Abstract:
We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme. The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schroedinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U. Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications. We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states.

Abstract:
The Schroedinger equation with an energy-dependent complex absorbing potential, associated with a scattering system, can be reduced for a special choice of the energy-dependence to a harmonic inversion problem of a discrete pseudo-time correlation function. An efficient formula for Green's function matrix elements is also derived. Since the exact propagation up to time 2t can be done with only t real matrix-vector products, this gives an unprecedently efficient scheme for accurate calculations of quantum spectra for possibly very large systems.

Abstract:
A Gaussian resolution method for the computation of equilibrium density matrices rho(T) for a general multidimensional quantum problem is presented. The variational principle applied to the ``imaginary time'' Schroedinger equation provides the equations of motion for Gaussians in a resolution of rho(T) described by their width matrix, center and scale factor, all treated as dynamical variables. The method is computationally very inexpensive, has favorable scaling with the system size and is surprisingly accurate in a wide temperature range, even for cases involving quantum tunneling. Incorporation of symmetry constraints, such as reflection or particle statistics, is also discussed.

Abstract:
Semiclassical spectra weighted with products of diagonal matrix elements of operators A_{alpha}, i.e., g_{alpha alpha'}(E) = sum_n /(E-E_n) are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.

Abstract:
In semiclassical theories for chaotic systems such as Gutzwiller's periodic orbit theory the energy eigenvalues and resonances are obtained as poles of a non-convergent series g(w)=sum_n A_n exp(i s_n w). We present a general method for the analytic continuation of such a non-convergent series by harmonic inversion of the "time" signal, which is the Fourier transform of g(w). We demonstrate the general applicability and accuracy of the method on two different systems with completely different properties: the Riemann zeta function and the three disk scattering system. The Riemann zeta function serves as a mathematical model for a bound system. We demonstrate that the method of harmonic inversion by filter-diagonalization yields several thousand zeros of the zeta function to about 12 digit precision as eigenvalues of small matrices. However, the method is not restricted to bound and ergodic systems, and does not require the knowledge of the mean staircase function, i.e., the Weyl term in dynamical systems, which is a prerequisite in many semiclassical quantization conditions. It can therefore be applied to open systems as well. We demonstrate this on the three disk scattering system, as a physical example. The general applicability of the method is emphasized by the fact that one does not have to resort a symbolic dynamics, which is, in turn, the basic requirement for the application of cycle expansion techniques.

Abstract:
We calculated all 2967 even and odd bound states of the adiabatic ground state of NO_2, using a modification of the ab initio potential energy surface of Leonardi et al. [J. Chem. Phys. 105, 9051 (1996)]. The calculation was performed by harmonic inversion of the Chebyshev correlation function generated by a DVR Hamiltonian in Radau coordinates. The relative error for the computed eigenenergies is $10^{-4}$ or better. Near the dissociation threshold the density of states is about 0.3cm$^{-1}$. Statistical analysis of the states shows some interesting structure of the rigidity parameter $\Delta_3$ as a function of energy.

Abstract:
The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the two-species PASEP. In this model, there are two species of particles, one "heavy" and one "light", on a one-dimensional finite lattice with open boundaries. Both particles can swap places with adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can swap places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our main result is a combinatorial interpretation for the stationary distribution at $q=0$ in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and thus directly explains the connection between the two. Finally, we give a conjecture that extends our formula for the stationary distribution to the $q=1$ case, using certain two-species alternative tableau.

Abstract:
We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with $\alpha$'s and $\beta$'s that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with $n$ sites, precisely $k$ of the sites are occupied by particles. This formula is an $\alpha\ /\ \beta$ generalization of the Narayana numbers.

Abstract:
The Totally Asymmetric Simple Exclusion Process (TASEP) is a non-equilibrium particle model on a finite one-dimensional lattice with open boundaries. In our earlier paper, we obtained a determinantal formula that computes the steady state probabilities of this process by the enumeration of "Catalan alternative tableaux", which are certain fillings of Young diagrams. Here, we present a new, more illuminating bijective proof of this determinantal formula using the Lindstr\"{o}m-Gessel-Viennot Lemma.

Abstract:
We study a generalization of the partially asymmetric exclusion process (PASEP) in which there are $k$ species of particles of varying weights hopping right and left on a one-dimensional lattice of $n$ sites with open boundaries. In this process, only the heaviest particle type can enter on the left of the lattice and exit from the right of the lattice. In the bulk, two adjacent particles of different weights can swap places. We prove a Matrix Ansatz for this model, in which different rates for the swaps are allowed. Based on this Matrix Ansatz, we define a combinatorial object which we call a $k$-rhombic alternative tableau, which we use to give formulas for the steady state probabilities of the states of this $k$-species PASEP. We also describe a Markov chain on the 2-rhombic alternative tableaux that projects to the 2-species PASEP.