Expansions in the limits of strong and weak interactions

Expansion of the energy per particle, units of $\hbar^2/2ma_{1D}^2$:

$$\begin{array}{\vert c\vert c\vert c\vert c\vert c\vert} \hline te... ...}&\displaystyle n\vert a_{1D}\vert~^{(5)}&\displaystyle &\\ \hline \end{array}$$

References:
(1) V. Ya. Krivnov, A. A. Ovchinnikov, Sov. Phys. JETP 40, 781 (1975), [Zh. Eksp. Teor. Fiz. 67 1568 (1974)]
(2) M. Casas, C. Esebbag, A. Extremera, J. M. Getino, M. de Llano, A. Plastino, and H. Rubio, Phys. Rev. A 44, 4915 (1991)
(3) Here we correct result obtained in (2)
(4) New result
(5) A. Recati, P. O. Fedichev, W. Zwerger, and P. Zoller Phys. Rev. Lett. 90, 020401 (2003)

Expansion of the chemical potential, units of $\hbar^2/2ma_{1D}^2$:

$$\begin{array}{\vert c\vert c\vert c\vert c\vert c\vert} \hline te... ...})^2}{4}&\displaystyle 2\vert na_{1D}\vert&\displaystyle&\\ \hline \end{array}$$

Speed of sound, units of $\pi\hbar n/m$

$$\begin{array}{\vert c\vert c\vert c\vert c\vert c\vert} \hline te... ...}{2}&\displaystyle \frac{1}{\pi^2na_{1D}}&\displaystyle &\\ \hline \end{array}$$

The frequency of the oscillations $\frac{\omega^2}{\omega_z^2} =4(1+\triangle\omega)$

$$\begin{array}{\vert l\vert c\vert} \hline $limit$&\displaystyle \... ...splaystyle -\frac{8}{3\pi^2}/\frac{\sqrt{N}a_{1D}}{a_{z}}\\ \hline \end{array}$$

Energy and chemical potential (LDA), units of $N\hbar\omega_z$

$$\begin{array}{\vert l\vert c\vert c\vert} \hline $limit$&\display... ...eft(1-\frac{32}{3\pi^2}/\frac{\sqrt{N}a_{1D}}{a_z}\right)\\ \hline \end{array}$$

Size of the condensate and mean $z^2$, units of $a_z^2$

$$\begin{array}{\vert l\vert c\vert c\vert} \hline $limit$&\display... ...eft(1-\frac{16}{3\pi^2}/\frac{\sqrt{N}a_{1D}}{a_z}\right)\\ \hline \end{array}$$