Quantum scattering of spinless particles in Riemannian manifolds

Quantum mechanics is sensitive to the geometry of the underlying space. Here,\nwe present a framework for quantum scattering of a non-relativistic particle\nconfined to a two-dimensional space. When the motion manifold hosts localized\ncurvature modulations, scattering occurs from an emergent geometric potential\nand the metric tensor field. Analytical and full numerical simulations identify\nthe geometric potential as the primary source for low-energy scattering, while\nthe metric tensor field of the curved space governs high-energy diffraction.\nCompared to flat spaces, important differences in the validity range of\nperturbation approaches are found and demonstrated by full numerical\nsimulations using combined finite element and boundary element methods. As an\nillustration, we consider a Gaussian-shaped dent leading to effects known as\ngravitational lensing. Experimentally, the considered setup is realizable based\non geometrically engineered 2D materials.\n