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We seek for a pair of a well and barrier potentials such that the real partsof the complex energies of the resonances transmitted through the barrier equalthe energies of the states bound within the well and find the hyperbolicPoeschl-Teller barrier, ~sech^2\rho, and the trigonometric Scarf well, ~\sec^2\chi. The potentials are shown to be conformally symmetric by the aid ofthe de Sitter space time, dS4, related to flat conformal space time by aconformal map. Namely, we transform the quantum mechanical wave equations withthe above potentials to free quantum motions on the respective open time likehyperbolic and the closed space like hyper spherical, S3, geodesics of dS4, theformer by itself is related to Minkowski space time by a conformal map.Weformulate a conformal symmetry respecting classification scheme for mesons seeneither as resonances in scattering, or as states bound within a potential,according to trajectories in which the total spin of the meson, l-dependslinearly on the first power of the invariant mass, M, and not as in thecanonical Regge formalism on the squared mass.We analyze 71 reported mesons inthis scheme and in finding good agreement with data predict the masses of 12missing mesons. We observe that on S3 the color quantum number of mesons islimited to neutral. To the amount physics is independent of the choice of theset of coordinates, this property remains valid in the flat geometry too. Weconclude on the usefulness of conformal maps of flat to curved space times astools for modelling and interpreting the color neutrality of hadrons requiredby the confinement phenomenon and on the relevance of trigonometric andhyperbolic potentials for constituent quark models. Finally, all involvedpotentials have been motivated by Wilson loops with cusps.

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space time dS4