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I am studying quantum scattering and stumbled upon the "scattering channel" and "single- and multi-channel scattering" terms. However, I didn't manage to find any sufficiently formal definitions of these, could someone please explain me them?

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Okay, I think I got it after reading a lot of scattering-related papers.

I will omit the wavefunctions normalization and the constants like $\hbar$, $m$, $\pi$, etc., where possible since the precise values are not so relevant.

Consider a 2D rectangular pipe (for simplicity), infinite in $x$ direction and of finite height $H$ in $y$ direction.

Under the 'impenetrable walls' boundary conditions (that is, $\psi(x, 0) = 0$ and $\psi(x, H) = 0$), the stationary Schrodinger equation is separable in the so-called 'transport' direction ($x$ variable) and the 'confinement' direction ($y$ variable): $\psi(x, y) = X(x) Y(y)$.

The Schrodinger equation in the $x$ direction now describes 1D free particle, which takes energies $E^X \in (0, \infty)$ with eigenfunctions $X_E(x) = e^{\pm i \sqrt{E} x}$. For simplicity, we may impose boundary condition of scattering from the left and leave only functions of form $e^{i \sqrt{E} x}$.

The Schrodinger equation in the $y$ direction now describes 1D infinite potential well of width $H$. It is known that energy takes discrete values $E^Y_n$ proportional to $\frac{n^2}{H^2}$ for each positive integer $n$ with eigenfunctions $Y_n(x) = \sin(\frac{\pi n}{H} x)$ (these are sometimes called transversal modes)

If we fix the total energy $E$ and the transversal mode $n$, we can write down the corresponding wavefunction $\psi(x, y)$ as:

$$\psi_n(x, y) = \sin(\frac{\pi n}{H} x) e^{i \sqrt{E - E^Y_n} x}$$

, which is called scattering state, or scattering channel.

If we now start scattering (it's hardly scattering in the current setup, but anyway) waves of energy $E$ from the left, energy splits among the confinement and transport wavefunctions: $E = E^X + E^Y$. There exists some finite $N$ such that $E^Y_N < E < E^Y_{N + 1}$, so we can write the function $\psi(x, y)$ in the form:

$$\psi(x, y) = \sum\limits_{n = 1}^{N} \psi_n(x, y) = \sum\limits_{n = 1}^{N} \sin(\frac{\pi n}{H} x) e^{i \sqrt{E - E^Y_n} x}$$

There can't be summands with $n > N$ since the energy left for the $x$ direction will be negative, which is not the case of free particle. It is said that $N$ channels are open for transport for the current scattering problem.

The difference between the single- and multi-channel scattering just depends on whether we consider the scattering problem for a particular channel, or all open channels at the same time (it is important while computing the transmission coefficient, for instance).

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