Search results for "Hermitian matrix"
showing 10 items of 44 documents
On the arithmetic and geometry of binary Hamiltonian forms
2011
Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.
Quantum like modelling of decision making: quantifying uncertainty with the aid of the Heisenberg-Robertson inequality
2018
This paper contributes to quantum-like modeling of decision making (DM) under uncertainty through application of Heisenberg’s uncertainty principle (in the form of the Robertson inequality). In this paper we apply this instrument to quantify uncertainty in DM performed by quantum-like agents. As an example, we apply the Heisenberg uncertainty principle to the determination of mutual interrelation of uncertainties for “incompatible questions” used to be asked in political opinion pools. We also consider the problem of representation of decision problems, e.g., in the form of questions, by Hermitian operators, commuting and noncommuting, corresponding to compatible and incompatible questions …
(Approximate) Low-Mode Averaging with a new Multigrid Eigensolver
2015
We present a multigrid based eigensolver for computing low-modes of the Hermitian Wilson Dirac operator. For the non-Hermitian case multigrid methods have already replaced conventional Krylov subspace solvers in many lattice QCD computations. Since the $\gamma_5$-preserving aggregation based interpolation used in our multigrid method is valid for both, the Hermitian and the non-Hermitian case, inversions of very ill-conditioned shifted systems with the Hermitian operator become feasible. This enables the use of multigrid within shift-and-invert type eigensolvers. We show numerical results from our MPI-C implementation of a Rayleigh quotient iteration with multigrid. For state-of-the-art lat…
Volumes of certain small geodesic balls and almost-Hermitian geometry
1984
Let D be the characteristic connection of an almost-Hermitian manifold, V D m (r) the volume of a small geodesic ball for the connection D and C C D 1 the first non-trivial term of the Taylor expansion of V D m (r). NK-manifolds are characterized in terms of C C D 1 and a family of Hermitian manifolds for which ∫ M C C D 1 dvol is a spectral invariant is given and one proves that C C D 1 and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.
Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one
2016
This paper provides various “contractivity” results for linear operators of the form I−C where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces Lp(μ), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. A(λ−A)−1x≤x for all x∈L+p(μ)(λ>0), provided that p⩾2. We show also that this result is not true for 1 ⩽ p<2. We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X+. We deduce from this result that, in such spaces, I−C is a contraction on X+ for any positive projection…
Estimating the two-particle $K$-matrix for multiple partial waves and decay channels from finite-volume energies
2017
An implementation of estimating the two-to-two $K$-matrix from finite-volume energies based on the L\"uscher formalism and involving a Hermitian matrix known as the "box matrix" is described. The method includes higher partial waves and multiple decay channels. Two fitting procedures for estimating the $K$-matrix parameters, which properly incorporate all statistical covariances, are discussed. Formulas and software for handling total spins up to $S=2$ and orbital angular momenta up to $L=6$ are obtained for total momenta in several directions. First tests involving $\rho$-meson decay to two pions include the $L=3$ and $L=5$ partial waves, and the contributions from these higher waves are f…
Evanescent wave approximation for non-Hermitian Hamiltonians
2020
The counterpart of the rotating wave approximation for non-Hermitian Hamiltonians is considered, which allows for the derivation of a suitable effective Hamiltonian for systems with some states undergoing decay. In the limit of very high decay rates, on the basis of this effective description we can predict the occurrence of a quantum Zeno dynamics, which is interpreted as the removal of some coupling terms and the vanishing of an operatorial pseudo-Lamb shift.
Inverse eigenvalue problem for normal J-hamiltonian matrices
2015
[EN] A complex square matrix A is called J-hamiltonian if AT is hermitian where J is a normal real matrix such that J(2) = -I-n. In this paper we solve the problem of finding J-hamiltonian normal solutions for the inverse eigenvalue problem. (C) 2015 Elsevier Ltd. All rights reserved.
A chain of solvable non-Hermitian Hamiltonians constructed by a series of metric operators
2021
We show how, given a non-Hermitian Hamiltonian $H$, we can generate new non-Hermitian operators sequentially, producing a virtually infinite chain of non-Hermitian Hamiltonians which are isospectral to $H$ and $H^\dagger$ and whose eigenvectors we can easily deduce in an almost automatic way; no ingredients are necessary other than $H$ and its eigensystem. To set off the chain and keep it running, we use, for the first time in our knowledge, a series of maps all connected to different metric operators. We show how the procedure works in several physically relevant systems. In particular, we apply our method to various versions of the Hatano-Nelson model and to some PT-symmetric Hamiltonians.
The discretized harmonic oscillator: Mathieu functions and a new class of generalized Hermite polynomials
2003
We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansa…