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

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

A case for resource-conscious out-of-order processors

Modern out-of-order processors tolerate long-latency memory operations by supporting a large number of in-flight instructions. This is achieved in part through proper sizing of critical resources, such as register files or instruction queues. In light of the increasing gap between processor speed and memory latency, tolerating upcoming latencies in this way would require impractical sizes of such critical resources.To tackle this scalability problem, we make a case for resource-conscious out-of-order processors. We present quantitative evidence that critical resources are increasingly underutilized in these processors. We advocate that better use of such resources should be a priority in future research in processor architectures.Peer ReviewedPostprint (published version

Concilii Vaticani II in ordinem redigens schemata cum relationibus necnon Patrum orationes atque animadversiones Constitutio Pastoralis de Ecclesia in mundo huius temporis Gaudium et Spes. [Reseña]

Reseña de Francisco GIL HELLÍN, Concilii Vaticani II in ordinem redigens schemata cum relationibus necnon Patrum orationes atque animadversiones Constitutio Pastoralis de Ecclesia in mundo huius temporis Gaudium et Spes, Pontificia Universitas SanctaeSanctae Crucis-Libreria Editrice Vaticana, Roma-Città del Vaticano 2003, 1658 pp., 18 x 25, ISBN 88-2097-523-8

Saint Grégoire de Narek Théologien et Mystique, colloque international tenu à l'Institut Pontifical Oriental. [Reseña]

Reseña de J.-P. MAHÉ y B.L. ZEIOYAN (eds.), Saint Grégoire de Narek Théologien et Mystique, colloque international tenu à l'Institut Pontifical Oriental, 20 - 22 2005, Pontificio Istituto Orientale («Orientalia Christiana Analecta», 275), Roma 2006, 377 pp., 17 X 24, ISBN 978-88-7210-350-9