Quantum mechanics of a moving bound state in the lab frame Consider a non-relativistic quantum-mechanical problem of two bodies interacting via a confining potential $V(\vec{\mathbf{r}})$ (say, a hydrogen atom) moving with a constant speed $\vec{v}$ relative to the lab frame. Alternatively, we could consider a single-body problem with reduced mass $M$ and time-dependent potential $V(\vec{\mathbf{r}}-\vec{\mathbf{v}}t)$.
I. What is the spectrum/wavefunctions of this system in the lab frame?
Attempt to answer: as for the wave functions, we simply use the Galilean invariance and say that any $\psi_k(\vec{x},t)$ becomes $\psi_k(\vec{x}-\vec{v}t,t)$, and the corresponding energies $E_k$ become $E_k + M_{}v^2/2$. But... how can we even talk about eigenenergies without the separation of variables?
My guess is that the answer can be given directly in terms of properties of the $\bigl[i\hbar\partial_t-\hat{H}(x,t)\bigr]$ operator.
II. OK, let's imagine that some kind of sensible answer to I. exists, and that one managed to interpret the results in the lab frame. How would one obtain those without switching to the comoving frame, directly in the lab frame? At  least, in principle.
BTW, a related question has never been answered.
 A: Actually, your two examples here are subtly different:

Consider a non-relativistic quantum-mechanical problem of two bodies interacting via a confining potential (⃗ ) (say, a hydrogen atom) moving with a constant speed ⃗  relative to the lab frame. Alternatively, we could consider a single-body problem with reduced mass  and time-dependent potential (⃗ −⃗ ).

In the first case, you indeed have Galilean invariance. You can perform a change of variables into the center of mass $r_{com}=(m_1 x_1 + m_2 x_2)/(m_1+m_2)$ and relative coordinate between the two particles $r_{rel}=x_2-x_1$. You will find that there is a separation of variables in the wavefunction between these two variables. The center of mass will obey a free particle Schrodinger equation. The center of mass energy is $\frac{1}{2}(m_1 + m_2) v^2$, where $v$ is the velocity of the center of mass. The relative coordinate energies, meanwhile, will only depend on $V(r_{rel})$, which does not depend on the relative velocity. Note that the potential -- which describes the strength of the interaction between the two particles -- is not time dependent.
In the second case, the presence of an external potential actually breaks Galilean invariance. It might be simpler first to realize that the potential breaks translation invariance. The minimum of the potential shifts if you perform a translation, so the Hamiltonian does not commute with the translation operator. Similarly, the potential defines a preferred frame (where the potential is at rest).
Still, you can make some progress. Let's write the full Schrodinger equation as
\begin{equation}
\left[-i \hbar \frac{\partial}{\partial t} + \frac{\hbar^2}{2m} \nabla^2 \right]\Psi = V(r - vt) \Psi
\end{equation}
If you perform a change of coordinates on the right hand side to make $V$ time independent (by defining $r'=r-vt$), then the left hand side has the same form in the $r', t$ coordinates. So you can solve the Schrodinger equation in the $r', t$ coordinates using standard methods, and then re-express the solutions back in terms of $r, t$ coordinates. However eigenstates, which are separable in the $r;, t$ coordinates, will not be separable in the $r, t$ coordinates.
Let's pretend we didn't know about the $r', t$ coordinates. In the $r, t$ coordinates, you apparently have a general time-dependent Hamiltonian, with a non-conserved energy. You cannot solve this (directly) as an eigenvalue problem; as you said you cannot separate variables directly in the $r, t$ coordinates. So, you must solve the general time-dependent partial differential equation. However, even without knowing about the $r', t$ coordinates you could (at least in principle) discover that there is an operator whose eigenstates are separable. Of course, we know that this operator is the Hamiltonian in the $r', t$ coordinates. So an alternative way to solve the problem is to find the eigenstates of this operator (which would appear quite mysterious if you didn't know about Galilean transformations). Solving for the eigenstates of this operator will amount to doing the transformation into the rest frame of the potential.
