Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. II. Quartic and Sextic Anharmonicity Cases

In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials $V(r)= r^2 + g^{2(m-1)}\, r^{2m}, m=2,3$, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8-12 figures for any $D=1,2,3\ldots$ and $g \geq 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r \geq 0$.Comment: 52 pages, 17 figures, 3 appendice

Radial power-like potentials: from the Bohr-Sommerfeld SS-state energies to the exact ones

Following our previous study of the Bohr-Sommerfeld (B-S) quantization condition for one-dimensional case (del Valle \& Turbiner (2021) \cite{First}), we extend it to $d$-dimensional power-like radial potentials. The B-S quantization condition for $S$-states of the $d$-dimensional radial Schr\"odinger equation is proposed. Based on numerical results obtained for the spectra of power-like potentials, $V(r)=r^m$ with $m \in [-1, \infty)$, the correctness of the proposed B-S quantization condition is established for various dimensions $d$. It is demonstrated that by introducing the {\it WKB correction} $\gamma$ (supposedly coming from the higher order WKB terms) into the r.h.s. of the B-S quantization condition leads to the so-called {\it exact WKB quantization condition}, which reproduces the exact energies, while $\gamma$ remains always very small. For $m=2$ (any integer $d$) and for $m=-1$ (at $d=2$) the WKB correction $\gamma=0$: for $S$ states the B-S spectra coincides with the exact ones. Concrete calculations for physically important cases of linear, cubic, quartic, and sextic oscillators, as well as Coulomb and logarithmic potentials in dimensions $d=2,3,6$ are presented. Radial quartic anharmonic oscillator is considered briefly.Comment: 15 pages, 4 figures, 4 tables; extended, some typos fixed, to be published in IJMP

Demodulation of Spatial Carrier Images: Performance Analysis of Several Algorithms Using a Single Image

http://link.springer.com/article/10.1007%2Fs11340-013-9741-6#Optical full-field techniques have a great importance in modern experimental mechanics. Even if they are reasonably spread among the university laboratories, their diffusion in industrial companies remains very narrow for several reasons, especially a lack of metrological performance assessment. A full-field measurement can be characterized by its resolution, bias, measuring range, and by a specific quantity, the spatial resolution. The present paper proposes an original procedure to estimate in one single step the resolution, bias and spatial resolution for a given operator (decoding algorithms such as image correlation, low-pass filters, derivation tools ...). This procedure is based on the construction of a particular multi-frequential field, and a Bode diagram representation of the results. This analysis is applied to various phase demodulating algorithms suited to estimate in-plane displacements.GDR CNRS 2519 “Mesures de Champs et Identification en Mécanique des Solide

Status of neutrino oscillations 2018: first hint for normal mass ordering and improved CP sensitivity

We present a new global fit of neutrino oscillation parameters within the simplest three-neutrino picture, including new data which appeared since our previous analysis~\cite{Forero:2014bxa}. In this update we include new long-baseline neutrino data involving the antineutrino channel in T2K, as well as new data in the neutrino channel, data from NO$\nu$A, as well as new reactor data, such as the Daya Bay 1230 days electron antineutrino disappearance spectrum data and the 1500 live days prompt spectrum from RENO, as well as new Double Chooz data. We also include atmospheric neutrino data from the IceCube DeepCore and ANTARES neutrino telescopes and from Super-Kamiokande. Finally, we also update our solar oscillation analysis by including the 2055-day day/night spectrum from the fourth phase of the Super-Kamiokande experiment. With the new data we find a preference for the atmospheric angle in the upper octant for both neutrino mass orderings, with maximal mixing allowed at $\Delta\chi^2 = 1.6 \, (3.2)$ for normal (inverted) ordering. We also obtain a strong preference for values of the CP phase $\delta$ in the range $[\pi,2\pi]$, excluding values close to $\pi/2$ at more than 4$\sigma$. More remarkably, our global analysis shows for the first time hints in favour of the normal mass ordering over the inverted one at more than 3$\sigma$. We discuss in detail the origin of the mass ordering, CP violation and octant sensitivities, analyzing the interplay among the different neutrino data samples.Comment: Updated neutrino oscillation analysis using the most recent results from T2K, NO$\nu$A, RENO and Super-Kamiokande. 17 pages, 8 figures, 1 tabl

A non-resonant dark-side solution to the solar neutrino problem

We re-analyse spin-flavour precession solutions to the solar neutrino problem in the light of the recent SNO CC result as well as the 1258--day Super-Kamiokande data and the upper limit on solar anti-neutrinos. In a self-consistent magneto-hydrodynamics approach the resulting scheme has only 3 effective parameters: $\Delta m^2$, $\mu B_\perp$ and the neutrino mixing angle $\theta$. We show how a rates-only analysis for fixed $\mu B_\perp$ slightly favours spin-flavour precession (SFP) solutions over oscillations (OSC). In addition to the resonant solution (RSFP for short), there is a new non-resonant solution (NRSFP) in the ``dark-side''. Both RSFP and NRSFP lead to flat recoil energy spectra in excellent agreement with the latest SuperKamiokande data. We also show that in the presence of a neutrino transition magnetic moment of $10^{-11}$ Bohr magneton, a magnetic field of 80 KGauss eliminates all large mixing solutions other than the so-called LMA solution.Comment: 12 pages, 3 postscript figures, using elsart.cls. Published versio

Neutrino Electron Scattering and Electroweak Gauge Structure: Future Tests

Low-energy high-resolution neutrino-electron scattering experiments may play an important role in testing the gauge structure of the electroweak interaction. We propose the use of radioactive neutrino sources (e.g. $^{51}$Cr) in underground experiments such as BOREXINO, HELLAZ and LAMA. As an illustration, we display the sensitivity of these detectors in testing the possible existence of extra neutral gauge bosons, both in the framework of E_6 models and of models with left-right symmetry.Comment: 22 pages, revtex, 4 figures included, accepted for publication in Phys. Rev.