Is the Schrödinger equation still valid if we use a non-Hermitian Hamiltonian with it? By this I mean does:
$$\hat{H}\psi(t) = i\hbar\frac{\partial}{\partial t}\psi(t)$$
if $\hat{H}$ is not Hermitian?
Answer
If you have some arbitrary linear operator $\hat A$, there's nothing stopping you from formulating the differential equation $$ i\partial_t \Psi = \hat A \Psi, $$ but you also have no guarantee that its solutions will play nicely or even exist.
In the simplest case, you can take $\hat A=-ia\mathbb I$, and your Schrödinger equation reads $\partial_t \Psi = -a\Psi$, giving you exponential decays of the form $\Psi(t) = e^{-at}\Psi(0)$. If you have a postive real part, as in e.g. $\hat A=+ia\mathbb I$, then you'll have an exponential growth as $\Psi(t) = e^{+at}\Psi(0)$, which isn't terrible. You can also have mixtures between these, such as e.g. a two-by-two matrix $$ A=\begin{pmatrix} ia&0\\0&ib\end{pmatrix}, $$ and you'll get different decay constants for the different coordinates. From here it's easy to extend to arbitrary finite complex matrices, where the solutions will obviously be a bit more complex. However, you need to be careful, because if you break the premise of hermiticity you also lose the guarantee that your operator will be diagonalizable, such as a Jordan block of the form $$ A=\begin{pmatrix} ia&1\\0&ia\end{pmatrix}, $$ which cannot be reduced further; as such, you probably want to demand that your operator be normal, or some similar guarantee of niceness.
You should be double wary, of course, in infinite dimensions, particularly if there is the chance that $\hat A$ will have a definite spectrum whose imaginary part is unbounded from above. A simple example of that type is $$ i\partial_t \Psi(x,t) = ix\Psi(x,t), $$ which is just about solvable as $\Psi(x,t) = e^{xt}\Psi(x,0)$, but here if your initial condition is at large positive $x$ the rate of growth becomes unbounded. From there, it isn't hard to envision the possibility that with slightly more pathological operators you could completely lose the existence of the solutions.
That said, non-hermitian hamiltonians are used with some frequency in the literature, particularly if you're dealing with resonances in a continuum or decaying states. The book referenced here might be a good starting point if you want to read about those kinds of methods. At a more gritty level, you can ask Google Scholar and it will yield ~20k results, with many of those related to something called PT-symmetric quantum mechanics, but that's probably rather more information than what you're after at the moment.
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