Change of variables as Borel resummation of semiclassical series

Abstract

The 1D Schrödinger equation with meromorphic potentials are considered. It is shown that any fundamental solution constructed for such potentials has a property to depend in some particular way on an arbitrary meromorphic function f(x). A family of fundamental solutions created in this way can be reproduced by multiplying some fixed member of the family by appropriate -dependent constants. This freedom allows for a proper construction of fundamental solutions at simple and double poles of considered potentials. It expresses the fact that any change of variable keeping the form of 1D Schrödinger equation keeps invariant the form of the fundamental solutions as well. In fact there is one-to-one correspondence between the functions f(x) and the Schwarzian of y(x). Since any fundamental solution is Borel summable it means also that the effect of the change of variable in fundamental solutions can be recoverd by multiplying the `initial' fundamental solutions by suitably chosen -dependent constants and Borel resumming the corresponding semiclassical expansions. This explains why a change of variable can improve JWKB formulae. However, it is shown also that a change of variable itself cannot provide us with the exact JWKB formulae.