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In this paper we investigate the bound state problem of nonrelativisticquantum particles on a conical surface. This kind of surface appears as atopological defect in ordinary semiconductors as well as in graphene sheets.Specifically, we compare and discuss the results stemming from two differentapproaches. In the first one, it is assumed that the charge carriers are boundto the surface by a constraining potential, while the second one is based onthe Klein-Gordon type equation on surfaces, without the constraining potential.The main difference between both theories is the presence/absence of apotential which contains the mean curvature of a given surface. This factchanges the dependence of the bound states on the angular momentum $l$.Moreover, there are bound states that are absent in the Klein-Gordon theory,which instead appear in the Schr\"{o}dinger one.

Nonrelativistic quantum dynamics on a cone with and without a constraining potential