David K. Hoffman

General, energy‐separable Faber polynomial representation of operator functions: Theory and application in quantum scattering

A general, uniformly convergent series representation of operator‐valued functions in terms of Faber polynomials is presented. The method can be used to evaluate the action of any operator‐valued function which is analytic in a simply connected region enclosed by a curve, Lγ. The three most important examples include the time‐independent Green’s operator, G+(E)=1/[E−(H−ie)], where H may be Hermitian or may also contain a negative imaginary absorbing potential, the time‐dependent Green’s or evolution operator, exp(−iHt/ℏ), and the generalized collision operator from nonequilibrium statistical mechanics, 1/[E−(L−ie)], where L is the Liouvillian operator for the Hamiltonian. The particular uniformly convergent Faber polynomial expansion employed is determined by the conformal mapping between the simply connected region external to the curve Lγ, which encloses the spectrum of H−ie (or L−ie), and the region external to a disk of radius γ. A locally smoothed conformal mapping is introduced containing a finite n...

A general time-to-energy transform of wavepackets. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Abstract Recently, new time-independent wavepacket-Schrodinger and wavepacket-Lippmann—Schwinger equations have been derived making use of absorbing potentials. We show that these equations, which are characterized by the occurrence of an initial ¢L2-wavepacket source of scattered waves, can be gotten without introducing the absorbing potential. We also show that a powerful method of solving the equations can be based on a Chebychev representation of the causal full Green function, combined with the distributed approximating function representation of the system Hamiltonian, and we present two example applications illustrating approach.

Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions

Abstract An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating δ( E – H ), the spectral density operator (SDO), the projection operator that projects out of any L 2 wavepacket the eigenstate (s) of H having energy E . If applied to an L 2 wavepacket which overlaps the interaction, it yields either scattering-type (improper) eigenstates or proper bound eigenstates. For negative energies, the exact SDO yields zero away from an eigenvalue, and yields the energy eigenstate (times a constant) when E equals an eigenvalue. The finite orthogonal polynomial expansion of the SDO, acting on an L 2 wavepacket, yields approximately zero for E not equal to an eigenvalue, and becomes nonzero in the neighborhood of an eigenvalue.

On jz‐preserving propensities in molecular collisions. I. Quantal coupled states and classical impulsive approximations

The occurrence of jz‐preserving propensities in atom–linear molecule collisions is considered within the contexts of the quantum mechanical CS approximation and of a classical model collision system. The latter involves an impulsive interaction which is the extreme limit of the class of potentials for which the CS approximation is expected to be valid. The classical model results in exact conservation of jz along a ’’kinematic apse.’’ Quantum mechanically, the CS approximation is reformulated in a manner that clearly shows the relationship between the l choice and the degree and direction of jz preservation. Away from the forward direction, the simplest choice obeying time reversal symmetry l=(l+l′)/2, is shown to result in a propensity for preserving jz along a ’’geometric apse’’ which coincides with the kinematic apse in the energy sudden limit, and for nonenergy sudden systems only differs significantly from it close to the forward direction.

Time-to-energy transform of wavepackets using absorbing potentials. Time-independent wavepacket-Schrödinger and wavepacket-Lippmann—Schwinger equations

Abstract It is shown that one may use an L 2 basis, matrix representation of the Hamiltonian, including a negative imaginary absorbing potential, to carry out arbitrarily long-time evolution of wavepackets. The time-to-energy Fourier transform of the wavepacket is carried out analytically, yielding a new type of time-independent scattering equation in which the “source” of scattered waves is the initial ( t =0) L 2 wavepacket used in the time-dependent propagation. Alternatively, one obtains the analogous time-independent, inhomogeneous wavepacket-Schrodinger equation. A banded representation of the Hamiltonian is achieved by the use of distributed approximating function theory to evaluate the kinetic energy. The resulting new time-independent wavepacket equations are solved both by matrix diagonalization of the Hamiltonian, and as inhomogeneous linear algebraic equations. The approach is illustrated by application to electron scattering (in one dimension) by a double barrier potential.

Variational principles for the time‐independent wave‐packet‐Schrödinger and wave‐packet‐Lippmann–Schwinger equations

Several variational principles, whose Euler equations are the recently derived time‐independent wave‐packet‐Schrodinger or wave‐packet‐Lippmann–Schwinger equations, are presented. A particularly attractive wave‐packet‐Kohn variational principle for either the T‐ or S‐matrix is given which yields inhomogeneous algebraic equations whose ‘‘universal inhomogeneity’’ does not depend explicitly on the collision energy. The validity of the approach is demonstrated with calculations for two simple one dimensional scattering problems and for the collinear H+H2 reactive scattering problem.

A comparative study of time dependent quantum mechanical wave packet evolution methods

We present a detailed comparison of the efficiency and accuracy of the second‐ and third‐order split operator methods, a time dependent modified Cayley method, and the Chebychev polynomial expansion method for solving the time dependent Schrodinger equation in the one‐dimensional double well potential energy function. We also examine the efficiency and accuracy of the split operator and modified Cayley methods for the imaginary time propagation.

Further analysis of solutions to the time‐independent wave packet equations of quantum dynamics. II. Scattering as a continuous function of energy using finite, discrete approximate Hamiltonians

We consider further how scattering information (the S‐matrix) can be obtained, as a continuous function of energy, by studying wave packet dynamics on a finite grid of restricted size. Solutions are expanded using recursively generated basis functions for calculating Green’s functions and the spectral density operator. These basis functions allow one to construct a general solution to both the standard homogeneous Schrodinger’s equation and the time‐independent wave packet, inhomogeneous Schrodinger equation, in the non‐interacting region (away from the boundaries and the interaction region) from which the scattering solution obeying the desired boundary conditions can be constructed. In addition, we derive new expressions for a ‘‘remainder or error term,’’ which can hopefully be used to optimize the choice of grid points at which the scattering information is evaluated. Problems with reflections at finite boundaries are dealt with using a Hamiltonian which is damped in the boundary region as was done by ...

A general, energy-separable polynomial representation of the time-independent full Green operator with application to time-independent wavepacket forms of Schrödinger and Lippmann—Schwinger equations

Abstract A general, energy-separable Faber polynomial representation of the full time-independent Green operator is presented. Non-Hermitian Hamiltonians are included, allowing treatment of negative imaginary absorbing potentials. A connection between the Faber polynomial expansion and our earlier Chebychev polynomial expansion (Chem. Phys. Letters 206 (1993) 96) is established, thereby generalizing the Chebychev expansion to the complex energy plane. The method is applied to collinear H + H2 reactive scattering.

Image denoising using a tight frame

We present a general mathematical theory for lifting frames that allows us to modify existing filters to construct new ones that form Parseval frames. We apply our theory to design nonseparable Parseval frames from separable (tensor) products of a piecewise linear spline tight frame. These new frame systems incorporate the weighted average operator, the Sobel operator, and the Laplacian operator in directions that are integer multiples of 45/spl deg/. A new image denoising algorithm is then proposed, tailored to the specific properties of these new frame filters. We demonstrate the performance of our algorithm on a diverse set of images with very encouraging results.