The pseudopotential method

As it was discussed above, in Secs. 1.3.2.1-1.3.2.3, scattering on different short-ranged potentials in the low-energy limit is universal, i.e. depends essentially on one parameter, the scattering length and the particular shape of the potential is of no large importance. Thus it is very useful to relate scattering on all those potentials to a scattering on a simple $\delta$-potential. In other words instead of considering a particular shape of the interaction potential, we give the description it terms of a free scattering solution at $\vert\r\vert>0$ with an appropriate boundary condition at $r=0$, which takes properly into account the scattering length and, thus, the interaction potential.

In one dimensional case the application of this scheme is straight as the Schrödinger equation for two particles can be directly solved, as it is explained in the Sec. 1.3.3.2. In three dimensions the situation is more complicated as the behavior of the solution (1.39) is not compatible with scattering on a $\delta$-potential and special adjustments should be made.

Let us revise the solution of the Schrödinger equation in the limit of low energy scattering. From the definition of the three-dimensional scattering length (1.44) it follows that the scattering solution vanishes at the distance $r = a_{3D}$. Thus we define the scattering function of the pseuodopotential in such a way that it satisfies the free scattering equation in the region $r>0$:

$$(\Delta+k^2) f(r) = 0,\qquad r>0$$ (1.81)

We will use the expression (1.45) to approach the $r\to 0$ limit:

$$f(r) \to \chi \left(1-\frac{a_{3D}}{r}\right)$$ (1.82)

where the constant $\chi$ can be related to the scattering length by multiplying (1.82) by $\r$ and differentiating

$$\chi = \lim\limits_{r\to 0} \frac{\partial}{\partial r}(rf(r))$$ (1.83)

We can now modify Eq. 1.81 in such a way that it satisfies the correct boundary condition (1.82). By inserting (1.82) into (1.81) we obtain^1.8

$$(\Delta+k^2)\, \chi \left(1-\frac{a_{3D}}{r}\right) = -4\pi \delta(r) \frac{\partial}{\partial r}(rf(r))$$ (1.84)

The operator $\delta(\r)\frac{\partial}{\partial r}(r\cdot)$ is called the pseudopotential. Going back to energy units we obtain the relation between the strength of the pseudopotential $g_{3D}$ (coupling constant) and the three-dimensional scattering length

$$V_{int} = g_{3D} \delta(\r)\frac{\partial}{\partial r}(r\cdot)$$ (1.85)

$$g_{3D} = \frac{4\pi\hbar^2}{m}a_{3D},$$ (1.86)

where we considered the case of equal-mass particles $\mu = m/2$.

The pseudopotential (1.85,1.86) was used by Olshanii [Ols98] to solve quasi one dimensional scattering problem in a tight harmonic transverse confinement.

Finally, the wave function $f(r)$ satisfies the equation^1.9:

$$\left(-\frac{\hbar^2}{2m}\Delta_1 -\frac{\hbar^2}{2m}\Delta_2 + \... ...12}~\cdot~)\right) f({\vec r_1},{\vec r_2}) = {\cal E}f({\vec r_1},{\vec r_2}),$$ (1.87)