Search results for "eigenvalues"
showing 10 items of 315 documents
Generalized modulational instability in multimode fibers: Wideband multimode parametric amplification
2015
In this paper intermodal modulational instability (IM-MI) is analyzed in a multimode fiber where several spatial and polarization modes propagate. The coupled nonlinear Schr\"odinger equations describing the modal evolution in the fiber are linearized and reduced to an eigenvalue problem. As a result, the amplification of each mode can be described by means of the eigenvalues and eigenvectors of a matrix that stores the information about the dispersion properties of the modes and the modal power distribution of the pump. Some useful analytical formulas are also provided that estimate the modal amplification as function of the system parameters. Finally, the impact of third-order dispersion …
Wave Propagation in a 3-D Optical Waveguide
2003
In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.\cite{MS} The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavio…
Operators on Partial Inner Product Spaces: Towards a Spectral Analysis
2014
Given a LHS (Lattice of Hilbert spaces) $V_J$ and a symmetric operator $A$ in $V_J$, in the sense of partial inner product spaces, we define a generalized resolvent for $A$ and study the corresponding spectral properties. In particular, we examine, with help of the KLMN theorem, the question of generalized eigenvalues associated to points of the continuous (Hilbertian) spectrum. We give some examples, including so-called frame multipliers.
Numerical range and positive block matrices
2020
We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate, $$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$ between the diameters of the numerical ranges for the full matrix and its partial trace.
Flavor versus mass eigenstates in neutrino asymmetries: implications for cosmology
2017
We show that, if they exist, lepton number asymmetries ($L_\alpha$) of neutrino flavors should be distinguished from the ones ($L_i$) of mass eigenstates, since Big Bang Nucleosynthesis (BBN) bounds on the flavor eigenstates cannot be directly applied to the mass eigenstates. Similarly, Cosmic Microwave Background (CMB) constraints on mass eigenstates do not directly constrain flavor asymmetries. Due to the difference of mass and flavor eigenstates, the cosmological constraint on the asymmetries of neutrino flavors can be much stronger than conventional expectation, but not uniquely determined unless at least the asymmetry of the heaviest neutrino is well constrained. Cosmological constrain…
Scaling property of variational perturbation expansion for a general anharmonic oscillator with xp-potential
1995
We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.
Connecting Berry's phase and the pumped charge in a Cooper pair pump
2003
The properties of the tunnelling-charging Hamiltonian of a Cooper pair pump are well understood in the regime of weak and intermediate Josephson coupling, i.e. when $E_{\mathrm{J}}\lesssim E_{\mathrm{C}}$. It is also known that Berry's phase is related to the pumped charge induced by the adiabatical variation of the eigenstates. We show explicitly that pumped charge in Cooper pair pump can be understood as a partial derivative of Berry's phase with respect to the phase difference $\phi$ across the array. The phase fluctuations always present in real experiments can also be taken into account, although only approximately. Thus the measurement of the pumped current gives reliable, yet indirec…
On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method
2017
The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson's method. …
Extension of the Launay Quantum Reactive Scattering Code and Direct Computation of Time Delays.
2019
Scattering computations, particularly within the realm of molecular physics, have seen an increase in study since the development of powerful quantum methods. These dynamical processes can be analyzed via (among other quantities) the duration of the collision process and the lifetime of the intermediate complex. We use the Smith matrix Q = -iℏS†dS/dE calculated from the scattering matrix S and its derivative with respect to the total energy. Its real part contains the state-to-state time delays, and its eigenvalues give the lifetimes of the metastable states [ Smith Phys. Rev. 1960 , 118 , 349 - 356 ]. We propose an extension of the Launay HYP3D code [ Launay and Le Dourneuf Chem. Phys. Let…
Solution for an arbitrary number of coupled identical oscillators.
1992
We propose a solution to the problem of solving the Schr\"odinger equation for an arbitrary number of identical one-dimensional harmonically coupled oscillators raised by Fan Hong-yi [Phys. Rev. A 42, 4377 (1990)]. The relationship between the Fock spaces associated with the uncoupled and coupled oscillators is given as well as the coordinate representation of the eigenstates. In view of further applications, the Lie algebraic properties of the model are examined, and the generalization to three spatial dimensions is made.