# Complex valued energy

I'm working on a 2D He superfluid system with vortices. I was asked to calculate the kinetic energy of vortex-(anti-)vortex pairs and compare the two situations. One finds in literature that the vortex-vortex situation is unstable, and the vortex-antivortex pair can be stable. This is a crucial observation in what follows.

In the calculation of the energies, I came across a striking fact, which may or may not have any physical implications: the (kinetic) energy of the vortex-vortex situation had an imaginary component, while in the vortex-antivortex energy, everything imaginary nicely cancelled. It seems my results are consistent with what is expected, and certainly scale as they should for the system (energy logarithmically in the size of the system, and in the distance between the vorteces).

For now, I just "forgot" about the imaginary part of the energy, but my question is this: Can the imaginary part of the energy be interpreted as the lifetime of an unstable state in the case of a vortex-vortex situation (comparable to the lifetime of resonances in S-wave scattering, which also comes into play as the complex part of the energy of the resonance). Or is it merely a remnant of my approximate calculation and the result of logarithms in the complex plane and have I been staring at this too long?

Thanks!

PS: for those interested, the integration I performed can be viewed as a question on math.stackexchange.com. Please ignore the question itself, because it is not really an example of my intellect in this field, just some conjectures/"fleeting hope" ;) https://math.stackexchange.com/questions/32286/choosing-the-branch-of-a-logarithm

• To someone with enough rep: please add the tags "vortex", "superfluid", and "lifetime". Thanks! – rubenvb Apr 11 '11 at 15:01