Sinc Method For Time Fractional Schr?dinger Equation

In recent years,quantum mechanics has become one of the hottest physics theories.Schr?dinger equation is one of the most basic equations of quantum mechanics.The time fractional Schr?dinger equation,as its generalization equation,is widely used to describe many phenomena,such as the non-Markov evolution of free particles in quantum physics,the fractional dynamics of quantum mechanics,and the fractional Planck quantum energy relationship.At present,there is still a large space for the numerical solution of time fractional Schr?dinger equation in numerical accuracy,convergence,theoretical analysis and so on.In this paper,the time fractional high-order discretization methods and the exponential convergence Sinc method are used to solve the one-dimensional and two-dimensional time fractional Schr?dinger equation,and the corresponding theoretical analysis results are given.The specific tasks are as follows:(1)A series of high order time semi-discretization schemes for time fractional Schr?dinger equation are established.First,on the basis of the second-order and third-order Weighted and Shifted Grunwald-Letnikov(WSGL)difference operators,the fourth-order WSGL difference operator is derived,and the corresponding time semi-discrete schemes are established,and their stability are analyzed by Z-transform.Then,two fourth-order time semi-discrete schemes are established based on the Weighted and Shifted Lubich difference(WSLD)operators.Lastly,the second-order,third-order and fourth-order time semi-discrete schemes are established based on the Lubich difference operator,and the corresponding stability theorem are given by Z-transform.(2)The Sinc-Galerkin fully discrete schemes for time fractional Schr?dinger equation.For the one-dimensional and two-dimensional time fractional Schr?dinger equations,based on the time semi-discrete schemes,the spatial operators are discretized by the Sinc-Galerkin method,and eight full discrete schemes for solving time fractional Schr?dinger equations are established.In particular,for solving the discrete scheme of two-dimensional time fractional Schr?dinger equation,the Kronecker product is used to rewrite the discrete scheme into a large sparse discrete system,and then solved as a Sylvester equation form.Finally,the availability of the established schemes are verified by taking different parameters by numerical examples.At the same time,the high order convergence in time and exponential order convergence in space are verified.(3)The Sinc-Collocation fully discrete schemes for time fractional Schr?dinger equation.For the time operators are still discretized by the proposed time semi-discrete schemes and the space operators are discretized by the Sinc-Collocation method,the corresponding full-discrete scheme are established for the time fractional Schr?dinger equation of one and two dimensions,respectively.Finally,the validity of the proposed schemes are verified by numerical examples in one and two dimensions,respectively.The results show that the Sinc-Collocation method is not only exponentially convergent,but also has good accuracy for the singularity problem.