Semiclassical Approximations for Fractional Quantum Mechanics

Fractional quantum mechanics generalizes standard quantum mechanics to include nonlocality, memory effects, and fractal structures. It retains core quantum characteristics while introducing nonlocal dynamics, which poses substantial challenges for mathematical analysis. The semiclassical analysis, which is among the most significant mathematical frameworks for investigating conventional quantum mechanics, is lacking for fractional quantum mechanics, due to the unmatched difficulties raised by the nonlocal dynamics. In this talk, we will present recent progress toward the development of semiclassical analysis for fractional quantum mechanics, including both analytical and numerical investigations. Semiclassical approximations for the fractional Schrodinger equations will be presented, where ansatz in the form based on Wentzel–Kramers–Brillouin-Jeffreys (WKBJ), Hankel functions, and Fox-H functions are derived. In the semiclassical approximations, the phase and amplitude are proved to be determined by Hamilon-Jacobi type partial differential equations. The semiclassical approximations, as well as the Hamilton-Jacobi type partial differential equations, reduce consistently to those in the semiclassical analysis for standard quantum mechanics when the fractional order approaches integer order, which justifies that the derived semiclassical approximations generalizes those for standard quantum mechanics. The performance of different types of approximations will be compared, showing that the Fox-H function based approximation can achieve uniform accuracy near point sources, in constrast to those based on WKBJ or Hankel functions. Numerical experiments are performed to further justify the derived semiclassical apporixmations.