835 research outputs found
Quantum ergodicity in mixed and KAM Hamiltonian systems
In this thesis, we investigate quantum ergodicity for two classes of
Hamiltonian systems satisfying intermediate dynamical hypotheses between the
well understood extremes of ergodic flow and quantum completely integrable
flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian
systems.
Hamiltonian systems with mixed phase space decompose into finitely many
invariant subsets, only some of which are of ergodic character. It has been
conjectured by Percival that the eigenfunctions of the quantisation of this
system decompose into associated families of analogous character. The first
project in this thesis proves a weak form of this conjecture for a class of
dynamical billiards, namely the mushroom billiards of Bunimovich for a full
measure subset of a shape parameter .
KAM Hamiltonian systems arise as perturbations of completely integrable
Hamiltonian systems. The dynamics of these systems are well understood and have
near-integrable character. The classical-quantum correspondence suggests that
the quantisation of KAM systems will not have quantum ergodic character. The
second project in this thesis proves an initial negative quantum ergodicity
result for a class of positive Gevrey perturbations of a Gevrey Hamiltonian
that satisfy a mild "slow torus" condition
A Shape Dynamical Approach to Holographic Renormalization
We provide a bottom-up argument to derive some known results from holographic
renormalization using the classical bulk-bulk equivalence of General Relativity
and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The
purpose of this paper is twofold: 1) to advertise the simple classical
mechanism: trading of gauge symmetries, that underlies the bulk-bulk
equivalence of General Relativity and Shape Dynamics to readers interested in
dualities of the type of AdS/CFT; and 2) to highlight that this mechanism can
be used to explain certain results of holographic renormalization, providing an
alternative to the AdS/CFT conjecture for these cases. To make contact with
usual the semiclassical AdS/CFT correspondence, we provide, in addition, a
heuristic argument that makes it plausible why the classical equivalence
between General Relativity and Shape Dynamics turns into a duality between
radial evolution in gravity and the renormalization group flow of a conformal
field theory. We believe that Shape Dynamics provides a new perspective on
gravity by giving conformal structure a primary role within the theory. It is
hoped that this work provides the first steps towards understanding what this
new perspective may be able to teach us about holographic dualities.Comment: 27 pages, no figures. Version to appear in EPJC. Title changed. Minor
corrections to tex
Propagation of singularities and Fredholm analysis for the time-dependent Schr\"odinger equation
We study the time-dependent Schr\"odinger operator
acting on functions defined on , where, using coordinates and , denotes ,
is the positive Laplacian with respect to a time dependent family of
non-trapping metrics on which is equal
to the Euclidean metric outside of a compact set in spacetime, and is a potential function which is also compactly supported in spacetime. In
this paper we introduce a new approach to studying , by finding pairs of
Hilbert spaces between which the operator acts invertibly.
Using this invertibility it is straightforward to solve the `final state
problem' for the time-dependent Schr\"odinger equation, that is, find a global
solution of having prescribed asymptotics as .
These asymptotics are of the form where , the `final
state' or outgoing data, is an arbitrary element of a suitable function space
; here is a regularity parameter
simultaneously measuring smoothness and decay at infinity. We can of course
equally well prescribe asymptotics as ; this leads to incoming
data . We consider the `Poisson operators'
and precisely characterize the range of these operators on
spaces. Finally we show that the scattering
matrix, mapping to , preserves these spaces.Comment: 63 pages, 3 figure
Einstein gravity as a 3D conformally invariant theory
We give an alternative description of the physical content of general
relativity that does not require a Lorentz invariant spacetime. Instead, we
find that gravity admits a dual description in terms of a theory where local
size is irrelevant. The dual theory is invariant under foliation preserving
3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume
(for the spatially compact case). Locally, this symmetry is identical to that
of Horava-Lifshitz gravity in the high energy limit but our theory is
equivalent to Einstein gravity. Specifically, we find that the solutions of
general relativity, in a gauge where the spatial hypersurfaces have constant
mean extrinsic curvature, can be mapped to solutions of a particular gauge
fixing of the dual theory. Moreover, this duality is not accidental. We provide
a general geometric picture for our procedure that allows us to trade foliation
invariance for conformal invariance. The dual theory provides a new proposal
for the theory space of quantum gravity.Comment: 27 pages. Published version (minor changes and corrections
Two-photon Lithography for 3D Magnetic Nanostructure Fabrication
Ferromagnetic materials have been utilised as recording media within data
storage devices for many decades. Confinement of the material to a two
dimensional plane is a significant bottleneck in achieving ultra-high recording
densities and this has led to the proposition of three dimensional (3D)
racetrack memories that utilise domain wall propagation along nanowires.
However, the fabrication of 3D magnetic nanostructures of complex geometry is
highly challenging and not easily achievable with standard lithography
techniques. Here, by using a combination of two-photon lithography and
electrochemical deposition, we show a new approach to construct 3D magnetic
nanostructures of complex geometry. The magnetic properties are found to be
intimately related to the 3D geometry of the structure and magnetic imaging
experiments provide evidence of domain wall pinning at a 3D nanostructured
junction
- …