I have recently been reading about simulating the dynamics of many body Hamiltonians by means of quantum computers and I am a bit confused about some terminology. I understand that if you are able to compute the eigenvalues and eigenvectors of an specific Hamiltonian, then its ground state and dynamics are given by those. Thus, I understand that a model is solvable if one is able to diagonalize it by some method. However, some times the discussions on those are not very precise or they restrict to finding the ground state (which is nice by itself, but it would not describe the whole dynamics as far as I understand). Thus, am I right in understanding that a model is solvable if the exact diagonalization is known?
Furthermore, I was wondering if there are Hamiltonians for which the energies (eigenvalues) can be effectively computed, but not the eigenstates. I mean if you know the eigenvalues of a huge Hamiltonian (imagine $100$ qubits) but you require to solve the $H|\psi\rangle=\lambda|\psi\rangle$ equation, then it is cannot be computed, in a reasonable amount of time ($2^{100}$ equations). Is this reasonable? If so, what could be an example to this case?
Finally, I did read that the Bethe ansatz does solve (in my notion of solvable, i.e. exactly diagonalize) some Ising and Heisenberg chain Hamiltonians in 1D. However, I did not find any clear classification on which of those could be solved. For example, I found in https://arxiv.org/pdf/1012.0653, that the transverse field Ising model is exactly solvable in 1D, but not if there is an additional longitudinal field. Thus, are there any known references on which 1D models are exactly solvable? Or whose eigenvalues are known but not their eigenvectors? I also ask this because is somehow counterintuitive for me to see that the transverse field Ising models is solved in 1D and then find quantum computing people simulating it by means of quantum computers. I know that 2D models are much more complex, that's why I am asking this regarding 1D models.
