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

Conceptual Problems in Scattering from Localized non-Hermitian Potentials

We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square-wells or delta-functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly-defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering centre, and the mechanism for this is the non-locality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.Comment: The emphasis has been changed from v1, recognizing that it makes physical sense that the wave functions of scattering states are fundamentally changed in the quasi-Hermitian framework. In contrast, bound states should not be significantly affected by the introduction of a distant non-Hermitian scattering potentia

A comparison of measured and theoretical predictions for STS ascent and entry sonic booms

Sonic boom measurements have been obtained during the flights of STS-1 through 5. During STS-1, 2, and 4, entry sonic boom measurements were obtained and ascent measurements were made on STS-5. The objectives of this measurement program were (1) to define the sonic boom characteristics of the Space Transportation System (STS), (2) provide a realistic assessment of the validity of xisting theoretical prediction techniques, and (3) establish a level of confidence for predicting future STS configuration sonic boom environments. Detail evaluation and reporting of the results of this program are in progress. This paper will address only the significant results, mainly those data obtained during the entry of STS-1 at Edwards Air Force Base (EAFB), and the ascent of STS-5 from Kennedy Space Center (KSC). The theoretical prediction technique employed in this analysis is the so called Thomas Program. This prediction technique is a semi-empirical method that required definition of the near field signatures, detailed trajectory characteristics, and the prevailing meteorological characteristics as an input. This analytical procedure then extrapolates the near field signatures from the flight altitude to an altitude consistent with each measurement location

Analysis of unidirectional non-paraxial invisibility of purely reflective PT-symmetric volume gratings

We study the diffraction produced by a slab of purely reflective PT-symmetric volume Bragg grating that combines modulations of refractive index and gain/loss of the same periodicity with a quarter-period shift between them. Such a complex grating has a directional coupling between the different diffraction orders, which allows us to find an analytic solution for the first three orders of the full Maxwell equations without resorting to the paraxial approximation. This is important, because only with the full equations can the boundary conditions, allowing for the reflections, be properly implemented. Using our solution we analyze unidirectional invisibility of such a grating in a wide variety of configurations.Comment: 17 pages, 11 figures. arXiv admin note: text overlap with arXiv:1412.050

Full-potential modeling of blade-vortex interactions

A comparison is made of four different models for predicting the unsteady loading induced by a vortex passing close to an airfoil. (1) The first model approximates the vortex effect as a change in the airfoil angle of attack. (2) The second model is related to the first but, instead of imposing only a constant velocity on the airfoil, the distributed effect of the vortex is computed and used. This is analogous to a lifting surface method. (3) The third model is to specify a branch cut discontinuity in the potential field. The vortex is modeled as a jump in potential across the branch cut, the edge of which represents the center of the vortex. (4) The fourth method models the vortex expressing the potential as the sum of a known potential due to the vortex and an unknown perturbation due to the airfoil. The purpose of the current study is to investigate the four vortex models described above and to determine their relative merits and suitability for use in large three-dimensional codes