Given two harmonic oscillators with frequencies $\Omega$ and $\Omega'$, the eigenstates themselves are exactly known. Let's call them $\Psi_n$ and $\Psi'_n$.
Is there a compact expression for the basis transformation that relates the $\Psi_n$ to the $\Psi'_n$?
For the ground state, I think I have an idea of how to do that: The matrix element $\langle \Psi'_0 | \Psi_0\rangle$ is easy to calculate as it's just a Gaussian integral, and for the matrix elements of the type $\langle \Psi'_n | \Psi_0 \rangle$ I can use $$\langle \Psi'_n | a | \Psi_0 \rangle = 0$$ and then use the fact that the operator $a$ must be related to operator $a'$ for the other oscillator via a unitary transformation of the type $a = u a' + v a'^\dagger$.
That gives me a recursive relation for the coefficients of a relatively simple form that I can solve.
But I'm not sure if that's the most elegant way, and how I'd generalized that to get all the matrix elements.