An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential

The potential -x^4, which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integral systems, was subsequently constructed for the general class of potentials -(ix)^N. For PT-symmetric but non-Hermitian Hamiltonians the natural PT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m^2x^2+igx^3. However, until now, the -x^4 potential has proved intractable. In the present paper we give explicit, closed-form expressions for Q and h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.Comment: New reference [10] added and discussed. Minor typographical correction

Dynamical evolution of a doubly-quantized vortex imprinted in a Bose-Einstein Condensate

The recent experiment by Y. Shin \emph{et al.} [Phys. Rev. Lett. \textbf{93}, 160406 (2004)] on the decay of a doubly quantized vortex imprinted in $^{23}$Na condensates is analyzed by numerically solving the Gross-Pitaevskii equation. Our results, which are in very good quantitative agreement with the experiment, demonstrate that the vortex decay is mainly a consequence of dynamical instability. Despite apparent contradictions, the local density approach is consistent with the experimental results. The monotonic increase observed in the vortex lifetimes is a consequence of the fact that, for large condensates, the measured lifetimes incorporate the time it takes for the initial perturbation to reach the central slice. When considered locally, the splitting occurs approximately at the same time in every condensate, regardless of its size.Comment: 5 pages, 4 figure

An Extinction Map and Color Magnitude Diagram for the Globular Cluster NGC 3201

Differential $E_{V-I}$ variations of up to $\sim 0.2$ mag on a scale of arcminutes across NGC 3201 are presented in the form of an extinction map. This map, created by calculating average $E_{V-I}$ values for stars in small subregions of the field with respect to a fiducial region, greatly improves the appearance of the CMD of the cluster. We describe how we implemented this technique in detail with our data for NGC 3201. A comparison between our map and that of the same region extracted from the COBE/DIRBE reddening maps published by Schlegel, Finkbeiner, & Davis (1998) (SFD) displays larger-scale similarities between the two maps as well as smaller-scale features which show up in our map but not in the SFD map. Several methods of determining an $E_{V-I}$ zeropoint to add to our differential extinction map are presented. Isochrone fitting proved to be the most successful one, but it produces an average $E_{V-I}$ for the cluster which is smaller than previously published values by $\sim 1.5\sigma$. Finally, our results seem to support the statement by Arce & Goodman (1999) that the SFD maps overestimate the reddening in regions of high extinction.Comment: 19 pages, 12 figures, 1 table, accepted for publication in AJ (March 2001). Full resolution version may be obtained at http://www.astro.lsa.umich.edu/users/kaspar/html/ngc3201.pdf (PDF) and at http://www.astro.lsa.umich.edu/users/kaspar/html/ngc3201.ps.gz (PS

Enlarging instruction streams

The stream fetch engine is a high-performance fetch architecture based on the concept of an instruction stream. We call a sequence of instructions from the target of a taken branch to the next taken branch, potentially containing multiple basic blocks, a stream. The long length of instruction streams makes it possible for the stream fetch engine to provide a high fetch bandwidth and to hide the branch predictor access latency, leading to performance results close to a trace cache at a lower implementation cost and complexity. Therefore, enlarging instruction streams is an excellent way to improve the stream fetch engine. In this paper, we present several hardware and software mechanisms focused on enlarging those streams that finalize at particular branch types. However, our results point out that focusing on particular branch types is not a good strategy due to Amdahl's law. Consequently, we propose the multiple-stream predictor, a novel mechanism that deals with all branch types by combining single streams into long virtual streams. This proposal tolerates the prediction table access latency without requiring the complexity caused by additional hardware mechanisms like prediction overriding. Moreover, it provides high-performance results which are comparable to state-of-the-art fetch architectures but with a simpler design that consumes less energy.Peer ReviewedPostprint (published version

On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex PTPT-Symmetric Quartic Hamiltonian

It can be shown using operator techniques that the non-Hermitian $PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential $-gx^4$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naive derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.Comment: LaTeX, 12 pages, no figure

Static locality analysis for cache management

Most memory references in numerical codes correspond to array references whose indices are affine functions of surrounding loop indices. These array references follow a regular predictable memory pattern that can be analysed at compile time. This analysis can provide valuable information like the locality exhibited by the program, which can be used to implement more intelligent caching strategy. In this paper we propose a static locality analysis oriented to the management of data caches. We show that previous proposals on locality analysis are not appropriate when the proposals have a high conflict miss ratio. This paper examines those proposals by introducing a compile-time interference analysis that significantly improve the performance of them. We first show how this analysis can be used to characterize the dynamic locality properties of numerical codes. This evaluation show for instance that a large percentage of references exhibit any type of locality. This motivates the use of a dual data cache, which has a module specialized to exploit temporal locality, and a selective cache respectively. Then, the performance provided by these two cache organizations is evaluated. In both organizations, the static locality analysis is responsible for tagging each memory instruction accordingly to the particular type(s) of locality that it exhibits.Peer ReviewedPostprint (published version