14,934 research outputs found
A variant of Horn's problem and derivative principle
Identifying the spectrum of the sum of two given Hermitian matrices with
fixed eigenvalues is the famous Horn's problem.In this note, we investigate a
variant of Horn's problem, i.e., we identify the probability density function
(abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with
given spectra. We then use it to re-derive the pdf of the eigenvalues of the
sum of two random Hermitian matrices with given eigenvalues via
\emph{derivative principle}, a powerful tool used to get the exact probability
distribution by reducing to the corresponding distribution of diagonal
entries.We can recover Jean-Bernard Zuber's recent results on the pdf of the
eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover,
as an illustration, we derive the analytical expressions of eigenvalues of the
sum of two random Hermitian matrices from \rG\rU\rE(n) or Wishart ensemble by
derivative principle, respectively.We also investigate the statistics of
exponential of random matrices and connect them with Golden-Thompson
inequality, and partly answer a question proposed by Forrester. Some potential
applications in quantum information theory, such as uniform average quantum
Jensen-Shannon divergence and average coherence of uniform mixture of two
orbits,are discussed.Comment: 24 pages, LaTeX; a new result, i.e., Theorem 3.7, is added and
several references are include
Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio
We consider the asymptotic solutions of an interface problem corresponding to
an elliptic partial differential equation with Dirich- let boundary condition
and transmission condition, subject to the small geometric perturbation and the
high contrast ratio of the conductivity. We consider two types of
perturbations: the first corresponds to a thin layer coating a fixed bounded
domain and the second is the per perturbation of the interface. As the
perturbation size tends to zero and the ratio of the conductivities in two
subdomains tends to zero, the two-parameter asymptotic expansions on the fixed
reference domain are derived to any order after the single parameter expansions
are solved be- forehand. Our main tool is the asymptotic analysis based on the
Taylor expansions for the properly extended solutions on fixed domains. The
Neumann boundary condition and Robin boundary condition arise in two-parameter
expansions, depending on the relation of the geometric perturbation size and
the contrast ratio
X(1576) and the Final State Interaction Effect
We study whether the broad peak X(1576) observed by BES Collaboration arises
from the final state interaction effect of decays. The
interference effect could produce an enhancement around 1540 MeV in the
spectrum with typical interference phases. However, the branching
ratio from the final state interaction effect is far less than the
experimental data.Comment: 6 pages, 4 figures. Some typos corrected, more discussion and
references adde
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