129 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
Ullemar's formula for the Jacobian of the complex moment mapping
The complex moment sequence m(P) is assigned to a univalent polynomial P by
the Cauchy transform of the P(D), where D is the unit disk. We establish the
representation of the Jacobian det dm(P) in terms of roots of the derivative
P'. Combining this result with the special decomposition for the Hurwitz
determinants, we prove a formula for the Jacobian which was previously
conjectured by C. Ullemar. As a consequence, we show that the boundary of the
class of all locally univalent polynomials in is contained in the union of
three irreducible algebraic surfaces.Comment: 14 pages, submitted for "Complex Variables. Theory and Application
Multivariate truncated moments problems and maximum entropy
We characterize the existence of the Lebesgue integrable solutions of the
truncated problem of moments in several variables on unbounded supports by the
existence of some maximum entropy -- type representing densities and discuss a
few topics on their approximation in a particular case, of two variables and
4th order moments.Comment: Revised version, to appear in Analysis and Mathematical Physic
Restrictions and extensions of semibounded operators
We study restriction and extension theory for semibounded Hermitian operators
in the Hardy space of analytic functions on the disk D. Starting with the
operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D)
of measure zero, there is a densely defined Hermitian restriction of zd/dz
corresponding to boundary functions vanishing on F. For every such restriction
operator, we classify all its selfadjoint extension, and for each we present a
complete spectral picture.
We prove that different sets F with the same cardinality can lead to quite
different boundary-value problems, inequivalent selfadjoint extension
operators, and quite different spectral configurations. As a tool in our
analysis, we prove that the von Neumann deficiency spaces, for a fixed set F,
have a natural presentation as reproducing kernel Hilbert spaces, with a
Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure
Electromagnetic waves in an axion-active relativistic plasma non-minimally coupled to gravity
We consider cosmological applications of a new self-consistent system of
equations, accounting for a nonminimal coupling of the gravitational,
electromagnetic and pseudoscalar (axion) fields in a relativistic plasma. We
focus on dispersion relations for electromagnetic perturbations in an initially
isotropic ultrarelativistic plasma coupled to the gravitational and axion
fields in the framework of isotropic homogeneous cosmological model of the de
Sitter type. We classify the longitudinal and transversal electromagnetic modes
in an axionically active plasma and distinguish between waves (damping,
instable or running), and nonharmonic perturbations (damping or instable). We
show that for the special choice of the guiding model parameters the
transversal electromagnetic waves in the axionically active plasma,
nonminimally coupled to gravity, can propagate with the phase velocity less
than speed of light in vacuum, thus displaying a possibility for a new type of
resonant particle-wave interactions.Comment: 19 pages, 9 figures, published versio
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
Effect of Contour Shape of Nervous System Electromagnetic Stimulation Coils on the Induced Electrical Field Distribution
BACKGROUND: Electromagnetic stimulation of the nervous system has the advantage of reduced discomfort in activating nerves. For brain structures stimulation, it has become a clinically accepted modality. Coil designs usually consider factors such as optimization of induced power, focussing, field shape etc. In this study we are attempting to find the effect of the coil contour shape on the electrical field distribution for magnetic stimulation. METHOD AND RESULTS: We use the maximum of the induced electric field stimulation in the region of interest as the optimization criterion. This choice required the application of the calculus of variation, with the contour perimeter taken as a pre-set condition. Four types of coils are studied and compared: circular, square, triangular and an 'optimally' shaped contour. The latter yields higher values of the induced electrical field in depths up to about 30 mm, but for depths around 100 mm, the circular shape has a slight advantage. The validity of the model results was checked by experimental measurements in a tank with saline solution, where differences of about 12% were found. In view the accuracy limitations of the computational and measurement methods used, such differences are considered acceptable. CONCLUSION: We applied an optimization approach, using the calculus of variation, which allows to obtain a coil contour shape corresponding to a selected criterion. In this case, the optimal contour showed higher intensities for a longer line along the depth-axis. The method allows modifying the induced field structure and focussing the field to a selected zone or line
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
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
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