摘要
By considering a cigar-shaped trapping potential elongated in a proper curvilinear coordinate, we discover a new form of wave localization that arises from the interplay of geometry and topological protection. The potential is undulated in its shape such that local curvature introduces a geometrical potential. The curvature varying along the trap curvilinear axis encodes a topological Harper modulation. The varying geometry maps our system in a one-dimensional Andre-Aubry-Harper grating. We show that a mobility edge exists and topologically protected states arise. These states are extremely robust against disorder in the shape of the string. The results may be relevant to localization phenomena in Bose-Einstein condensates, optical fibers and waveguides, and new laser devices.