Bloch Decomposition-Based Stochastic Galerkin/Collocation Method for Schrodinger Equation with Random Inputs
Department of Mathematical Sciences, Tsinghua University
2017-04-11 10:30-11:30 Science Building No. 1 1418
Abstract: In this talk, we focus on the analysis and numerical methods for the Schrodinger equation with lattice potential and random inputs. This is an important model in solid state physics where randomness is involved to describe some complicated phenomena that are not exactly known. Here we recall the well-known Bloch decomposition-based split-step pseudospectral method where we diagonalize the periodic part of the Hamilton operator so that the effects from dispersion and periodic lattice potential are computed together. Meanwhile, for the random nonperiodic external potential, we utilize the generalize polynomial chaos with Galerkin procedure to form an ode system which can be solved analytically. Further more, we analyse the convergence theory of the stochastic collocation method for the linear Schrodinger equation with random inputs. Based on the interpolation theories, the convergence rate depends on the regularity of the solution with respect to the random variables. Hence, we investigate the dependence of the regularity of the solution on that of the random potential and initial data. We provide sufficient conditions on the random potential and initial data to ensure the spectral convergence.