In this article we present a new method for reconstructing three-dimensional ͑3D͒ images with cylindrical symmetry from their two-dimensional projections. The method is based on expanding the projection in a basis set of functions that are analytical projections of known well-behaved functions. The original 3D image can then be reconstructed as a linear combination of these well-behaved functions, which have a Gaussian-like shape, with the same expansion coefficients as the projection. In the process of finding the expansion coefficients, regularization is used to achieve a more reliable reconstruction of noisy projections. The method is efficient and computationally cheap and is particularly well suited for transforming projections obtained in photoion and photoelectron imaging experiments. It can be used for any image with cylindrical symmetry, requires minimal user's input, and provides a reliable reconstruction in certain cases when the commonly used Fourier-Hankel Abel transform method fails.

New methods of high resolution spectral analysis of short time signals are presented. These methods utilize the filter-diagonalization approach of Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] that extracts the complex frequencies ωk and amplitudes dk from a signal C(t)=∑kdke−itωk in a small frequency interval by recasting the harmonic inversion problem as the one of a small matrix diagonalization. The present methods are rigorously adapted to the conventional case of the signal available on a sparse equidistant time grid and use a more efficient boxlike filter. Various applications are discussed, such as iterative diagonalization of large Hamiltonian matrices for calculating bound and resonance states, scattering calculations in the presence of narrow resonances, etc. For the scattering problem the harmonic inversion is directly applied to the signal cn=(χf,Tn(Ĥ)χi), generated by the dynamical system governed by a modified Chebyshev recursion, avoiding the usual recasting the problem to the time domain. Some challenging numerical examples are presented. The general filter-diagonalization method is shown to be stable and efficient for the extraction of thousands of complex frequencies ωk and amplitudes dk from a signal. When the model signal is “spoiled” by a moderate amount of an additive Gaussian noise the obtained spectral estimate is still superior to the conventional Fourier spectrum.

The stabilization method is used to calculate the density of resonance states and when applied to isolated resonances yields a most simple method for extracting the resonance energy and width.PACS numbers: 34.10.+zThe stabilization method is usually presented as a method for obtaining resonance parameters, resonance energy £^res5 ^^^ total width F from £^ bound state type calculations [1][2][3][4][5][6][7][8][9][10]. For the purposes of obtaining exclusively Eres and r scattering methods are known to be more difficult than necessary.The simple stabilization method repeatedly diagonalizes the Hamiltonian in the basis sets of ever larger extension (L) from what is believed to be the region where the resonance wave function is localized. {Q space [11], which is here taken as the localized space, operator Q projecting onto it. P space contains the asymptotes and is the orthogonal complement space.) Here for simplicity we use a basis set complete over the energy range of interest in a box of size L (see, e.g., [2, 3, 5]). The result is a stabilization diagram of the eigenenergies Ej (L) vs L as seen in Fig. 1. The physical origin of the flat region hinges on the fact that the resonance scattering wave function, unlike other continuum solutions, is localized at short range and, as such, its energy converges, i.e., is stabilized, at L beyond the Q region. £^ states mimicking nonresonant scattering states do not have this property FIG. 1. Eigenenergies Ej as a function of the box size L for the potential in Eq. (10). The potential is shown as the thick curve. and their energies generally decrease with L.In this paper we give a particularly simple way to extract from the stabilization diagram the density of resonance states p^{E) which for isolated resonances gives Eres and F. The method does not need the wave functions [4,10], nor does it use analytic continuation of the energy in the complex plane [7][8][9] or arguments about the asymptotic form of the wave functions [1-3]; it also does not have to calculate p^ by first finding (in a scattering computation) the collision lifetime matrix [12] then using the formula [13,14] p«=7r-iTrQ.(1) (2)The method does not use imaging techniques [15] or complex potentials [16].In the following we will present a novel approach for calculation of the density of resonance states using the stabilization method and then demonstrate its use on one-and two-channel model problems.Calculation of the density of resonance states.-The method is based on the density of states obtained at a value of L. This density should have a contribution from two regions which we symbolize by writing PUE) = P'1{E) + PZ{E).(3) p^{E) is the expected resonant part which stabiHzes, becoming independent of L, for L outside the Q region and which for a case of an isolated resonance is expected to be (see, e.g.,Parenthetically, for overlapping resonances, where -Eres and F cannot be defined, p^ will stabilize with a shape that varies from problem to problem but which can be Fourier analyzed in various resoluti...

Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time

The new recently introduced [J. Chem. Phys 102, 7390 (1995)] empirical recursion formula for the scattering solution is here proved to yield an exact polynomial expansion of the operator [E−(Ĥ+Γ̂)]−1, Γ̂ being a simple complex optical potential. The expansion is energy separable and converges uniformly in the real energy domain. The scaling of the Hamiltonian is trivial and does not involve complex analysis. Formal use of the energy-to-time Fourier transform of the ABC (absorbing boundary conditions) Green’s function leads to a recursion polynomial expansion of the ABC time evolution operator that is global in time. Results at any energy and any time can be accumulated simultaneously from a single iterative procedure; no actual Fourier transform is needed since the expansion coefficients are known analytically. The approach can be also used to obtain a perturbation series for the Green’s function. The new iterative methods should be of a great use in the area of the reactive scattering calculations and other related fields.

A new method of implementing scattering calculations is presented. For the S-matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal ℒ2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−Ĥ)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.

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A new version of the filter diagonalization method of diagonalizing large real symmetric Hamiltonian matrices is presented. Our previous version would first produce a small set of adapted basis functions by applying the Chebyshev polynomial expansion of the Green’s function on a generic initial vector χ. The small Hamiltonian, H, and overlap, S, matrices would then be evaluated in this adapted basis and the corresponding generalized eigenvalue problem would be solved yielding the desired spectral information. Here in analogy to a recent work by Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] H and S are computed directly using only the Chebyshev coefficients cn=〈χ|Tn(Ĥ)|χ〉, calculation of which requires a minimal storage if the Ĥ matrix is sparse. The expressions for H and S are analytically simple, computationally very inexpensive and stable. The method can be used to obtain all the eigenvalues of Ĥ using the same sequence {cn}. We present an application of the method to a realistic quantum dynamics problem of calculating all bound state energies of H3+ molecule. Since the sequence {cn} is the only input required to obtain all the eigenenergies, the present method can be reformulated for the problem of spectral analysis of a real symmetric time signal defined on an equidistant time grid. The numerical example considers a model signal C(tn)=∑kdk cos(tnωk) generated by a set of N=100 000 frequencies and amplitudes, (ωk,dk). It is demonstrated that all the ωk’s and dk’s can be obtained to very high precision using the minimal information, i.e., 200 000 sampling points.