The idea of the paper cited by the OP is to exploit the so-called intertwining property of the wave (or Møller) operators $\Omega^{\pm}(H,H_0)$ of scattering theory. Such operators have the property:
$$H\Omega^{\pm}=\Omega^{\pm}H_0\; .$$
Therefore, given they are invertible with inverse $\Omega^{\pm}(H_0,H)$, we have that
$$\Omega^{\pm}(H_0,H)H\Omega^{\pm}(H,H_0)=H_0\; .$$
However, the wave operators are not unitary in general. In addition, they do not exist if $H_0$ and $H$ have purely discrete spectrum. In fact, the wave operator $\Omega^{\pm}(A,B)$ is defined as the strong limit, when $t\to \pm \infty$, of $e^{itA}e^{-itB}$. The strong limit means that one should have
$$\lim_{t\to\pm\infty}\lVert (e^{itA}e^{-itB}-\Omega^{\pm}(A,B))\psi\rVert=0$$
for any $\psi$ in the Hilbert space. However, suppose that $\psi=\psi_\lambda$ is an eigenvector of $B$ with value $\lambda$. Then the limit $t\to\pm\infty$ exists only if $\psi_\lambda$ is also an eigenvector of $A$ with the same eigenvalue. This is because if else the mutual oscillations do not cancel to give a well-defined limit. However this last condition is not realistic, since a perturbation of $H_0$ would change the spectrum.
In fact, the wave operators are usually defined with the projection on the continuous spectrum of the operator acting on the right.
Therefore, they cannot be used to define the change of basis that the OP is looking for.
