The Quantum Mechanics Problem of the Schroedinger Equation with the Trigonometric Rosen-Morse Potential

We present the quantum mechanics problem of the one-dimensional Schroedinger equation with the trigonometric Rosen-Morse potential. This potential is of possible interest to quark physics in so far as it captures the essentials of the QCD quark-gluon dynamics and (i) interpolates between a Coulomb-like potential (associated with one-gluon exchange) and the infinite wall potential (associated with asymptotic freedom), (ii) reproduces in the intermediary region the linear confinement potential (associated with multi-gluon self-interactions) as established by lattice QCD calculations of hadron properties. Moreover, its exact real solutions given here display a new class of real orthogonal polynomials and thereby interesting mathematical entities in their own.