Diffusion and localization of ultra-cold particles on rough substrates

Diffusion and localization of ultra-cold particles moving along randomly co rrugated substrates is analyzed quasianalytically. The particles are either bound to the substrate or pressed to it by the external holding field. The localization length and diffusion coefficient are expressed explicitly via the correlation radius of surface inhomogeneities. This quantum bounding b all problem with a random rough wall is solved analytically in three limiti ng cases of longwave particles, large aps between bound states, and single- state occupancy. Elsewhere, the diffusion coefficient and localization leng th are evaluated numerically for Gaussian correlation of inhomogeneities. T he results are applied to ultra-cold neutrons in the gravitational trap, el ectrons on helium and hydrogen surfaces, and hydrogen particles bound to he lium surface. Experimental observation of weak 2D localization for neutrons and electrons requires further cooling and substrate preparation.