Deborah K. Watson

Frequency response characteristics of electroreceptors in the weakly electric fish,Gymnotus carapo

SummaryFrequency response characteristics of phasic electroreceptors in the weakly electric pulse fish,Gymnotus carapo, were related to the character of a receptor signal (RS) recorded externally over the receptor pore. Receptors that were equally sensitive to a broad range of frequencies produced RSs of a single hump, whereas sharply tuned receptors produced oscillatory RSs (Fig. 4A). Furthermore, the frequency of the recorded oscillation provided a reliable estimate of the receptors best frequency (Fig. 4B).The two major classes of electroreceptors, burst duration coders and pulse markers, could not be distinguished on the basis of best frequency (BF), Q5, or threshold at BF (Fig. 1A-C). These parameters were widely distributed in both classes. Pulse markers, however, never responded to a pulse stimulus with more than one spike and tended to have shorter latencies than burst duration coders (Fig. 2).Power spectra were calculated for local EOD waveforms recorded with three different electrode configurations (Figs. 6B, 7A-J). The local spectral peak power measured transepidermally in the rostral half of the body was 540±52 Hz, whereas the spectral peak power of the H-T waveform was 1127±90 Hz. Electroreceptors were not sharply tuned to the fishs own EOD, but showed a decrease in median BF from the rostral region where it was 950 Hz to the middle region where it was 350 Hz (Fig. 8A-D).

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...

Determination of partial widths using properly normalised resonance wavefunctions

A procedure based on the definition of a resonance state in terms of the poles of the total Greens function is used to obtain properly normalised resonance wavefunctions and the corresponding partial widths for a model multi-channel problem. The author uses a recently introduced resonance method which searches the complex momentum plane for the poles of the Schwinger T matrix. The residues of the resonant T matrix are determined and a new expression for the normalisation factor for the resonance wavefunction is derived.

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.

Schwinger Variational Methods

Publisher Summary This chapter reviews the development of the Schwinger variational (SV) method from its introduction in 1947. The first application of the SV principle was published in 1949 by Blatt and Jackson. They successfully studied neutron–proton scattering below 10 MeV by using the SV principle to obtain an expansion for the phase shift in powers of the energy. The re-introduction of the SV method into atomic physics occurred slightly later and for different reasons than the nuclear physics studies. The major impetus did not come from nuclear physics or from efforts to find an anomaly-free variational method, but rather from attempts to use discrete-basis-set methods to describe scattering phenomena. The determination of resonance energies and widths using the SV principle is a fairly recent application of this variational principle and offers an intriguing alternative to the many Hamiltonian-based resonance methods. The diversity of resonance methods reflects the different manifestations of a resonance including the rapid variation of the phase shift used by Breit–Wigner methods, the purely outgoing asymptotic behavior of the resonance wave function employed by the complex rotation, and complex R matrix method, as well as the vanishing of the T matrix denominator used by the Schwinger method.

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.

Determination of accurate quantum defects and wavefunctions for alkali Rydberg states with high principal quantum numbers

Quantum defects and the corresponding wavefunctions are determined for lithium, sodium and potassium using a new method that searches for the poles of the Schwinger T matrix along the negative real energy axis. A Coulomb Greens function is used to factor out the effect of the long-range potential. Model potentials are employed to include the effects of core polarisation and correlation. Quantum defects accurate to 1% are obtained with small basis sets and small grids. The method has been tested and found to be numerically stable for states with principal quantum number n as high as 80.

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.

Bose-Einstein condensation in variable dimensionality

We introduce dimensional perturbation techniques to Bose-Einstein condensation of inhomogeneous alkali-metal gases. The perturbation parameter is δ= 1/κ, where K depends on the effective dimensionality of the condensate and on the angular momentum quantum number. We derive a simple approximation that is more accurate and flexible than the N→∞ Thomas-Fermi ground-state approximation of the Gross-Pitaevskii equation. The approximation presented here is well suited for calculating properties of states in three dimensions and in low-effective dimensionality, such as vortex states in a highly anisotropic trap.

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...