you forgot to take the root. Phonons allways have a linear dispersion relation for particals these are quadratic. Then the equation reads alternatively E=hbar*omega For the rest the statement is correct.

Thanks a lot Josselin.

Do you know how to calculate the energy of a vortex dipole, actually it is done for diametric case (Phy. Rev. A 70,043619(2004)) but will it be same for symmetrically distributed (with respect to origin) vortices ?

There is an analytic hydrodynamic prediction for the energy of a rectilinear vortex dipole in a superfluid. The energy of the pair per unit length is:

E= (n κ^2 / 2π) ln (d / \xi) + A

where n is the particle density, kappa is the quantum of circulation, d is the separation of the pair, xi is the healing length and A is a constant representing the rest mass of the vortices.

Details of this prediction should be in the book "Quantized Vortices in Helium II" by Russell J. Donnelly.

This analytic form agrees very well with numerical results of the Gross-Pitaevskii equation for separations over a few healing lengths (see, e.g. the comparison on page 142 of the document at http://massey.dur.ac.uk/np/chapter8.pdf)

Of course, this is all based on a homogeneous infinite system but it may hold some predictive power in a trapped BEC, providing the BEC density varies little over the lengthscale of the pair and it is well away from the BEC boundary.

You may also get an elegant discussion on the energy of vortices in the following book.

S.J. Putterman, Superfluid Hydrodynamics North -Holland,

Amsterdam (1974).