4,260 research outputs found
Intermediate wave-function statistics
We calculate statistical properties of the eigenfunctions of two quantum
systems that exhibit intermediate spectral statistics: star graphs and Seba
billiards. First, we show that these eigenfunctions are not quantum ergodic,
and calculate the corresponding limit distribution. Second, we find that they
can be strongly scarred by short periodic orbits, and construct sequences of
states which have such a limit. Our results are illustrated by numerical
computations.Comment: 4 pages, 3 figures. Final versio
Value distribution of the eigenfunctions and spectral determinants of quantum star graphs
We compute the value distributions of the eigenfunctions and spectral
determinant of the Schrodinger operator on families of star graphs. The values
of the spectral determinant are shown to have a Cauchy distribution with
respect both to averages over bond lengths in the limit as the wavenumber tends
to infinity and to averages over wavenumber when the bond lengths are fixed and
not rationally related. This is in contrast to the spectral determinants of
random matrices, for which the logarithm is known to satisfy a Gaussian limit
distribution. The value distribution of the eigenfunctions also differs from
the corresponding random matrix result. We argue that the value distributions
of the spectral determinant and of the eigenfunctions should coincide with
those of Seba-type billiards.Comment: 32 pages, 9 figures. Final version incorporating referee's comments.
Typos corrected, appendix adde
No quantum ergodicity for star graphs
We investigate statistical properties of the eigenfunctions of the
Schrodinger operator on families of star graphs with incommensurate bond
lengths. We show that these eigenfunctions are not quantum ergodic in the limit
as the number of bonds tends to infinity by finding an observable for which the
quantum matrix elements do not converge to the classical average. We further
show that for a given fixed graph there are subsequences of eigenfunctions
which localise on pairs of bonds. We describe how to construct such
subsequences explicitly. These constructions are analogous to scars on short
unstable periodic orbits.Comment: 26 pages, 5 figure
Spectral Statistics of "Cellular" Billiards
For a bounded planar domain whose boundary contains a number of
flat pieces we consider a family of non-symmetric billiards
constructed by patching several copies of along 's. It is
demonstrated that the length spectrum of the periodic orbits in is
degenerate with the multiplicities determined by a matrix group . We study
the energy spectrum of the corresponding quantum billiard problem in
and show that it can be split in a number of uncorrelated subspectra
corresponding to a set of irreducible representations of . Assuming
that the classical dynamics in are chaotic, we derive a
semiclassical trace formula for each spectral component and show that their
energy level statistics are the same as in standard Random Matrix ensembles.
Depending on whether is real, pseudo-real or complex, the spectrum
has either Gaussian Orthogonal, Gaussian Symplectic or Gaussian Unitary types
of statistics, respectively.Comment: 18 pages, 4 figure
Autocorrelation of Random Matrix Polynomials
We calculate the autocorrelation functions (or shifted moments) of the
characteristic polynomials of matrices drawn uniformly with respect to Haar
measure from the groups U(N), O(2N) and USp(2N). In each case the result can be
expressed in three equivalent forms: as a determinant sum (and hence in terms
of symmetric polynomials), as a combinatorial sum, and as a multiple contour
integral. These formulae are analogous to those previously obtained for the
Gaussian ensembles of Random Matrix Theory, but in this case are identities for
any size of matrix, rather than large-matrix asymptotic approximations. They
also mirror exactly autocorrelation formulae conjectured to hold for
L-functions in a companion paper. This then provides further evidence in
support of the connection between Random Matrix Theory and the theory of
L-functions
The absent-present researcher: data analysis of pre-recorded parent-driven campaign videos
In recent years, there has been a proliferation of sophisticated, user-friendly and accessible instruments of video data collection (e.g. mobile/cell phones and tablets) which facilitate video-based research and analysis. This paper reports on the opportunities and challenges of undertaking video analysis by reporting on the qualitative video analysis of a subset of 30 purposively selected videos from #notanurse_but, a parent-driven video campaign initiated by WellChild, a UK charity. This paper provides insight into one way of conducting video analysis, appreciating that a variety of approaches exist and that methodological reflections on analytical work with video recordings are limited. The authors critically consider researcher subjectivity; the everydayness of video data; making assumptions; and the incomplete picture provided by video data. Despite notable limitations to the approach of video analysis as a standalone method, the authors conclude that video analysis is capable of eliciting data that may not otherwise be obtained
COMPASS: a 2.6m telescope for CMBR polarization studies
COMPASS (COsmic Microwave Polarization at Small Scale) is an experiment devoted to measuring the polarization of the CMBR. Its design and characteristics are presented
Random matrix theory, the exceptional Lie groups, and L-functions
There has recently been interest in relating properties of matrices drawn at
random from the classical compact groups to statistical characteristics of
number-theoretical L-functions. One example is the relationship conjectured to
hold between the value distributions of the characteristic polynomials of such
matrices and value distributions within families of L-functions. These
connections are here extended to non-classical groups. We focus on an explicit
example: the exceptional Lie group G_2. The value distributions for
characteristic polynomials associated with the 7- and 14-dimensional
representations of G_2, defined with respect to the uniform invariant (Haar)
measure, are calculated using two of the Macdonald constant term identities. A
one parameter family of L-functions over a finite field is described whose
value distribution in the limit as the size of the finite field grows is
related to that of the characteristic polynomials associated with the
7-dimensional representation of G_2. The random matrix calculations extend to
all exceptional Lie groupsComment: 14 page
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 page
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