So, I was studying quantum mechanics and reached the point where perturbation theory is discussed. It is my first time in this topic, and something called my attention: it was said that we need perturbation theory when there is this hamiltonian that we cannot solve and them we can write:
$$H = H_0 + \lambda V$$
where $H_0$ is a hamiltonian that we can solve - or better put, is a hamiltonian that we KNOW how to solve -, and $V$ is another one that we can solve or not, that we do know how to solve or we do not, the question is that their sum, $H$, is the one that we cannot solve.
Ok, $H$ may be written in something way that is really hard to solve, or impossible, so we perturb the system and solve an easier problem. My question is: if we know how to solve $H_0$ and $V$, why we cannot solve $H$ = $H_0 + \lambda V$? The Schrödinger equation is linear.
What am I missing?