Abstract: Using the Liouville-Green method of solution of ODE's, in physics known as WKB, we estimate the asymptotic behaviour of the perturbation potential matrix W in the basis of the oscillator, H_0= -d_x^2 +Q^alpha with alpha greater or equal to one, on a halfline with Dirichlet or Neumann boundary condition. This behaviour is crucial for application such as KAM, adiabatic regularisation, etc., which are used to study time-dependent systems with Hamiltonians H(t):= H_0 + f(t)W, where f is a given scalar function.