**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.