Determination of matrix potential from scattering matrix

(i) For the matrix Schr\"{o}dinger operator on the half line, it is shown that if the potential exponentially decreases fast enough then only the scattering matrix uniquely determines the self-adjoint potential and the boundary condition. (ii) For the matrix Schr\"{o}dinger operator on the full line, it is shown that if the potential exponentially decreases fast enough then the scattering matrix (or equivalently, the transmission coefficient and reflection coefficient) uniquely determine the potential. If the potential vanishes on $(-\infty,0)$ then only the left reflection coefficient uniquely determine the potential.Comment: 9 page

Inverse spectral problems for the Sturm-Liouville operator with discontinuity

In this work, we consider the Sturm-Liouville operator on a finite interval $[0,1]$ with discontinuous conditions at $1/2$. We prove that if the potential is known a priori on a subinterval $[b,1]$ with $b\ge1/2$, then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions. For the case $b<1/2$, parts of either one or two spectra can uniquely determine the potential and a part of parameters.Comment: 13 page

Solvability of the inverse scattering problem for the selfadjoint matrix Schrodinger operator on the half line

In this work we study the inverse scattering problem for the selfadjoint matrix Schrodinger operator on the half line. We provide the necessary and sufficient conditions for the solvability of the inverse scattering problem.Comment: 29 page

Inverse resonance problems for the Schroedinger operator on the real line with mixed given data

In this work, we study inverse resonance problems for the Schr\"odinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential. (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $a<1/2$, all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential. (iii) It is also shown that all eigenvalues and resonances, together with a set of logarithmic derivative values of eigenfunctions and wave-functions at $1/2$, can uniquely determine the potential.Comment: 12 page

The diffuse gamma-ray flux associated with sub-PeV/PeV neutrinos from starburst galaxies

One attractive scenario for the excess of sub-PeV/PeV neutrinos recently reported by IceCube is that they are produced by cosmic rays in starburst galaxies colliding with the dense interstellar medium. These proton-proton ($pp$) collisions also produce high-energy gamma-rays, which finally contribute to the diffuse high-energy gamma-ray background. We calculate the diffuse gamma-ray flux with a semi-analytic approach and consider that the very high energy gamma-rays will be absorbed in the galaxies and converted into electron-position pairs, which then lose almost all their energy through synchrotron radiation in the strong magnetic fields in the starburst region. Since the synchrotron emission goes into energies below GeV, this synchrotron loss reduces the diffuse high-energy gamma-ray flux by a factor of about two, thus leaving more room for other sources to contribute to the gamma-ray background. For a $E_\nu^{-2}$ neutrino spectrum, we find that the diffuse gamma-ray flux contributes about 20% of the observed diffuse gamma-ray background in the 100 GeV range. However, for a steeper neutrino spectrum, this synchrotron loss effect is less important, since the energy fraction in absorbed gamma-rays becomes lower.Comment: Accepted by ApJ, one figure added, small revisions in text, results and conclusions unchange

Indecomposability of entanglement witnesses constructed from any permutations

Let $n\geq 2$ and $\Phi_{n,t,\pi}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})$ be a linear map defined by $\Phi_{n,t,\pi}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,\pi(i)}AE_{i,\pi(i)}^\dag-A$, where $0\leq t\leq n$, $E_{ij}$s are the matrix units and $\pi$ is a non-identity permutation of $(1,2,\cdots,n)$. Denote by $\{{ F}_s: s=1,2\ldots, k\}$ the set of all minimal cycles of $\pi$ and $l(\pi)=\max\{\# { F}_s: s=1,2,\ldots,k\}$ the length of $\pi$. It is shown that the Hermitian matrix $W_{n,t,\pi}$ induced by $\Phi_{n,t,\pi}$ is an indecomposable entanglement witness if and only if $\pi^2\not={\rm id}$ (the identity permutation) and $0<t\leq\frac{n}{l(\pi)}$. Some new bounded entangled states are detected by such witnesses that cannot be distinguished by PPT criterion, realignment criterion, etc..Comment: 11 page

A parallel space-time domain decomposition method for unsteady source inversion problems

In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknowns. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications

Distinguishing the right-handed up/charm quarks from top quark via discrete symmetries in the standard model extensions

We propose a class of the two Higgs doublet Standard models (SMs) with a SM singlet and a class of supersymmetric SMs with two pairs of Higgs doublets, where the right-handed up/charm quarks and the right-handed top quark have different quantum numbers under extra discrete symmetries. Thus, the right-handed up and charm quarks couple to one Higgs doublet field, while the right-handed top quark couples to another Higgs doublet. The quark CKM mixings can be generated from the down-type quark sector. As one of phenomenological consequences in our models, we explore whether one can accommodate the observed direct CP asymmetry difference in singly Cabibbo-suppressed D decays. We show that it is possible to explain the measured values of CP violation under relevant experimental constraints.Comment: 20 pages; matches published versio

Two-level space-time domain decomposition methods for unsteady inverse problems

As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a novel space-time coupled algorithm for solving an inverse problem associated with the time-dependent convection-diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space-time parallel domain decomposition preconditioner for the Karush-Kuhn-Tucker (KKT) system induced from reformulating the inverse problem as an output least-squares optimization problem in the space-time domain. The new full space approach eliminates the sequential steps of the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We report strong scalability results obtained on a supercomputer with more than 1,000 processors

The Super-Natural Supersymmetry and Its Classic Example: M-Theory Inspired NMSSM

We briefly review the super-natural supersymmetry (SUSY), which provides a most promising solution to the SUSY electroweak fine-tuning problem. In particular, we address its subtle issues as well. Unlike the Minimal Supersymmetric Standard model (MSSM), the Next to MSSM (NMSSM) can be scale invariant and has no mass parameter in its Lagrangian before SUSY and gauge symmetry breakings. Therefore, the NMSSM is a perfect framework for super-natural SUSY. To give the SUSY breaking soft mass to the singlet, we consider the moduli and dilaton dominant SUSY breaking scenarios in M-theory on $S^1/Z_2$. In these scenarios, SUSY is broken by one and only one $F$-term of moduli or dilaton, and the SUSY breaking soft terms can be determined via the K\"ahler potential and superpotential from Calabi-Yau compactification of M-theory on $S^1/Z_2$. Thus, as predicted by super-natural SUSY, the SUSY electroweak fine-tuning measure is of unity order. In the moduli dominant SUSY breaking scenario, the right-handed sleptons are relatively light around 1 TeV, stau can be even as light as 580 GeV and degenerate with the lightest neutralino, chargino masses are larger than 1 TeV, the light stop masses are around 2 TeV or larger, the first two-generation squark masses are about 3 TeV or larger, and gluinos are heavier than squarks. In the dilaton dominant SUSY breaking scenario, the qualitative picture remain the same but we have heavier spectra as compared to moduli dominant SUSY breaking scenario. In addition to it, we have Higgs $H_{2}/A_{1}$-resonance solutions for dark matter (DM). In both scenarios, the minimal value of DM relic density is about 0.2. To obtain the observed DM relic density, we can consider the dilution effect from supercritical string cosmology or introduce the axino as the lightest supersymmetric particle.Comment: 35 pages, 12 figures and two table