Inverse characterization and non-uniqueness of the Gaussian configurational partition function

Given a configurational partition function, in this work we investigate the inverse problem of reconstructing the one-dimensional potential. For the case of the Gaussian partition function, the configurational density of states (CDOS) is univocally determined, being the harmonic potential uniquely recovered due to its symmetric single–well feature. When multiples inverse branches are considered, the uniqueness of the potential is broken and more information is needed in order to obtain a unique potential. The branch topology imposes alternating orientations, thus allowing distinct spatial realizations. Finally, the Gaussian partition function only determines the CDOS but not the potential, manifesting in this way the precise scope and limitations of the inverse characterization of the Gaussian configurational partition function.