A novel spectral method for the semi-classical Schrodinger equation based on the Gaussian wave-packet transform
报告题目(Title)：A novel spectral method for the semi-classical Schrodinger equation based on the Gaussian wave-packet transform
报告人(Speaker)：周珍楠（北京大学）
地点(Place)：电子楼103
时间(Time)：11月30日（星期一），3:00pm-4:00pm
邀请人(Inviter)：蔡勇勇
报告摘要
In this work, we develop a new spectral method to solve the semi-classical Schrodinger equationbased on the Gaussian wave-packet transform (GWPT) and Hagedorn’s semi-classical wave-packet(HWP) method. The GWPT equivalently recasts the high oscillatory wave equation as a much lessoscillatory wave equation (w equation) coupled with a set of ordinary differential equations governingthe dynamics of the so-called GWPT parameters. The Hamiltonian of the w equation consists of a quadratic part and a small non-quadratic perturbation, which is of orderO(√ε). By expanding the solution of the w equation as superpositions of Hagedorn’s wave-packets, which are naturally the eigenfunctions of the quadratic part, we construct a spectral method while theO(√ε) perturbationpart is treated by Galerkin approximation. This numerical implementation of the GWPT avoid artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional case, we establish how the error of solving the semi-classical Schrodinger equation with GWPT is determined by the error of solving the w equation and the GWPT parameters. We prove that this scheme has spectral convergence with respect to the number of Hagedorn’s wave-packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method. With proper choice of wave-packets, numerical tests strongly suggest that the scheme has less computational complexity than the original GWPT method in high dimensions.
