On a transformation operator

We prove the existence of a transformation operator that takes the solution of the equation $y''=\lambda^{2n}y$ to the solution of the equation
$$ y''-\bigl(q_0(x)+\lambda q_1(x)+\dots+\lambda^{n-1}q_{n-1}(x)\bigr)y=\lambda^{2n}y $$
with a condition at infinity. Some properties of the kernel of this operator are studied.