Inverse quantum potential reconstruction via generalized Bertlmann–Martin inequalities

Reconstructing a radial (1D) quantum potential, [Formula: see text], from a few bound-state energies is a long-standing inverse problem because limited spectral data must constrain an entire potential. We present a Laplace-moment reconstruction pipeline that links the Bertlmann–Martin gap bound to generalized Bertlmann–Martin (GBM) even-moment ladders, continues the Laplace transform with Padé approximants, and inverts the transform to recover [Formula: see text] and [Formula: see text]. Odd moments are supplied by a physically consistent interpolation scheme. Benchmark settings and diagnostics for Coulomb, harmonic oscillator, Hulthén, Kratzer, and hyperbolic-well cases are stated so each approximation stage can be assessed under a common empirical basis. The conclusions are therefore limited to the reported benchmark settings rather than offered as universal method claims.