# Schrodinger's equation over a randomized grid

I am interested in solutions to $$\frac{d}{dt} \Psi = -iH \Psi$$ for $H$ hermitian and time independent. This boils down to evaluating $$\Psi(t) = e^{-iHt}\Psi_0$$ at points of interest $t_n$. I want to quickly compute $\Psi(t)$ up to a large final time $T$ by spacing the $t_n$ randomly and advancing the propagator over the nonuniform grid.

To be specific, fix $\Delta t$ and draw $t_n$ uniformly at random from $\Delta t, 2\Delta t, ..., N\Delta t$. The gaps $t_{n+1}-t_{n}$ will be distributed geometrically (ie we will have large gaps).

Any ideas how I could compute $e^{-iHt}$ accurately and efficiently when $t$ may be large?

Fast approximations that I know of (eg Strang splitting, Trotter product) are very efficient but I have found them to be horribly inaccurate and unstable for large gaps.

Machine precision accuracy is achieved by expanding $e^{-iHt}$ in Chebyshev polynomials, but the computational cost (measured by counting applications of $H$) scales linearly with the timestep size and we get to $T$ no faster.

I am beginning to think that spectrally accurate approximations of $e^{-iHt}$ provably require linearly more applications of $H$ as $T$ increases. Does anyone know if this is proven?

Thanks for solving my thesis problem,

Ryan

• Is there a compelling reason for spacing the $t_n$ randomly?
– j.c.
Nov 7, 2010 at 12:57
• Yes, it would let you reconstruct the spectrum of $H$ from few $t_n$. However, unless there is something very creative out there (cmon theoretical physics!) my approximation problem would be the same one that a person interested in $\Delta t$ large would face. Nov 7, 2010 at 20:10
• Have you looked at "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later" www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf? Nov 8, 2010 at 17:57
• Yes, but I have a little more structure then that paper deals with. One difference I have is the $iH$ as opposed from simply $H$. Also, I think I can just diagonalize $H$ and reduce the problem to one of approximating $e^{-ix}$. Chebyshev polynomials are orthogonal wrt a weighted norm and I would not be surprised if what I currently have is optimal. Nov 8, 2010 at 22:47

E.g. $exp(-ihH) = exp(-ih/2 H) exp(-ih/2 H)$, and then approximate the r.h.s. exponentials with some splitting. Think of it as intermediate stages if you don't like the idea of using a smaller time-step.