Exactness of conventional and supersymmetric JWKB formulae and their relation to the shape invariance property of potentials

Change of variables as Borel resummation of semiclassical series

“…The latter possibility (i.e. the exact results), however, can realize only due to particular properties of considered potentials reflected in global structures of their respective Stokes graphs [11]. This is in fact the subject of the original approach of Maslov and collaborators.…”

“…the JWKB formula provided in this way by (5.4) is always only an approximation. Some additional symmetry conditions have to be satisfied by the initial q(x) to provide us with such an exact JWKB formula [11].…”

“…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].…”

“…

It is shown that a change of variable in 1-dim Schrödinger equation applied to the Borel summable fundamental solutions [8,9] is equivalent to Borel resummation of the fundamental solutions multiplied by suitably chosenh-dependent constant. This explains why change of variable can improve JWKB formulae [11]. It is shown also that a change of variable alone cannot provide us with the exact JWKB formulae.

…”

“…Sometimes, a suitable change of variable provides us with JWKB-like formulae solving the problem of energy spectra even exactly [4]. No doubts, however, that the latter possibility depends totally on a potential considered and changing variable plays in such cases only an auxiliary role [11].…”