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Information-entropic measures for non-zero l states of confined hydrogen-like ions
Relative Fisher information (IR), which is a measure of correlative
fluctuation between two probability densities, has been pursued for a number of
quantum systems, such as, 1D quantum harmonic oscillator (QHO) and a few
central potentials namely, 3D isotropic QHO, hydrogen atom and pseudoharmonic
potential (PHP) in both position () and momentum () spaces. In the 1D
case, the state is chosen as reference, whereas for a central potential,
the respective circular or node-less (corresponding to lowest radial quantum
number ) state of a given quantum number, is selected. Starting from
their exact wave functions, expressions of IR in both and spaces are
obtained in closed analytical forms in all these systems. A careful analysis
reveals that, for the 1D QHO, IR in both coordinate spaces increase linearly
with quantum number . Likewise, for 3D QHO and PHP, it varies with single
power of radial quantum number in both spaces. But, in H atom they
depend on both principal () and azimuthal () quantum numbers. However, at
a fixed , IR (in conjugate spaces) initially advance with rise of and
then falls off; also for a given , it always decreases with
Two-Baryon Systems with Twisted Boundary Conditions
We explore the use of twisted boundary conditions in extracting the nucleon
mass and the binding energy of two-baryon systems, such as the deuteron, from
Lattice QCD calculations. Averaging the results of calculations performed with
periodic and anti-periodic boundary conditions imposed upon the light-quark
fields, or other pair-wise averages, improves the volume dependence of the
deuteron binding energy from ~exp(-kappa*L)/L to ~exp(-sqrt(2)kappa*L)/L.
However, a twist angle of pi/2 in each of the spatial directions improves the
volume dependence from ~exp(-kappa*L)/L to ~exp(-2kappa*L)/L. Twist averaging
the binding energy with a random sampling of twist angles improves the volume
dependence from ~exp^(-kappa*L)/L to ~exp(-2kappa*L)/L, but with a standard
deviation of ~exp(-kappa*L)/L, introducing a signal-to-noise issue in modest
lattice volumes. Using the experimentally determined phase shifts and mixing
angles, we determine the expected energies of the deuteron states over a range
of cubic lattice volumes for a selection of twisted boundary conditions.Comment: 20 pages, 3 figure
On the relation between states and maps in infinite dimensions
Relations between states and maps, which are known for quantum systems in
finite-dimensional Hilbert spaces, are formulated rigorously in geometrical
terms with no use of coordinate (matrix) interpretation. In a tensor product
realization they are represented simply by a permutation of factors. This leads
to natural generalizations for infinite-dimensional Hilbert spaces and a simple
proof of a generalized Choi Theorem. The natural framework is based on spaces
of Hilbert-Schmidt operators and
the corresponding tensor products of
Hilbert spaces. It is proved that the corresponding isomorphisms cannot be
naturally extended to compact (or bounded) operators, nor reduced to the
trace-class operators. On the other hand, it is proven that there is a natural
continuous map
from trace-class operators on
(with the nuclear norm) into compact operators mapping the space of all bounded
operators on into trace class operators on
(with the operator-norm). Also in the infinite-dimensional context, the Schmidt
measure of entanglement and multipartite generalizations of state-maps
relations are considered in the paper.Comment: 19 page
Quasisplit Hecke algebras and symmetric spaces
Let (G,K) be a symmetric pair over an algebraically closed field of
characteristic different of 2 and let sigma be an automorphism with square 1 of
G preserving K. In this paper we consider the set of pairs (O,L) where O is a
sigma-stable K-orbit on the flag manifold of G and L is an irreducible
K-equivariant local system on O which is "fixed" by sigma. Given two such pairs
(O,L), (O',L'), with O' in the closure \bar O of O, the multiplicity space of
L' in the a cohomology sheaf of the intersection cohomology of \bar O with
coefficients in L (restricted to O') carries an involution induced by sigma and
we are interested in computing the dimensions of its +1 and -1 eigenspaces. We
show that this computation can be done in terms of a certain module structure
over a quasisplit Hecke algebra on a space spanned by the pairs (O,L) as above.Comment: 46 pages. Version 2 reorganizes the explicit calculation of the Hecke
module, includes details about computing \bar, and corrects small misprints.
Version 3 adds two pages relating this paper to unitary representation
theory, corrects misprints, and displays more equations. Version 4 corrects
misprints, and adds two cases previously neglected at the end of 7.
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