On the simpleness of zeros of Stokes multipliers

The aim of this paper is to discuss the simpleness of zeros of Stokes multipliers associated with the differential equation -Φ″(X)+W(X)Φ(X)=0, where W(X)=Xm+a1Xm-1+⋯+am is a real monic polynomial. We show that, under a suitable hypothesis on the coefficients ak, all the zeros of the Stokes multipliers are simple.