Heun quantization of the Lorentzian potential well

The one-dimensional Lorentzian potential well admits local solutions in terms of confluent Heun functions, whereas its bound-state spectrum is determined by a global connection problem at infinity. We develop a parity-resolved Heun-to-Jost formulation in which finite-point Wronskians define exact even- and odd-parity spectral functions. Their zeros give the complete bound-state spectrum, and an energy-differentiated Wronskian proves that every zero is simple. An explicit asymptotic recurrence for the Jost solutions yields the first finite-width correction beyond the inverse-square tail. High-precision calculations, matching-radius stability tests, a local Frobenius verification, and independent finite-box diagonalization confirm the construction. In the supercritical regime, the near-threshold levels exhibit the universal geometric accumulation ratio of the inverse-square problem. The critical threshold and an equivalent spheroidal representation are also obtained.