How is the scattering length in 2d defined? Scattering length is 3d is well-defined. In the literature, one can also see scattering length in 2d. How is it defined? Can we even generalize it to 1d?
 A: Scattering length $l_s$ of a particle scattering on a potential $\hat{V}(\textbf{x})$ can actually be defined in a general $d$ dimensions case, using Green's functions formalism.
Studying a scattering problem consists basically in solving the Schrödinger equation (taking $\hbar=m=1$):
$$
\left[-\frac{\Delta}{2}+\hat{V}(\textbf{x})\right]|\Psi\rangle=\mathrm{i}\frac{\partial}{\partial t}|\Psi\rangle
$$
where the general solution reads :
$$
\Psi(\textbf{x},t)=\langle\textbf{x}|\Psi(t)\rangle=\int\mathrm{d}\textbf{x}'\,G(\textbf{x}',\textbf{x},t)\,\Psi(\textbf{x}',0)
$$
where $G(\textbf{x}',\textbf{x},t)=\langle\textbf{x}|e^{-\mathrm{i}tH}|\textbf{x}'\rangle$ is the Green's function associated to the equation. Solving the equation or calculate $G$ is then equivalent.
It's then possible to associate a Green's operator to such $G$ function, simply defined as :
$$
\hat{G}(t)=-\mathrm{i}\,\Theta(t)\,e^{-\mathrm{i}t\hat{H}}
$$
where $\Theta(t)$ stands for the Heaviside step function. The $-\mathrm{i}\,\Theta(t)$ term has been introduced to ensure causality of the solution $\Psi(\textbf{x},t)$.
Most of the time, it's more convenient to discuss in terms of spectrum rather than time, so we will considere the Fourier tranform of $\hat{G}(t)$, which is nothing but the associated resolvant :
$$
\hat{G}(\epsilon)=\frac{1}{\epsilon-\hat{H}+\mathrm{i}0}
$$
For convenience, we will define :
$$
\hat{G}_0(\epsilon)=\frac{1}{\epsilon-\hat{H}_0+\mathrm{i}0}\quad\text{with}\quad\hat{H}_0=-\frac{\Delta}{2}\equiv\frac{k^2}{2}
$$
so called free Green operator. By cleverly playing the two previous expressions, one can derive the Dyson equation :
$$
\hat{G}=\hat{G}_0+\hat{G}_0\hat{V}\hat{G}
$$
which solution is a power law expansion in the potential $\hat{V}$ :
$$
\hat{G}=\hat{G}_0\left[\sum_{n=0}^\infty(\hat{V}\hat{G}_0)^n\right]=\hat{G}_0+\hat{G}_0\hat{\Sigma}\hat{G}_0
$$
where $\hat{\Sigma}$ is the self energy of the particle. 
Note that the Born approximation consists here in taking :
$$
\hat{G}\simeq\hat{G}_0+\hat{G}_0\hat{V}\hat{G}_0\hat{V}\hat{G}_0
$$
i.e. the non trivial first order in the Dyson equation solution.
Finally, one can show that the imaginary part of the self energy provides the typical life time of a particle state under the effect of $\hat{V}(\textbf{x})$, which is nothing more than the scattering time $\tau_s$, provided :
$$
-\frac{1}{2\tau_s}=\Im{\langle\textbf{k}|\hat{\Sigma}(\epsilon)|\textbf{k}_i\rangle}
$$
with $|\textbf{k}_i\rangle$ is the initial wave vector before the scattering event, and $|\textbf{k}\rangle$ the wave vector after the scattering event. In the nice case when the scattering is elastic, i.e. $\|\textbf{k}_i\|=\|\textbf{k}\|=k$, the scattering length is then simply deduced from $\tau_s$ : 
$$
l_s=k\tau_s
$$
As you can see, the above derivation still valid for any dimension problem. The main issue here is to compute $\Im{\langle\textbf{k}|\hat{\Sigma}(\epsilon)|\textbf{k}_i\rangle}$ which is a $d$ dimension integral and can be more or less complex following the approximation you've chosed to compute $\hat{G}$.
A: One has to be careful when extending the concept of scattering length to lower dimensions. A straightforward extension of the 3D methodology to 2D is prone to lead to logarithmic divergences. Reason being that in 2D the radial Schrödinger's equation for the s-wave includes a negative centrifugal potential.
When carefully defining the scattering length as the hard-cylinder potential that gives the same low-wave number scattering solution phase shifts, one arrives at scattering lengths as defined by Verhaar et al: 

Scattering length and effective range in two dimensions: application to adsorbed hydrogen atoms,
  B J Verhaar, J P H W van den Eijnde, M A J Voermans and M M J Schaffrath
  J. Physics A, vol. 17 (1984).

A: We want to find what with length the center of a incoming wave packet appreciate variations. The scattering approach prefer to talk about time and momentum in detriment of space, but we can talk about space after.
The scatering state always can be written as 
$$
\psi^{(+)}_g(t)=\int d\alpha\, e^{-iE_\alpha t} g(\alpha) \phi_\alpha + \int d\alpha\,\int d\beta\,  \frac{e^{-iE_\alpha t} g(\alpha)T_{\beta \alpha}}{E_\alpha -E_\beta + i\epsilon} \phi_\beta
$$
Where $g(\alpha)$ represent a suitable superposition of free-states $\phi_\alpha$. Note that when we send $ t \rightarrow - \infty $ the state $\psi^{(\pm)}_g(t)$ becomes $\int d\alpha\, e^{-iE_\alpha t} g^{(\pm)}(\alpha) \phi_\alpha$ because the pole in $\frac{1}{E_\alpha -E_\beta \pm i\epsilon}$. The term $T_{\beta \alpha}$ is responsible for include interactions. If we have $H=H_0+V$ then $T_{\beta \alpha}=\langle \beta|V|\alpha^{+}\rangle$, where 
$$
|\alpha^{+}\rangle=|\alpha\rangle+\frac{V}{E_\alpha -H_0 \pm i\epsilon}|\alpha^{+}\rangle .
$$
We can calculate the integral on $\alpha$ in the second term by simple analysis of  poles contained in $T_{\beta \alpha}$ as function of complex $\alpha$. This is because we can deforming the integral for a large semi circle in the upper or lower complex plane and the integral in this contour is damped by $\frac{e^{-iE_\alpha t}}{E_\alpha - E_\beta+i\epsilon}$. Then only the poles and cuts of $T_{\beta \alpha}$ in complex plane contribute to the integral. For positive times, in the lower, for negative times, in the upper.
The most close pole to the real axis define the typical energy and time of scattering.
For calculate the typical length is simple take the momentum and mass of the in-state and confront to the typical time. Typical time is the inverse of the distance of the pole in units of $\hbar$. This relation of energy and time is a result of the damping mechanism of $\frac{e^{-iE_\alpha t}}{E_\alpha - E_\beta+i\epsilon}$. 
In terms of Green's Functions we can calculate this pole. We simply take the unperturbed Green's Function $ G_0(\alpha)= \frac{1}{E_\alpha-H_0 \pm \epsilon}$ and calculate the time of an in-particle stay unperturbed. 
