515 research outputs found
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
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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
Thermal noise and dephasing due to electron interactions in non-trivial geometries
We study Johnson-Nyquist noise in macroscopically inhomogeneous disordered
metals and give a microscopic derivation of the correlation function of the
scalar electric potentials in real space. Starting from the interacting
Hamiltonian for electrons in a metal and the random phase approximation, we
find a relation between the correlation function of the electric potentials and
the density fluctuations which is valid for arbitrary geometry and
dimensionality. We show that the potential fluctuations are proportional to the
solution of the diffusion equation, taken at zero frequency. As an example, we
consider networks of quasi-1D disordered wires and give an explicit expression
for the correlation function in a ring attached via arms to absorbing leads. We
use this result in order to develop a theory of dephasing by electronic noise
in multiply-connected systems.Comment: 9 pages, 6 figures (version submitted to PRB
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
One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
A one-dimensional model of classical diffusion in a random force field with a
weak concentration of absorbers is studied. The force field is taken as
a Gaussian white noise with \mean{\phi(x)}=0 and \mean{\phi(x)\phi(x')}=g
\delta(x-x'). Our analysis relies on the relation between the Fokker-Planck
operator and a quantum Hamiltonian in which absorption leads to breaking of
supersymmetry. Using a Lifshits argument, it is shown that the average return
probability is a power law \smean{P(x,t|x,0)}\sim{}t^{-\sqrt{2\rho/g}} (to be
compared with the usual Lifshits exponential decay in
the absence of the random force field). The localisation properties of the
underlying quantum Hamiltonian are discussed as well.Comment: 6 pages, LaTeX, 5 eps figure
Characterization of the nanophase precipitation in a metastable beta titanium-based alloy by electrical resistivity, dilatometry and neutron diffraction
The metastable beta Ti-6Mo-5Ta-4Fe (wt.%) alloys was synthesized by cold crucible levitation melting and then quenched in water from the beta phase field. In order to investigate the transformation sequence upon heating, thermal analysis methods such as electrical resistivity, dilatometry and neutron thermodiffraction were employed. By these methods, the different temperatures of transition were detected and solute partitioning was oberved to the beta matrix during the omega and alpha nanophase precipitatio
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
Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder
We study the distribution of the -th energy level for two different
one-dimensional random potentials. This distribution is shown to be related to
the distribution of the distance between two consecutive nodes of the wave
function.
We first consider the case of a white noise potential and study the
distributions of energy level both in the positive and the negative part of the
spectrum. It is demonstrated that, in the limit of a large system
(), the distribution of the -th energy level is given by a
scaling law which is shown to be related to the extreme value statistics of a
set of independent variables.
In the second part we consider the case of a supersymmetric random
Hamiltonian (potential ). We study first the case of
being a white noise with zero mean. It is in particular shown that
the ground state energy, which behaves on average like in
agreement with previous work, is not a self averaging quantity in the limit
as is seen in the case of diagonal disorder. Then we consider the
case when has a non zero mean value.Comment: LaTeX, 33 pages, 9 figure
Scattering for the Zakharov system in 3 dimensions
We prove global existence and scattering for small localized solutions of the
Cauchy problem for the Zakharov system in 3 space dimensions. The wave
component is shown to decay pointwise at the optimal rate of t^{-1}, whereas
the Schr\"odinger component decays almost at a rate of t^{-7/6}.Comment: Minor changes and referee's comments include
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