  Highly oriented solid-supported lipid membranes in stacks of controlled
number $N \simeq 16$ (oligo-membranes) have been prepared by spin-coating using
the uncharged lipid model system 1,2-dimyristoyl-sn-glycero-3-phosphocholine
(DMPC). The samples have been immersed in aqueous polymer solutions for control
of osmotic pressure and have been studied by X-ray reflectivity. The bilayer
structure and fluctuations have been determined by modelling the data over the
full q-range. Thermal fluctuations are described using the continuous smectic
Hamiltonian with the appropriate boundary conditions at the substrate and at
the free surface of the stack. The resulting fluctuation amplitudes and the
pressure-distance relation are discussed in view of the inter-bilayer
potential.
