= Pinned capillary-wave correlation
{title2=$\xi=\sqrt{\sigma/t}$}
Adding positive quadratic confinement $t\int h^2/2$ to the membrane energy changes its height spectrum to $k_BT/(\sigma q^2+t)$. The continuous height-shift <symmetry (physics)> is explicitly broken. <Connected correlation function> decay on the finite length $\xi$ and the height-difference <variance> saturates for every internal dimension, although its saturation value can depend on a microscopic cutoff. A negative confinement coefficient gives an unstable quadratic model rather than pinning.
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