Self-Consistent Effective Action for Quantum Particle with   Space-Dependent Mass
  We calculate the quantum corrections to the classical action of a particle with coordinate-dependent mass. The result is made self-consistent by a variational approach, thus making it applicable to strong-couplings and singular potentials. By including thermal fluctuations, the we obtain an effective action whose classical Euler-Lagrange equation describes the motion of a particle including quantum and thermal effects. 
M. E. S. Borelli and H. Kleinert