Curvature induced quantum potential on deformed surfaces

Abstract

We investigate the effect of curvature on the behaviour of a quantum particle bound to move on a surface. For the Gaussian bump we derive and discuss the quantum potential which results in the appearance of a bound state for particles with vanishing angular momentum. The Gaussian bump provides a characteristic length for the problem. For completeness we propose an inverse problem in differential geometry, i.e. what deformed surfaces produce prescribed curvature induced quantum potentials. We solve this inverse problem in the case of rotational surfaces. We also show that there exist rotational surfaces in the form of a circular strip around the axis of symmetry which allow particles with generic angular momentum to bind.