A High‐Order Integral Equation‐Based Solver for the Time‐Dependent Schrödinger Equation

We introduce a numerical method for the solution of the time-dependent Schrodinger equation 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. We present examples in one and two dimensions showing spectral accuracy in space and eighth-order accuracy in time for both periodic and free space problems.

[1]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[2]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[3]  J. Boyd.  Chebyshev and Fourier Spectral Methods , 1989 .

[4]  Time‐dependent treatment of scattering: Integral equation approaches using the time‐dependent amplitude density , 1990 .

[5]  Semigroup linearization for nonlinear parabolic equations , 1991 .

[6]  Claude Leforestier,et al.  A comparison of different propagation schemes for the time dependent Schro¨dinger equation , 1991 .

[7]  A. V. Popov,et al.  Implementation of transparent boundaries for numerical solution of the Schrödinger equation , 1991 .

[8]  Omar A. Sharafeddin,et al.  Time dependent integral equation approaches to quantum scattering: Comparative application to atom--rigid rotor multichannel scattering , 1992 .

[9]  Vladimir Rokhlin,et al.  Fast Fourier Transforms for Nonequispaced Data , 1993, SIAM J. Sci. Comput..

[10]  John Strain,et al.  Fast Adaptive Methods for the Free-Space Heat Equation , 1994, SIAM J. Sci. Comput..

[11]  R. Strichartz.  A Guide to Distribution Theory and Fourier Transforms , 1994 .

[12]  Igor V. Puzynin,et al.  Integral boundary conditions for the time-dependent Schrödinger equation: Atom in a laser field , 1999 .

[13]  Bradley K. Alpert,et al.  Hybrid Gauss-Trapezoidal Quadrature Rules , 1999, SIAM J. Sci. Comput..

[14]  Sergio Blanes,et al.  Splitting methods for the time-dependent Schrödinger equation , 2000 .

[15]  Leslie Greengard,et al.  Spectral Approximation of the Free-Space Heat Kernel , 2000 .

[16]  A. Iserles,et al.  Lie-group methods , 2000, Acta Numerica.

[17]  Fernando Casas,et al.  Improved High Order Integrators Based on the Magnus Expansion , 2000 .

[18]  Christophe Besse,et al.  CONSTRUCTION, STRUCTURE AND ASYMPTOTIC APPROXIMATIONS OF A MICRODIFFERENTIAL TRANSPARENT BOUNDARY CONDITION FOR THE LINEAR SCHRÖDINGER EQUATION , 2001 .

[19]  Christian Lubich,et al.  Fast Convolution for Nonreflecting Boundary Conditions , 2002, SIAM J. Sci. Comput..

[20]  Achim Schädle,et al.  Non-reflecting boundary conditions for the two-dimensional Schrödinger equation , 2002 .

[21]  Christian Lubich,et al.  Integrators for quantum dynamics : a numerical analyst ’ s brief review , 2002 .

[22]  Matthias Ehrhardt,et al.  Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability , 2003 .

[23]  A. Bourlioux,et al.  High-order multi-implicit spectral deferred correction methods for problems of reactive flow , 2003 .

[24]  Marlis Hochbruck,et al.  On Magnus Integrators for Time-Dependent Schrödinger Equations , 2003, SIAM J. Numer. Anal..

[25]  Angel Rubio,et al.  Propagators for the time-dependent Kohn-Sham equations. , 2004, The Journal of chemical physics.

[26]  Leslie Greengard,et al.  Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension , 2004 .

[27]  Houde Han,et al.  Exact artificial boundary conditions for the Schrödinger equation in $R ^2$ , 2004 .

[28]  Leslie Greengard,et al.  Accelerating the Nonuniform Fast Fourier Transform , 2004, SIAM Rev..

[29]  Christophe Besse,et al.  Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions , 2004, Math. Comput..

[30]  Houde Han,et al.  EXACT ARTIFICIAL BOUNDARY CONDITIONS FOR SCHRÖDINGER EQUATION IN R2∗ , 2004 .

[31]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[32]  André D. Bandrauk,et al.  Complex integration steps in decomposition of quantum exponential evolution operators , 2006 .

[33]  Thomas Hagstrom,et al.  On the spectral deferred correction of splitting methods for initial value problems , 2006 .

[34]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

[35]  Houde Han,et al.  Numerical solutions of Schrödinger equations in ℝ3 , 2007 .

[36]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[37]  Leslie Greengard,et al.  Efficient representation of nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions , 2008 .

[38]  Sverker Holmgren,et al.  Accurate time propagation for the Schrodinger equation with an explicitly time-dependent Hamiltonian. , 2008, The Journal of chemical physics.

[39]  Tobin A. Driscoll,et al.  The chebop system for automatic solution of differential equations , 2008 .

[40]  Germund Dahlquist,et al.  Numerical Methods in Scientific Computing: Volume 1 , 2008 .

[41]  A. Ostermann,et al.  High order splitting methods for analytic semigroups exist , 2009 .

[42]  Stéphane Descombes,et al.  Splitting methods with complex times for parabolic equations , 2009 .

[43]  André D. Bandrauk,et al.  Mathematical modeling of boundary conditions for laser‐molecule time‐dependent Schrödinger equations and some aspects of their numerical computation—One‐dimensional case , 2009 .

[44]  R M Feshchenko,et al.  Exact transparent boundary condition for the parabolic equation in a rectangular computational domain. , 2011, Journal of the Optical Society of America. A, Optics, image science, and vision.

[45]  Leslie Greengard,et al.  A new integral representation for quasi-periodic scattering problems in two dimensions , 2011 .

[46]  Matthias Ehrhardt,et al.  Discrete transparent boundary conditions for the Schr ¨ odinger equation on circular domains , 2012 .

[47]  Fernando Casas,et al.  Optimized high-order splitting methods for some classes of parabolic equations , 2011, Math. Comput..

[48]  Emmanuel Lorin,et al.  Atoms and molecules in intense laser fields: gauge invariance of theory and models , 2013, 1302.2932.

[49]  Exact transparent boundary condition for the three-dimensional Schrödinger equation in a rectangular cuboid computational domain. , 2013 .

[50]  Lloyd N. Trefethen,et al.  The Exponentially Convergent Trapezoidal Rule , 2014, SIAM Rev..

[51]  Angel Rubio,et al.  Modeling electron dynamics coupled to continuum states in finite volumes with absorbing boundaries , 2014 .

[52]  Yuan Liu,et al.  High order operator splitting methods based on an integral deferred correction framework , 2014, J. Comput. Phys..

[53]  Fernando Casas,et al.  An efficient algorithm based on splitting for the time integration of the Schrödinger equation , 2015, J. Comput. Phys..

[54]  V. Vaibhav.  Transparent boundary condition for numerical modeling of intense laser-molecule interaction , 2015, J. Comput. Phys..

[55]  Max Duarte,et al.  High order schemes based on operator splitting and deferred corrections for stiff time dependent PDEs , 2014, 1407.0195.

[56]  Exponential integrators in time-dependent density-functional calculations. , 2017, Physical review. E.

[57]  Armin Scrinzi,et al.  Perfect absorption in Schrödinger-like problems using non-equidistant complex grids , 2015, J. Comput. Phys..

[58]  S. Blanes,et al.  Symplectic time-average propagators for the Schrödinger equation with a time-dependent Hamiltonian. , 2017, The Journal of chemical physics.

[59]  Xavier Antoine,et al.  A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations , 2017 .

[60]  A. Popov,et al.  Exact transparent boundary conditions for the parabolic wave equations with linear and quadratic potentials , 2016, 1610.08967.

[61]  Xavier Antoine,et al.  Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains , 2018, Comput. Phys. Commun..

[62]  Arieh Iserles,et al.  Magnus-Lanczos Methods with Simplified Commutators for the Schrödinger Equation with a Time-Dependent Potential , 2018, SIAM J. Numer. Anal..

[63]  Sergio Blanes,et al.  Exponential propagators for the Schrödinger equation with a time-dependent potential. , 2018, The Journal of chemical physics.

[64]  L. Greengard,et al.  Transparent Boundary Conditions for the Time-Dependent Schr\"odinger Equation with a Vector Potential , 2018, 1812.04200.

[65]  Angel Rubio,et al.  Propagators for the Time-Dependent Kohn-Sham Equations: Multistep, Runge-Kutta, Exponential Runge-Kutta, and Commutator Free Magnus Methods. , 2018, Journal of chemical theory and computation.

[66]  Cheng Wang,et al.  On the Operator Splitting and Integral Equation Preconditioned Deferred Correction Methods for the “Good” Boussinesq Equation , 2018, J. Sci. Comput..

[67]  Lin Lin,et al.  A Mathematical Introduction to Electronic Structure Theory , 2019 .

[68]  Leslie Greengard,et al.  Fast integral equation methods for linear and semilinear heat equations in moving domains , 2019, ArXiv.

[69]  Jeremy F. Magland,et al.  A parallel non-uniform fast Fourier transform library based on an "exponential of semicircle" kernel , 2018, SIAM J. Sci. Comput..