A SPARSE SPECTRAL METHOD FOR HAMILTONIAN EIGENSYSTEMS RYAN COMPTON Abstract. We propose a sparse spectral method for Hamiltonian eigen- systems. 1. Introduction The use of signal processing to extract spectral information of quantum mechanical systems from time dependent simulations has proven to be an ef- ficient and robust methodology for nearly thirty years [13]. First advocated by Feit, Fleck and Steiger in 1982 spectral methods have become fundamen- tal in the development of many practical quantum mechanical algorithms [7][15]. Key to the spectral method is an FFT of the autocorrelation function, P (t), of a time dependent wavefunction, ψ(x,t), into the energy domain where eigenvalues of the Hamiltonian may be easily found. Reliable deter- mination of the eigenvalues depends on a detailed energy domain represen- tation and thus requires knowledge of the autocorrelation function at a high sample rate over a long time [8]. The accuracy of the energy domain representation, ˆ P (λ), is limited by the uncertainty principle in two ways. Highly resolved energy domain signals require P (t) be known for large t while small Δt is required if one is to avoid aliasing errors. Explicitly, the lower bound on the density of states that can be resolved from a signal P (t) is 2π T where T is the total propagation time while an upper bound on the timestep Δt is controlled as Δt< 2π ΔE where ΔE is the spectral radius of the Hamiltonian [1]. Overcoming the large T uncertainty principle and allowing for short prop- agation times while still resolving closely spaced eigenvalues is the achieve- ment of the Filter Diagonalization method introduced by Neuhauser in 1990 [23] [18] [14] [17]. The small Δt bound is traditionally dealt with by truncating the potential above some cutoff value and advancing ψ over a uniform grid spaced by 1 ΔE cutof f . The purpose of this work is to avoid the small Δt bound by exploiting the spare structure of ˆ P (λ) and making use of ideas from the theory of compressed sensing [3]. Date : Oct 13, 2010. 1