608 research outputs found
On the moment limit of quantum observables, with an application to the balanced homodyne detection
We consider the moment operators of the observable (i.e. a semispectral
measure or POM) associated with the balanced homodyne detection statistics,
with paying attention to the correct domains of these unbounded operators. We
show that the high amplitude limit, when performed on the moment operators,
actually determines uniquely the entire statistics of a rotated quadrature
amplitude of the signal field, thereby verifying the usual assumption that the
homodyne detection achieves a measurement of that observable. We also consider,
in a general setting, the possibility of constructing a measurement of a single
quantum observable from a sequence of observables by taking the limit on the
level of moment operators of these observables. In this context, we show that
under some natural conditions (each of which is satisfied by the homodyne
detector example), the existence of the moment limits ensures that the
underlying probability measures converge weakly to the probability measure of
the limiting observable. The moment approach naturally requires that the
observables be determined by their moment operator sequences (which does not
automatically happen), and it turns out, in particular, that this is the case
for the balanced homodyne detector.Comment: 22 pages, no figure
Taylor approximations of operator functions
This survey on approximations of perturbed operator functions addresses
recent advances and some of the successful methods.Comment: 12 page
On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis
A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
In the first (and abstract) part of this survey we prove the unitary
equivalence of the inverse of the Krein--von Neumann extension (on the
orthogonal complement of its kernel) of a densely defined, closed, strictly
positive operator, for some in a Hilbert space to an abstract buckling problem operator.
This establishes the Krein extension as a natural object in elasticity theory
(in analogy to the Friedrichs extension, which found natural applications in
quantum mechanics, elasticity, etc.).
In the second, and principal part of this survey, we study spectral
properties for , the Krein--von Neumann extension of the
perturbed Laplacian (in short, the perturbed Krein Laplacian)
defined on , where is measurable, bounded and
nonnegative, in a bounded open set belonging to a
class of nonsmooth domains which contains all convex domains, along with all
domains of class , .Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144
Sharpening the norm bound in the subspace perturbation theory
Let A be a self-adjoint operator on a Hilbert space H. Assume that {\sigma}
is an isolated component of the spectrum of A, i.e. dist({\sigma},{\Sigma})=d>0
where {\Sigma}=spec(A)\{\sigma}. Suppose that V is a bounded self-adjoint
operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral
projection of A associated with the spectral set {\sigma} and let Q be the
spectral projection of L corresponding to the closed ||V||-neighborhood of
{\sigma}. We prove a bound of the form arcsin(||P-Q||)\leq M(||V||/d), M:
[0,1/2)-->R^+, that is essentially stronger than the previously known estimates
for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||<1 and,
thus, that the spectral subspaces Ran(P) and Ran(Q) are in the acute-angle case
whenever ||V||<cd with c=0.454169... (the precise expression for c is also
given). Our proof of the above results is based on using the triangle
inequality for the maximal angle between subspaces and on employing the a
priori generic \sin2\theta estimate for the variation of a spectral subspace.
As an example, the boundedly perturbed quantum harmonic oscillator is
discussed.Comment: Some typos fixed; minor changes in the text; a new reference adde
On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary
We construct two-term asymptotics ?? k = ?m?2(M + ??k + O(?3/2)) of eigenvalues of a mixed boundary-value problem in ? R2 with many heavy (m > 2) concentrated masses near a straight part of the boundary ? . ? is a small positive parameter related to size and periodicity of the masses; k ? N. The main term M > 0
is common for all eigenvalues but the correction terms ?k , which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on , exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a "strongly" singular weight.This research work has been partially supported by Spanish MINECO, MTM2013-44883-P. Also, the research work of the first author has been partially supported by Russian Foundation of Basic research (project 15–01–02175)
Character of Christ: A Proposal for Excellence in Christian Character Education
Moral teaching programs, such as character education, have been implemented nationwide in order to curb the growing trend of violence, abuse, and moral relativism within schools, both public and private. These programs represent a variety of moral training philosophies, and current research is revealing some best practices within the field. However, these programs do little to address the needs of distinctively Christian educators who seek to train their students toward the character of Jesus Christ. The research in this study promotes the development of a curriculum to meet this need. The following research indicates that character education\u27s premise and many of its practices are worthy of consideration when developing a Christian character curriculum. However, the foundation of the character traits promoted by a Christian character curriculum must not be based on the consensus of a pluralistic society. The foundation must be established solely on the person of Christ. Best practices within the field of character education are emerging through current research. These practices and the theories behind them are also examined in light of the development of a Christian character curriculum. Recommendations and implications for a Christian character curriculum are made in both theory and practice
Search for new phenomena in final states with an energetic jet and large missing transverse momentum in pp collisions at √ s = 8 TeV with the ATLAS detector
Results of a search for new phenomena in final states with an energetic jet and large missing transverse momentum are reported. The search uses 20.3 fb−1 of √ s = 8 TeV data collected in 2012 with the ATLAS detector at the LHC. Events are required to have at least one jet with pT > 120 GeV and no leptons. Nine signal regions are considered with increasing missing transverse momentum requirements between Emiss T > 150 GeV and Emiss T > 700 GeV. Good agreement is observed between the number of events in data and Standard Model expectations. The results are translated into exclusion limits on models with either large extra spatial dimensions, pair production of weakly interacting dark matter candidates, or production of very light gravitinos in a gauge-mediated supersymmetric model. In addition, limits on the production of an invisibly decaying Higgs-like boson leading to similar topologies in the final state are presente
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