Multidimensional supersymmetric quantum mechanics: spurious states for the tensor sector two Hamiltonian.

We show that there exist spurious states for the sector two tensor Hamiltonian in multidimensional supersymmetric quantum mechanics. For one-dimensional supersymmetric quantum mechanics on an infinite domain, the sector one and two Hamiltonians have identical spectra with the exception of the ground state of the sector one. For tensorial multidimensional supersymmetric quantum mechanics, there exist normalizable spurious states for the sector two Hamiltonian with energy equal to the ground state energy of the sector one. These spurious states are annihilated by the adjoint charge operator, and hence, they do not correspond to physical states for the original Hamiltonian. The Hermitian property of the sector two Hamiltonian implies the orthogonality between spurious and physical states. In addition, we develop a method for construction of a specific form of the spurious states for any quantum system and also generate several spurious states for a two-dimensional anharmonic oscillator system and for the hydrogen atom.

Multidimensional supersymmetric quantum mechanics: a scalar Hamiltonian approach to excited states by the imaginary time propagation method.

Supersymmetric quantum mechanics (SUSY-QM) is shown to provide a novel approach to the construction of the initial states for the imaginary time propagation method to determine the first and second excited state energies and wave functions for a two-dimensional system. In addition, we show that all calculations are carried out in sector one and none are performed with the tensor sector two Hamiltonian. Through our tensorial approach to multidimensional supersymmetric quantum mechanics, we utilize the correspondence between the eigenstates of the sector one and two Hamiltonians to construct appropriate initial sector one states from sector two states for the imaginary time propagation method. The imaginary time version of the time-dependent Schrödinger equation is integrated to obtain the first and second excited state energies and wave functions using the split operator method for a two-dimensional anharmonic oscillator system and a two-dimensional double well potential. The computational results indicate that we can obtain the first two excited state energies and wave functions even when a quantum system does not exhibit any symmetry. Moreover, instead of dealing with the increasing computational complexity resulting from computations in the tensor sector two Hamiltonian, this study presents a new supersymmetric approach to calculations of accurate excited state energies and wave functions by directly using the scalar sector one Hamiltonian.

Quantum mechanical generalized phase-shift approach to atom-surface scattering: a Feshbach projection approach to dealing with closed channel effects.

We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schrödinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schrödinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.

New generalization of supersymmetric quantum mechanics to arbitrary dimensionality or number of distinguishable particles.

We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schrödinger operator can be factored using a 3N-component gradient operator and introducing vector "charge" operators: Q(1) and Q(1)(dagger). Using these operators, we can write the original (scalar) Hamiltonian as H(1) = Q(1)(dagger) x Q(1) + E(0)((1)), where E(0)((1)) is the ground-state energy. The second sector Hamiltonian is a tensor given by H(2) = Q(1)Q(1)(dagger) + E(0)((1)) and is isospectral with H(1). The vector ground state of sector 2, psi(0)((2)), canbe used with the charge operator Q(1)(dagger) to obtain the excited-state wave function of the first sector. In addition, we show that H(2) can also be factored in terms of a sector 2 vector superpotential with components W(2j) = -(partial partial differential ln psi(0j)((2)))/partial partial differentialx(j). Here psi(0j)((2)) is the jth component of psi(0)((2)). Then one obtains charge operators Q(2) and Q(2)(dagger) so that the second sector Hamiltonian can be written as H(2) = Q(2)(dagger)Q(2) + E(0)((2)). This allows us to define a third sector Hamiltonian which is a scalar, H(3) = Q(2) x Q(2)(dagger) + E(0)((2)). This prescription continues with the sector Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled 1D oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix.

Supersymmetric approach to excited states.

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrodinger equation. Since the ground state of each partner potential is nodeless, we avoid any "node" problem typically associated with the Monte Carlo technique. Although we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.

Supersymmetric quantum mechanics, excited state energies and wave functions, and the Rayleigh-Ritz variational principle: a proof of principle study.

In addition to ground state wave functions and energies, excited states and their energies are also obtained in a standard Rayleigh-Ritz variational calculation. However, their accuracy is generally much lower. Using the super-symmetric (SUSY) form of quantum mechanics, we show that better accuracy and more rapid convergence can be obtained by taking advantage of calculations of the ground states of higher sector SUSY Hamiltonians, followed by application of the SUSY "charge operators". Our proof of principle study uses a general family of one-dimensional anharmonic oscillator models. We first obtain the exact, analytic ground states for a general family of anharmonic systems. We give the general, factorized form of the Hamiltonian for the hierarchy that arises in SUSY theory. The "charge" operators can then be used to convert states among the sectors. We illustrate the approach with two specific anharmonic oscillator models. Using the ground state of the second sector Hamiltonian, we show that the corresponding excited state energies and wave functions of the first sector are accurately obtained by applying the charge operators, using significantly smaller basis sets than are required in a standard variational approach applied to the original Schrodinger equation. This is a consequence of the higher accuracy of the Rayleigh-Ritz variational method when applied for ground states.

The Heisenberg-Weyl algebra on the circle and a related quantum mechanical model for hindered rotation.

We discuss a periodic variant of the Heisenberg-Weyl algebra, associated with the group of translations and modulations on the circle. Our study of uncertainty minimizers leads to a periodic version of canonical coherent states. Unlike the canonical, Cartesian case, there are states for which the uncertainty product associated with the generators of the algebra vanishes. Next, we explore the supersymmetric (SUSY) quantum mechanical setting for the uncertainty-minimizing states and interpret them as leading to a family of "hindered rotors". Finally, we present a standard quantum mechanical treatment of one of these hindered rotor systems, including numerically generated eigenstates and energies.

Nanostructures with long-range order in monolayer self-assembly.

A key ingredient for continued expansion of nanotechnologies is the ability to create perfectly ordered arrays on a small scale with both site and size control. Self-assembly-i.e., the spontaneous formation of nanostructures-is a highly promising alternative to traditional fabrication methods. However, efforts to obtain perfect long-range order via self-assembly have been frustrated in practice as ensuing patterns contain defects. We use an idea based on the fundamental physics of pattern formation to introduce a strategy to consistently obtain perfect patterns.

Three-dimensional isotropic wavelets for post-acquisitional extraction of latent images of atherosclerotic plaque components from micro-computed tomography of human coronary arteries.

RATIONALE AND OBJECTIVES: The capability of wavelet transforms to separate signals into frequency bands is the basis for its use in image compression and storage, data management and transmission, and, recently, extraction of latent images of tissue components from noisy medical images. Analysis of temporal variations of radiofrequency backscatter of intravascular ultrasound with one-dimensional wavelets can detect lipid-laden plaque in coronary arteries with a sensitivity and specificity of >80%. In this study we evaluate the capability of a novel, 3-dimensional isotropic wavelet analysis to perform high resolution, non-directionally biased, statistically reliable, non-invasive discrimination between components of human coronary atherosclerotic plaques in micro-CT. MATERIALS AND METHODS: Coronary artery segments (5-15 mm) were excised at necropsy from 18 individuals with advanced coronary atherosclerosis. Specimens were imaged using a GE Locus SP ex vivo micro-CT scanner and processed for histological correlation (833 sections). The isotropic wavelet constructs were applied to the entire volume of CT data of each arterial segment to distinguish tissue textures of varying scales and intensities. Voxels were classified and plaque characterization achieved by comparing the relative magnitudes of these wavelet constituents to that of several reference plaque tissue components. RESULTS: Processing of micro-CT images via these isotropic wavelet algorithms permitted 3-D, color-coded, high resolution, digital discrimination between lumen, calcific deposits, lipid-rich deposits, and fibromuscular tissue providing detail not possible with conventional thresholding based on Hounsfield intensity units. Using the isotropic wavelets (with histology as the gold standard), lipid-rich pools approaching the size of the filter for the isotropic wavelet algorithm (0.25 mm [250 microns] in length) were identified with 81% sensitivity and 86% specificity. Calcific deposits, fibromuscular tissue, and lumen equal to or larger than the wavelet filter size were detected without error (100% sensitivity and specificity). CONCLUSION: Isotropic wavelet analysis permits high resolution, multi-dimensional identification of coronary atherosclerotic plaque components in micro-CT with sensitivity and specificity similar to that achieved with data obtained invasively (from IVUS in vivo) using one-dimensional wavelets. Further studies are necessary to test the applicability of this technology to clinical, multi-detector scanners.

Quantification of roughness of calcific deposits in computed tomography scans of human coronary arteries.

OBJECTIVES: The incidence of coronary artery disease has been shown to be greater in patients with calcific deposits than in those without. It has been suggested that the pattern of distribution of coronary calcific deposits within coronary arteries is of greater predictive value for acute coronary events than the overall quantity. Whether roughness of calcific deposits is a predictor of acute coronary events is not known. We derived and tested an algorithm, Voxel-Based Bosselation (VBB), for noninvasive quantification of roughness of calcific deposits in human coronary arteries imaged by computed tomography (CT). METHODS AND RESULTS: VBB was tested on 213 coronary calcific deposits from electron beam CT scans of 27 patients. This algorithm evaluates the 3-dimensional connectedness of surface voxels of each deposit: smooth masses have low VBB and rough masses high VBB. The algorithm was calibrated with artificially generated phantoms as well as background noise mimicking calcific deposits and surrounding heart tissue. The VBB algorithm is applicable to calcific deposits of all scales and gradations. The VBB values of the deposits in this study did not correlate with deposit size further supporting its validity as a measurement of roughness. The VBB index corresponded directly with visual reconstruction using Phong-shaded algorithms. CONCLUSIONS: The VBB index, derived here, is a noninvasive method of quantifying the roughness of calcific deposits in CT scan data which can now be used in future clinical studies to determine possible correlations with increased plaque vulnerability and major acute coronary events.

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 degrees. 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.

Usefulness of multidetector computed tomography for noninvasive evaluation of coronary arteries in asymptomatic patients.

This editorial addresses the capabilities, limitations, and potential of multidetector computed tomography (MDCT) for the noninvasive evaluation of coronary arteries in asymptomatic patients. The quantification of coronary calcium with MDCT correlates highly with that obtained by electron-beam computed tomography, but to date, neither has the capability of assessing the distribution of various morphologic patterns of calcium and their relation to other "soft" plaque components. Although MDCT can assess the thickness of the atherosclerotic wall and can readily identify calcific deposits, further plaque characterization (e.g., lipid pools and fibrous tissue), a prerequisite for the identification of most vulnerable lesions, is not yet a workable reality, even with the 64-slice machines in their current configuration. The noninvasive identification by MDCT of plaque components subtending vulnerable lesions will require additional improvement in the primary instrumentation, the use of hybrid constructs (e.g., with positron emission tomography and magnetic resonance imaging), the development of novel methods of post-acquisitional analysis to extract latent images of plaque components (e.g., signal analysis based on 3-dimensional wavelets), or the adaptation of molecular imaging techniques at the cell and gene levels to computed tomography. Such unique approaches may soon contribute a long list of additional parameters that could be evaluated on a noninvasive basis as predictors of acute coronary syndromes and overall patient vulnerability.

Statistical characterizations of spatiotemporal patterns generated in the Swift-Hohenberg model.

Two families of statistical measures are used for quantitative characterization of nonequilibrium patterns and their evolution. The first quantifies the disorder in labyrinthine patterns, and captures features like the domain size, defect density, variations in wave number, etc. The second class of characteristics can be used to quantify the disorder in more general nonequilibrium structures, including those observed during domain growth. The presence of distinct stages of relaxation in spatiotemporal dynamics under the Swift-Hohenberg equation is analyzed using both classes of measures.

Inverse scattering theory: renormalization of the Lippmann-Schwinger equation for acoustic scattering in one dimension.

The most robust treatment of the inverse acoustic scattering problem is based on the reversion of the Born-Neumann series solution of the Lippmann-Schwinger equation. An important issue for this approach to inversion is the radius of convergence of the Born-Neumann series for Fredholm integral kernels, and especially for acoustic scattering for which the interaction depends on the square of the frequency. By contrast, it is well known that the Born-Neumann series for the Volterra integral equations in quantum scattering are absolutely convergent, independent of the strength of the coupling characterizing the interaction. The transformation of the Lippmann-Schwinger equation from a Fredholm to a Volterra structure by renormalization has been considered previously for quantum scattering calculations and electromagnetic scattering. In this paper, we employ the renormalization technique to obtain a Volterra equation framework for the inverse acoustic scattering series, proving that this series also converges absolutely in the entire complex plane of coupling constant and frequency values. The present results are for acoustic scattering in one dimension, but the method is general. The approach is illustrated by applications to two simple one-dimensional models for acoustic scattering.

A method to Fourier filter textured images.

An algorithm is introduced to extract an underlying image from a class of textures. It is assumed that the image is bandwidth limited and the noise is broad-band. The initial step of the algorithm extends the signal to a larger periodic image using "Distributed Approximating Functionals." The second step introduces a low-pass filter which allows the identification and elimination of the high-frequency components of the noise. The periodicity of the resulting image allows it to be Fourier filtered without aliasing. The feasibility of the algorithm is demonstrated on several noisy patterns generated in experiments and model systems. (c) 2000 American Institute of Physics.