Phonon trial wave function ($\delta$-potential)

Here we shall construct a trial wave function which at distances short is a two-body solution of one-dimensional $\delta$-function scattering and has ``phonon'' like behavior at large distances (see [RC67]).

The trial wave function is chosen in the following form

$$f_2(z) = \left\{ {\begin{array}{ll} A \cos k(z-B), & z < R\\ \vert\sin^\alpha (\pi z/L)\vert, & z \ge R \end{array}} \right.$$ (2.55)

There are five undefined parameters $A,B,k,\alpha,R$ and four continuity equations. One parameter is left free. We will chose matching distance $R$ as the guiding parameter.

1)

Continuity condition at zero is the same as in the Lieb-Liniger trial wave function (1.66) and is given by the formula (2.52)

$$ka \mathop{\rm tg}\nolimits kB = 1$$ (2.56)

This equation fixes the value of the scattering momenta $\k$.

2)

Continuity condition at the matching point

a)

continuity of the wave function:

$$A \cos k(R-B) = \sin^\alpha (\pi R/L)$$ (2.57)

b)

continuity of the first derivative, which together with the condition a) means continuity of the logarithmic derivative:

$$-k \mathop{\rm tg}\nolimits k(R-B) = \alpha\frac{\pi}{L} \mathop{\rm ctg}\nolimits (\pi R/L)$$ (2.58)

c)

continuity of the second derivative or, together with a) and b) means continuity of the local energy:

$$-k^2 = \alpha \left(\frac{\pi}{L}\right)^2 [(\alpha-1)\mathop{\rm ctg}\nolimits ^2(\pi R/L)-1]$$ (2.59)

One can prove that following relation holds $kL/\pi\sin 2\pi R/L + \sin 2k(R-B)=0$. Another useful relation is $\mathop{\rm tg}\nolimits k(R-B) = (ka\sin kR-\cos kR) /(ka\cos kR+\sin kR)$.

The value of the scattering momenta is a solution of the equation

$$\frac{(\sin kR+ka\cos kR)(\cos kR - ka\sin kR)} {k((ka)^2+1)}=\frac{L}{2\pi} \sin \frac{2\pi R}{L}$$ (2.60)

The maximal value of the l.h.s. is reached at $k = 0$ and equals to $R+a$. For matching distance much smaller than $L$ the sinus function on the r.h.s. can be expanded and the condition for the existence of the solution is $a>0$ which is always fulfilled for the repulsive gas.

All other parameters can be found from the following formulae:

$$\left\{ \begin{array}{lll} B &=& \frac{1}{k} \mathop{\rm arcctg}\... ...ight)\\ A &=& \frac{\sin^\alpha (\pi R/L)}{\cos(k(R-B))}\\ \end{array}\right.$$ (2.61)

The contribution to the energy is given by

$$-\frac{f''}{f}+\left(\frac{f'}{f}\right)^2 = \left\{ {\begin{arra... ...hop{\rm ctg}\nolimits ^2 \frac{\pi x}{L}\right], & z \ge R \end{array}} \right.$$ (2.62)