Martin Dunn

A linear algebraic method for exact computation of the coefficients of the 1/D expansion of the Schrödinger equation

The 1/D expansion, where D is the dimensionality of space, offers a promising new approach for obtaining highly accurate solutions to the Schrodinger equation for atoms and molecules. The method typically employs an asymptotic expansion calculated to rather large order. Computation of the expansion coefficients has been feasible for very small systems, but extending the existing computational techniques to systems with more than three degrees of freedom has proved difficult. We present a new algorithm that greatly facilitates this computation. It yields exact values for expansion coefficients, with less roundoff error than the best alternative method. Our algorithm is formulated completely in terms of tensor arithmetic, which makes it easier to extend to systems with more than three degrees of freedom and to excited states, simplifies the development of computer codes, simplifies memory management, and makes it well suited for implementation on parallel computer architectures. We formulate the algorithm f...

N identical particles under quantum confinement: a many-body dimensional perturbation theory approach

In this paper we continue our development of a dimensional perturbation theory (DPT) treatment of N identical particles under quantum confinement. DPT is a beyond-mean-field method which is applicable to both weakly and strongly-interacting systems and can be used to connect both limits. In a previous paper we developed the formalism for low-order energies and excitation frequencies. This formalism has been applied to atoms, Bose-Einstein condensates and quantum dots. One major advantage of the method is that N appears as a parameter in the analytical expressions for the energy and so results for N up to a few thousand are easy to obtain. Other properties however, are also of interest, for example the density profile in the case of a BEC,and larger N results are desirable as well. The latter case requires us to go to higher orders in DPT. These calculations require as input zeroth-order wave functions and this paper, along with a subsequent paper, addresses this issue.

On the behavior of Padé approximants in the vicinity of avoided crossings

When linear Pade summation is applied to eigenvalue perturbation expansions near regions of parameter space where those eigenvalues undergo an avoided crossing, the Pade approximants may yield levels which cross diabatically, rather than displaying the proper avoided behavior. The purpose of this study is to elucidate the reasons for the peculiar behavior of Pade approximants in such situations. In particular, we demonstrate that the diabatic crossing is a natural consequence of using the (single‐valued) Pade rational approximant to successfully resum series expansions of the multivalued energy function over much of the parameter space. This is illustrated with a perturbative treatment of the Barbanis Hamiltonian.

A complete basis for a perturbation expansion of the general N-body problem

We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N-body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N, the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 25 for all N, i.e. the problem scales as N0, although one would initially expect an N6 scaling at first order. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, SN. The resulting perturbation series is order-by-order invariant under the N! operations of the SN point group which is responsible for the slower than exponential growth of the basis. In this paper, we demonstrate the completeness of the basis and perform the first test of this formalism through first order by comparing to an exactly solvable fully interacting problem of N particles with a two-body harmonic interaction potential.

On the use of group theoretical and graphical techniques toward the solution of the general N-body problem

Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the SN group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the SN group. This invariance under SN imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under SN. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N6 scaling of these terms at first order by effectively separating the N s...

Local optimization of the summation of divergent power series

A method of optimizing a sequence of economized rational approximants (ERAs) to produce a sequence of approximants with enhanced convergence properties is described. It is shown that such a technique improves upon the error of the Pade approximants at a chosen value of the independent variable, and in some cases leads to dramatic improvement, even in cases where Pade approximants behave erratically. The procedure is tested on six known functions, with improved convergence and accuracy in each case. The procedure is then applied to the problem of evaluating a perturbation series of an atomic system, diamagnetic hydrogen, with significant improvement in both convergence and accuracy as well.

Two-Electron Excited States

We report calculations of dimensional expansions for the energies of the three excited S states of helium that correspond to one quantum in either of the three normal models of the Langmuir vibrations. Very accurate energies are obtained for the 1s2s states, which arise from excitation of the stretching vibrations. Reasonable, but somewhat less accurate, results are found for the energy of the 2p2 resonance, which corresponds to excitation in the bending vibration; this is despite an infinite number of curve crossings due to the difference in ordering of the eigenvalues at D = 3 and D → ∞. More accurate results are obtained for the three corresponding Pe states, by virtue of the interdimensional degeneracies between D = 3 and D = 5.

Analysis of the growth in complexity of a symmetry-invariant perturbation method for large N-body systems

The work required to solve for the fully interacting N boson wavefunction, which is widely believed to scale exponentially with N, has been previously shown to scale as N0 when the problem is rearranged using analytic building blocks. The exponential complexity reappears in an exponential scaling with the order of our perturbation series allowing exact analytical calculations for very large N systems through low order. In this paper, we analyse the growth in complexity with order when a normal mode basis is used.

Branch-point structure and the energy level characterization of avoided crossings

The appearance of avoided crossings among energy levels as a system parameter is varied is signaled by the presence of square-root branch points in the complex parameter-plane. Even hidden crossings, which are so gradual as to be difficult to resolve experimentally, can be uncovered by the knowledge of the locations of these branch points. As shown in this paper, there are two different analytic structures that feature square-root branch points and give rise to avoided crossings in energy. Either may be present in an actual quantum-mechanical problem. This poses special problems in perturbation theory since the analytic structure of the energy is not readily apparent from the perturbation series, and yet the analytic structure must be known beforehand if the perturbation series is to be summed to high accuracy. Determining which analytic structure is present from the perturbation series is illustrated here with the example of a dimensional perturbation treatment of the diamagnetic hydrogen problem. The br...