# QM scattering in a finite-sized box

Background

Consider a non-relativistic particle in a one-dimensional box of length $L$ with (for definiteness) an attractive delta function at the origin:

$H = \frac{P^2}{2m} -|c|\delta(x), \qquad 0<x<L$

For any real momentum $k$, we have a scattering eigenstate with a particle coming from the left:

$\Psi^{(k)}_+ (x) = \Theta(-x)\left[ e^{i k x} + B(k) e^{-i k x}] + C(k)\ \Theta(x)e^{ikx}\right]$

Intuitively, I can imagine that the walls at $x=0$ and $x=L$ are perfect absorbers, and that the left wall is also a source that outputs a beam of particles with momentum $k$.

Questions

1. It seems that the Hilbert space includes physical states that have $\Psi(0)\ne\Psi(L)$, so therefore the momentum operator $P$ is non-Hermitian. Is that correct?

2. More generally, what is the correct way to mathematically describe the idea that the left wall is a source of particles with momentum $k$? (Maybe by a boundary condition at $x=0$?)

3. More generally still, what is the correct way to mathematically describe the idea that the left wall is a source of particles with momenta $k_1,\dots,k_n$? (Then, for example, we could take the $k_j$ to describe a filled Fermi sea.)