1. 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

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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
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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
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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
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Lévy path integral and fractional Schrödinger equation
Propagator:
(1)
fractional path integral measure:
(2)
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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)
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Lévy path integral and fractional Schrödinger equation
1D space-fractional Schrödinger equation:
(5)
Reisz fractional derivative operator:
(6)
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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)
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Path integration via summation of perturbation expansions
Introduce time-ordering operator,
(8)
Consider delta perturbations
(9)
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Path integration via summation of perturbation expansions
Energy-dependent Green’s function
unperturbed system
perturbed system
(10)
(11)
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Dirac delta potential
Consider free particle V = 0 with delta perturbation
Propagator for a free particle (Laskin, 2000)
(10)
Green’s function
(11)
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Dirac delta potential
Eigenenergies can be determined from:
(12)
Hence, we have the following
(13)
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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)
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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)
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Infinite square well with delta - perturbation
Propagator for an infinite square well (Dong, 2013)
(12)
Green’s function
(13)
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Infinite square well with delta - perturbation
Green’s function for the perturbed system
(14)
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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
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References
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References
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The end.
Thank you.
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