861 research outputs found
On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach
We consider a metric graph made of two graphs
and attached at one point. We derive a formula relating the
spectral determinant of the Laplace operator
in terms of the spectral
determinants of the two subgraphs. The result is generalized to describe the
attachment of graphs. The formulae are also valid for the spectral
determinant of the Schr\"odinger operator .Comment: LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and
ref adde
Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers
Some results on the ordered statistics of eigenvalues for one-dimensional
random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric
quantum mechanics with disorder, the existence of low energy delocalized states
induces eigenvalue correlations and makes the ordered statistics problem
nontrivial. The resulting distributions are used to analyze the problem of
classical diffusion in a random force field (Sinai problem) in the presence of
weakly concentrated absorbers. It is shown that the slowly decaying averaged
return probability of the Sinai problem, \mean{P(x,t|x,0)}\sim \ln^{-2}t, is
converted into a power law decay, \mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}},
where is the strength of the random force field and the density of
absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting
"Fundations and Applications of non-equilibrium statistical mechanics",
Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left
adde
Spectral determinants and zeta functions of Schr\"odinger operators on metric graphs
A derivation of the spectral determinant of the Schr\"odinger operator on a
metric graph is presented where the local matching conditions at the vertices
are of the general form classified according to the scheme of Kostrykin and
Schrader. To formulate the spectral determinant we first derive the spectral
zeta function of the Schr\"odinger operator using an appropriate secular
equation. The result obtained for the spectral determinant is along the lines
of the recent conjecture.Comment: 16 pages, 2 figure
Weak localization in multiterminal networks of diffusive wires
We study the quantum transport through networks of diffusive wires connected
to reservoirs in the Landauer-B\"uttiker formalism. The elements of the
conductance matrix are computed by the diagrammatic method. We recover the
combination of classical resistances and obtain the weak localization
corrections. For arbitrary networks, we show how the cooperon must be properly
weighted over the different wires. Its nonlocality is clearly analyzed. We
predict a new geometrical effect that may change the sign of the weak
localization correction in multiterminal geometries.Comment: 4 pages, LaTeX, 4 figures, 8 eps file
Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD
Extending our previous work in the strictly parabolic case, we show that a
linearly unstable Lax-type viscous shock solution of a general quasilinear
hyperbolic--parabolic system of conservation laws possesses a
translation-invariant center stable manifold within which it is nonlinearly
orbitally stable with respect to small perturbations, converging
time-asymptotically to a translate of the unperturbed wave. That is, for a
shock with unstable eigenvalues, we establish conditional stability on a
codimension- manifold of initial data, with sharp rates of decay in all
. For , we recover the result of unconditional stability obtained by
Mascia and Zumbrun. The main new difficulty in the hyperbolic--parabolic case
is to construct an invariant manifold in the absence of parabolic smoothing.Comment: 32p
Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
The concept of Lyapunov exponent has long occupied a central place in the
theory of Anderson localisation; its interest in this particular context is
that it provides a reasonable measure of the localisation length. The Lyapunov
exponent also features prominently in the theory of products of random matrices
pioneered by Furstenberg. After a brief historical survey, we describe some
recent work that exploits the close connections between these topics. We review
the known solvable cases of disordered quantum mechanics involving random point
scatterers and discuss a new solvable case. Finally, we point out some
limitations of the Lyapunov exponent as a means of studying localisation
properties.Comment: LaTeX, 23 pages, 3 pdf figures ; review for a special issue on
"Lyapunov analysis" ; v2 : typo corrected in eq.(3) & minor change
Sinai model in presence of dilute absorbers
We study the Sinai model for the diffusion of a particle in a one dimension
random potential in presence of a small concentration of perfect
absorbers using the asymptotically exact real space renormalization method. We
compute the survival probability, the averaged diffusion front and return
probability, the two particle meeting probability, the distribution of total
distance traveled before absorption and the averaged Green's function of the
associated Schrodinger operator. Our work confirms some recent results of
Texier and Hagendorf obtained by Dyson-Schmidt methods, and extends them to
other observables and in presence of a drift. In particular the power law
density of states is found to hold in all cases. Irrespective of the drift, the
asymptotic rescaled diffusion front of surviving particles is found to be a
symmetric step distribution, uniform for , where
is a new, survival length scale ( in the absence of
drift). Survival outside this sharp region is found to decay with a larger
exponent, continuously varying with the rescaled distance . A simple
physical picture based on a saddle point is given, and universality is
discussed.Comment: 21 pages, 2 figure
Scattering theory on graphs (2): the Friedel sum rule
We consider the Friedel sum rule in the context of the scattering theory for
the Schr\"odinger operator -\Dc_x^2+V(x) on graphs made of one-dimensional
wires connected to external leads. We generalize the Smith formula for graphs.
We give several examples of graphs where the state counting method given by the
Friedel sum rule is not working. The reason for the failure of the Friedel sum
rule to count the states is the existence of states localized in the graph and
not coupled to the leads, which occurs if the spectrum is degenerate and the
number of leads too small.Comment: 20 pages, LaTeX, 6 eps figure
Synthèse du colloque international sur la microfinance.
La microfinance a fait l’objet d’un colloque organisé par la Banque de France en juillet 2011. Les participants ont souligné son utilité comme outil d’inclusion financière, de développement économique et de lutte contre la pauvreté en s’appuyant sur les expériences des pays du Sud et ont proposé des pistes de réflexion pour le financement de son développement, ainsi que sur la pertinence et les modalités de sa régulation.microfinance, microcrédit, entrepreneuriat, lutte contre la pauvreté, régulation, réglementation, inclusion financière, développement économique.
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde
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