Lévy path integral approach to the fractional Schrödinger equation with δ-perturbed infinite square well Mary Madelynn Nayga and Jose Perico Esguerra Theoretical Physics Group National Institute of Physics University of the Philippines Diliman

7th Jagna International Workshop Outline Introduction Lévy path integral and fractional Schrödinger equation Path integration via summation of perturbation expansions Dirac delta potential Infinite square well with delta - perturbation Conclusions and possible work externsions January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Introduction Fractional quantum mechanics first introduced by Nick Laskin (2000) space-fractional Schrödinger equation (SFSE) containing the Reisz fractional derivative operator path integral over Brownian motions to Lévy flights time-fractional Schrödinger equation (Mark Naber) containing the Caputo fractional derivative operator space-time fractional Schrödinger equation (Wang and Xu) 1D Levy crystal – candidate for an experimental realization of space-fractional quantum mechanics (Stickler, 2013) Methods of solving SFSE piece-wise solution approach momentum representation method Lévy path integral approach January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Introduction Objectives use Lévy path integral method to SFSE with perturbative terms follow Grosche’s perturbation expansion scheme and obtain energy-dependent Green’s function in the case of delta perturbations solve for the eigenenergy of consider a delta-perturbed infinite square well January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Lévy path integral and fractional Schrödinger equation Propagator: (1) fractional path integral measure: (2) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Lévy path integral and fractional Schrödinger equation Levy probability distribution function in terms of Fox’s H function (3) Fox’s H function is defined as (4) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Lévy path integral and fractional Schrödinger equation 1D space-fractional Schrödinger equation: (5) Reisz fractional derivative operator: (6) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Path integration via summation of perturbation expansions Follow Grosche’s (1990, 1993) method for time-ordered perturbation expansions Assume a potential of the form Expand the propagator containing Ṽ(x) in a perturbation expansion about V(x) (7) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Path integration via summation of perturbation expansions Introduce time-ordering operator, (8) Consider delta perturbations (9) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Path integration via summation of perturbation expansions Energy-dependent Green’s function unperturbed system perturbed system (10) (11) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Dirac delta potential Consider free particle V = 0 with delta perturbation Propagator for a free particle (Laskin, 2000) (10) Green’s function (11) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Dirac delta potential Eigenenergies can be determined from: (12) Hence, we have the following (13) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Dirac delta potential Solving for the energy yields (12) where β(m,n) is a Beta function ( Re(m),Re(n) > 0 ) This can be rewritten in the following manner (13) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Dirac delta potential Solving for the energy yields (12) where β(m,n) is a Beta function ( Re(m),Re(n) > 0 ) This can be rewritten in the following manner (13) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Infinite square well with delta - perturbation Propagator for an infinite square well (Dong, 2013) (12) Green’s function (13) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Infinite square well with delta - perturbation Green’s function for the perturbed system (14) January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop Summary present non-trivial way of solving the space fractional Schrodinger equation with delta perturbations expand Levy path integral for the fractional quantum propagator in a perturbation series obtain energy-dependent Green’s function for a delta-perturbed infinite square well January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop References January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop References January 6, 2014 7th Jagna International Workshop

7th Jagna International Workshop The end. Thank you. January 6, 2014 7th Jagna International Workshop