Mass gap for gravity localized on Weyl thick branes

We consider thick brane configurations in a pure geometric Weyl integrable 5D space-time, a non-Riemannian generalization of Kaluza-Klein (KK) theory involving a geometric scalar field. Thus, the 5D theory describes gravity coupled to a self-interacting scalar field which gives rise to the structure of the thick branes. We continue the study of the properties of a previously found family of solutions which is smooth at the position of the brane but involves naked singularities in the fifth dimension. Analyzing their graviton spectrum, we find that a particularly interesting situation arises for a special case in which the 4D graviton is separated from the KK gravitons by a mass gap. The corresponding effective Schr\"odinger equation has a modified P\"oschl-Teller potential and can be solved exactly. Apart from the massless 4D graviton, it contains one massive KK bound state, and the continuum spectrum of delocalized KK modes. We also discuss the mass hierarchy problem, and explicitly compute the corrections to Newton's law in the thin brane limit.