On the Borel summability of WKB solutions of certain Schrödinger-type differential equations

Abstract

A class of Schrödinger-type second-order linear differential equations with a large parameter u is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of u. These formal series solutions are called the WKB solutions. We show that under mild conditions on the potential function of the equation, the WKB solutions are Borel summable with respect to the parameter u in large, unbounded domains of the independent variable. It is established that the formal series expansions are the asymptotic expansions, uniform with respect to the independent variable, of the Borel re-summed solutions and we supply computable bounds on their error terms. In addition, it is proved that the WKB solutions can be expressed using factorial series in the parameter, and that these expansions converge in half-planes, uniformly with respect to the independent variable. We illustrate our theory by application to a radial Schrödinger equation associated with the problem of a rotating harmonic oscillator and to the Bessel equation.