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Electronic properties of bilayer graphene catenoid bridge

We study the properties of an electron on a catenoid surface. The catenoid is understood as a realization of a bridge connecting two graphene layer by a smooth surface. The curvature induces a symmetrical reflectionless potential well around the bridge with one bound-state for $m=0$. For $m\neq 0$, a centrifugal potential barrier arises controlling the tunnelling between the layers. An external electric field breaks the parity symmetry and provides a barrier that controls the conductance from one layer to another. By applying a constant magnetic field the effective potential exhibits a confining double-well potential nearby the bridge. We obtain the corresponding bound states and study the effects of the curvature on the Landau levels.