Hermiticity of $id/dx$ operator In all quantum mechanics books there is a formal proof that: $ (\frac{d}{dx})$ is anti-hermitian operator and thus $(i\frac{d}{dx})$ is hermitian. While proving this we also consider the fact that $[\phi ^* \psi]^{\infty}_{-\infty} =0$ .
Now what I think is that books don't write two important points explicitely:

*

*the wavefunction must vanish at boundaries;


*$\phi $   &  $ \psi$ must be complex functions.
I think point 2 is necessary for point 1 to be true (can you please prove that mathematically) and also we know that hermitian operators have real eigenvalues but if we consider $\phi =e^{kx}$ as eigenfunction of $i\frac{d}{dx}$ then we get complex eigenvalue, which is a contradiction!!
So I want to be sure, are my reasoning 1 and 2 are correct or not? And what is a mathematical proof for resoning 2.
 A: In the study of quantum mechanics, we will be interested in functions defined over the full interval $-\infty \leq x\leq \infty$. They fall into two classes those that vanish as that labels these functions. It is clear that $K=-id/dx$ is Hermitian when sandwiched between two functions of the first class or a function from each, since in either case the surface term vanishes. When sandwiched between two functions two of the second class, the Hermiticity  hinges on whether
$$\left. e^{ikx}e^{-ik'x} \right|_{-\infty}^{\infty}=0$$
If $k=k'$, the contribution from one end cancels that from the other. If $k\not= k'$, the answer is unclear since $e^{i(k-k')x}$ oscillates, rather than approaching a limit as $|x|\rightarrow \infty$. Now,  there exists a way of defining a limit for such functions that cannot make up their minds: the limit as $|x|\rightarrow 0$ is defined to be the average over a large interval. According to this prescription, we have, say as $x\rightarrow \infty$,
$$\lim_{x\rightarrow \infty}  e^{ikx}e^{-ik'x}=   \lim \limits_{\substack{%
     L \to \infty\\
     \Delta \to \infty}}  \frac{1}{\Delta}\int_L^{L+\Delta}e^{i(k-k')x}dx=0 ,\ \ \ \ \mathrm{if}  \ \ k\not= k'$$
and so $K$ is Hermitian in this space.

The following discussion is taken from R. Shankar's book. The rest of the discussion is taken care of by @Vadim.
A: Regarding point 1: indeed, most quantum mechanics books are somewhat sloppy about the fact that a differential equation alone (Schrödinger equation or an eigenvalue equation) is not sufficient for having a mathematically well-posed problem. To be well-posed the problem needs to be supplemented by the boundary conditions (or initial conditions) and the appropriate constraints on the solutions, such as, e.g., normalizability (function $\phi(x)=e^{kx}$ is not normalizable on the whole real axis).
On the other hand, it is not required that the solutions be necessarily complex functions. I believe that your condition 1 already takes care of all the problematic cases (definitely of $\phi(x)=e^{kx}$).
