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 = 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.

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In this paper we study the corrections emergent from a Lorentz-violating CPT-odd extension of the complex scalar sector to the Bose-Einstein condensation and to the thermodynamics parameters. We initially discuss some features of the model to only then compute the corrections to the Bose-Einstein condensation. The calculations are done by computing the generating functional, from which we extract the thermodynamics parameters. We also obtain a Lorentz-violating correction for the critical temperature T c that sets the Bose-Einstein condensation.

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