Although the Hamilton-Jacobi equation could provide an
intellectually satisfactory alternative formal background for
quantum theory, formidable difficulties emerge during the process of
solution of its operator nonlinear partial differential form. There
exist ways of circumventing this difficulty (cf., e.g., M.
Roncadelli and L. S. Schulman, Phys. Rev. Lett. 99, 170406 (2007))
revealing specific merits of the approach. In particular, the action
variable can be used to find the exact bound-state energy levels of
a quantum system without solving the equation of motion for wave
functions themselves (ref. [28]). Ye\c{s}silta\c{s} and Sever
illustrate this possibility using several versions of Morse and
P\"{o}schl-Teller exactly solvable potentials in their real as well
as complex forms.
MR2386541 Yeşiltaş, Özlem; Sever, Ramazan Exponential type complex
and non-Hermitian potentials within quantum Hamilton-Jacobi
formalism. J. Math. Chem. 43 (2008), no. 3, 921--931. 81U15 (35F20)