1) is standard in a sense that any other one can be brought to (3.1) by redefinitions of the constants C k .…”

“…Therefore using the fundamental solution method in the 1-dim problems is completely equivalent to the corresponding Maslov one in the semiclassical regime of the problem but it has many obvious advantages over the latter with their use as the exact solutions to SE being the first one. Other important properties of the method have been mentioned and used in the main body of this as well as other papers [6,7,8,9,11].…”

“…As it has been shown by Milczarski and Giller [7] (see also [8]) such specific Borel summable solutions to SE are provided for meromorphic potentials by the F-F construction [3] in the form of so called fundamental solutions (FS) [8,9]. These are the only solutions with the Borel summability property among all the F-F-like solutions [7].…”

Analytic calculation of energies and wave functions of the quartic and pure quartic oscillators

“…[1] and references therein), the quartic anharmonic oscillator (1) also serves as a basic tool for checking various approximate and perturbative methods in quantum mechanics. Such an application appears in several recent field theoretical model studies [2,3,4,5,6,7,8,9,10,11,12,13,14]. It is well known [15,16] that for the quartic anharmonic oscillator the perturbation expansion diverges even for small couplings and becomes completely useless for strong coupling.…”