Dissertation/Thesis Abstract

Integral Equation-Based Numerical Methods for the Time-Dependent Schrödinger Equation
by Kaye, Jason, Ph.D., New York University, 2020, 137; 27666347
Abstract (Summary)

In the first part of this dissertation, we introduce a new numerical method for the solution of the time-dependent Schrodinger equation (TDSE) with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light-matter interaction.

The primary computational challenge in using the Volterra formulation is the application of a space-time history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex-frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT.

Our approach is related to pseudo-spectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive high-order implicit time marching schemes, and naturally includes time-dependent potentials.

In the second part, we explore a more traditional approach to the problem of artificial boundary conditions for the free space TDSE. We derive exact, nonlocal transparent boundary conditions (TBCs) for the TDSE set in free space, valid in arbitrary piecewise smooth computational domains. These conditions are written in terms of single and double layer Schrodinger potentials which depend on the full history of the boundary data. The Fourier recurrence scheme used for the Volterra integral operator may be applied to a certain far history part of the layer potentials. The near history part is computed by a scheme involving projection of the boundary data onto Chebyshev polynomials in time, and the butterfly algorithm. The result is an efficient method to compute exact TBCs which is applicable, for example, on a box-shaped domain with a Cartesian grid.

Indexing (document details)
Advisor: Greengard, Leslie
Commitee: Barnett, Alex, O'Neil, Mike, Cerfon, Antoine, Berger, Marsha
School: New York University
Department: Mathematics
School Location: United States -- New York
Source: DAI-B 81/10(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Applied Mathematics, Computational physics
Keywords: Contour deformation, Integral equations, Light-matter interaction, Pseudospectral methods, Time-dependent Schrödinger equation, Transparent boundary conditions
Publication Number: 27666347
ISBN: 9798607310318
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