38 research outputs found
Quantum Limits of Eisenstein Series and Scattering states
We identify the quantum limits of scattering states for the modular surface.
This is obtained through the study of quantum measures of non-holomorphic
Eisenstein series away from the critical line. We provide a range of stability
for the quantum unique ergodicity theorem of Luo and Sarnak.Comment: 12 pages, Corrects a typo and its ramification from previous versio
Random Walks Along the Streets and Canals in Compact Cities: Spectral analysis, Dynamical Modularity, Information, and Statistical Mechanics
Different models of random walks on the dual graphs of compact urban
structures are considered. Analysis of access times between streets helps to
detect the city modularity. The statistical mechanics approach to the ensembles
of lazy random walkers is developed. The complexity of city modularity can be
measured by an information-like parameter which plays the role of an individual
fingerprint of {\it Genius loci}.
Global structural properties of a city can be characterized by the
thermodynamical parameters calculated in the random walks problem.Comment: 44 pages, 22 figures, 2 table
Complex zeros of real ergodic eigenfunctions
We determine the limit distribution (as ) of complex
zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of
real eigenfunctions of the Laplacian on a real analytic compact Riemannian
manifold with ergodic geodesic flow. If is an
ergodic sequence of eigenfunctions, we prove the weak limit formula
\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial}
{\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of
integration over the complex zeros and where is with respect
to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and
corrected some typo
On The Isoperimetric Spectrum of Graphs and Its Approximations
In this paper we consider higher isoperimetric numbers of a (finite directed)
graph. In this regard we focus on the th mean isoperimetric constant of a
directed graph as the minimum of the mean outgoing normalized flows from a
given set of disjoint subsets of the vertex set of the graph. We show that
the second mean isoperimetric constant in this general setting, coincides with
(the mean version of) the classical Cheeger constant of the graph, while for
the rest of the spectrum we show that there is a fundamental difference between
the th isoperimetric constant and the number obtained by taking the minimum
over all -partitions. In this direction, we show that our definition is the
correct one in the sense that it satisfies a Federer-Fleming-type theorem, and
we also define and present examples for the concept of a supergeometric graph
as a graph whose mean isoperimetric constants are attained on partitions at all
levels. Moreover, considering the -completeness of the isoperimetric
problem on graphs, we address ourselves to the approximation problem where we
prove general spectral inequalities that give rise to a general Cheeger-type
inequality as well. On the other hand, we also consider some algorithmic
aspects of the problem where we show connections to orthogonal representations
of graphs and following J.~Malik and J.~Shi () we study the close
relationships to the well-known -means algorithm and normalized cuts method
Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
The n-particle periodic Toda chain is a well known example of an integrable
but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold
singularities of the Toda chain, ie points where there exist k independent
linear relations amongst the gradients of the integrals of motion, coincide
with points where there are k (doubly) degenerate eigenvalues of
representatives L and Lbar of the two inequivalent classes of Lax matrices
(corresponding to degenerate periodic or antiperiodic solutions of the
associated second-order difference equation). The singularities are shown to be
nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.
Sigma_k is shown to be of elliptic type, and the frequencies of transverse
oscillations under Hamiltonians which fix Sigma_k are computed in terms of
spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a
closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is
given by the product of the holonomies (equal to +/- 1) of the even- (or odd-)
indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio
Nonlinear Diffusion Through Large Complex Networks Containing Regular Subgraphs
Transport through generalized trees is considered. Trees contain the simple
nodes and supernodes, either well-structured regular subgraphs or those with
many triangles. We observe a superdiffusion for the highly connected nodes
while it is Brownian for the rest of the nodes. Transport within a supernode is
affected by the finite size effects vanishing as For the even
dimensions of space, , the finite size effects break down the
perturbation theory at small scales and can be regularized by using the
heat-kernel expansion.Comment: 21 pages, 2 figures include
Autocorrelation function of eigenstates in chaotic and mixed systems
We study the autocorrelation function of different types of eigenfunctions in
quantum mechanical systems with either chaotic or mixed classical limits. We
obtain an expansion of the autocorrelation function in terms of the correlation
length. For localized states, like bouncing ball modes or states living on
tori, a simple model using only classical input gives good agreement with the
exact result. In particular, a prediction for irregular eigenfunctions in mixed
systems is derived and tested. For chaotic systems, the expansion of the
autocorrelation function can be used to test quantum ergodicity on different
length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low
resolution only. For a version with pictures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab
Dirac and magnetic Schr\"odinger operators on fractals
In this paper we define (local) Dirac operators and magnetic Schr\"odinger
Hamiltonians on fractals and prove their (essential) self-adjointness. To do so
we use the concept of 1-forms and derivations associated with Dirichlet forms
as introduced by Cipriani and Sauvageot, and further studied by the authors
jointly with R\"ockner, Ionescu and Rogers. For simplicity our definitions and
results are formulated for the Sierpinski gasket with its standard self-similar
energy form. We point out how they may be generalized to other spaces, such as
the classical Sierpinski carpet
Evolution of Intermediate Water Masses Based on Argo Float Displacements
International audienc
Energy transfers between multidecadal and turbulent variability
One of the proposed mechanisms to explain the multidecadal variability observed in sea surface temperature of the North Atlantic consists of a large-scale low-frequency internal mode spontaneously developing because of the large-scale baroclinic instability of the time-mean circulation. Even though this mode has been extensively studied in terms of the buoyancy variance budget, its energetic properties remain poorly known. Here we perform the full mechanical energy budget including available potential energy (APE) and kinetic energy (KE) of this internal mode and decompose the budget into three frequency bands: mean, low frequency (LF) associated with the large-scale mode and high frequency (HF) associated with mesocale eddy turbulence. This decomposition allows us to diagnose the energy fluxes between the different reservoirs and to understand the sources and sinks. Due to the large-scale of the mode, most of its energy is contained in the APE. In our configuration, the only source of LF APE is the transfer from mean APE to LF APE that is attributed to the large-scale baroclinic instability. In return the sinks of LF APE are the parameterized diffusion, the flux toward HF APE and to a much lesser extent toward LF KE. The presence of an additional wind-stress component weakens multidecadal oscillations and modifies the energy fluxes between the different energy reservoirs. The KE transfer appears to only have a minor influence on the multidecadal mode compared to the other energy sources involving APE, in all experiments. These results highlight the utility of the full APE/ KE budget