Fluctuations and BEC at the free surface of He-4

There is some theoretical evidence that full Bose-Einstein condensation (BEC) is achieved at the free surface of superfluid He-4 where the density is small, We present new variational results of the BEC and of the density fluctuation spectrum in the surface region of a slab of He-4, We use both a standard wave function (wf) with bulk correlations and one body shape terms and a novel shadow wave function with a glue term (G-SWF), This last one describes the self- binding of He-4 only via interparticle correlations, In both cases we find that BEC increases from the bulk like value well inside the slab to a much larger value in the surface region but a striking different behavior is found in the low density region. With a standard wf the BEC reaches essentially 100% in the surface region, With the G-SWF the condensate only increases up to about 51%. Further out of the surface the condensate decreases and correspondingly there is an enhanced population of small momentum states, This different behavior is correlated with the presence in the G-SWF of Enhanced density fluctuations in the surface region due to the zero point motion of ripplons, This enhancement is absent in the case of a standard wf.