2,289 research outputs found
Regularity for eigenfunctions of Schr\"odinger operators
We prove a regularity result in weighted Sobolev spaces (or
Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator.
More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space
obtained by blowing up the set of singular points of the Coulomb type potential
V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N}
\frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u
in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution
sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0.
Our result extends to the case when b_j and c_{ij} are suitable bounded
functions on the blown-up space. In the single-electron, multi-nuclei case, we
obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy
Swing Options Valuation: a BSDE with Constrained Jumps Approach
We introduce a new probabilistic method for solving a class of impulse
control problems based on their representations as Backward Stochastic
Differential Equations (BSDEs for short) with constrained jumps. As an example,
our method is used for pricing Swing options. We deal with the jump constraint
by a penalization procedure and apply a discrete-time backward scheme to the
resulting penalized BSDE with jumps. We study the convergence of this numerical
method, with respect to the main approximation parameters: the jump intensity
, the penalization parameter and the time step. In particular,
we obtain a convergence rate of the error due to penalization of order
. Combining this approach with Monte Carlo techniques, we
then work out the valuation problem of (normalized) Swing options in the Black
and Scholes framework. We present numerical tests and compare our results with
a classical iteration method.Comment: 6 figure
Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low-temperature phase
We consider the low-temperature disorder-dominated phase of the
directed polymer in a random potentiel in dimension 1+1 (where )
and 1+3 (where ). To characterize the localization properties of
the polymer of length , we analyse the statistics of the weights of the last monomer as follows. We numerically compute the probability
distributions of the maximal weight , the probability distribution of the parameter as well as the average values of the higher order
moments . We find that there exists a
temperature such that (i) for , the distributions
and present the characteristic Derrida-Flyvbjerg
singularities at and for . In particular, there
exists a temperature-dependent exponent that governs the main
singularities and as well as the power-law decay of the moments . The exponent grows from the value
up to . (ii) for , the
distribution vanishes at some value , and accordingly the
moments decay exponentially as in . The
histograms of spatial correlations also display Derrida-Flyvbjerg singularities
for . Both below and above , the study of typical and
averaged correlations is in full agreement with the droplet scaling theory.Comment: 13 pages, 29 figure
New bounds for the free energy of directed polymers in dimension 1+1 and 1+2
We study the free energy of the directed polymer in random environment in
dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and
Vargas concerning very strong disorder by giving sharp estimates on the free
energy at high temperature. In dimension 2, we prove that very strong disorder
holds at all temperatures, thus solving a long standing conjecture in the
field.Comment: 31 pages, 4 figures, final version, accepted for publication in
Communications in Mathematical Physic
A new numerical approach to Anderson (de)localization
We develop a new approach for the Anderson localization problem. The
implementation of this method yields strong numerical evidence leading to a
(surprising to many) conjecture: The two dimensional discrete random
Schroedinger operator with small disorder allows states that are dynamically
delocalized with positive probability. This approach is based on a recent
result by Abakumov-Liaw-Poltoratski which is rooted in the study of spectral
behavior under rank-one perturbations, and states that every non-zero vector is
almost surely cyclic for the singular part of the operator.
The numerical work presented is rather simplistic compared to other numerical
approaches in the field. Further, this method eliminates effects due to
boundary conditions.
While we carried out the numerical experiment almost exclusively in the case
of the two dimensional discrete random Schroedinger operator, we include the
setup for the general class of Anderson models called Anderson-type
Hamiltonians.
We track the location of the energy when a wave packet initially located at
the origin is evolved according to the discrete random Schroedinger operator.
This method does not provide new insight on the energy regimes for which
diffusion occurs.Comment: 15 pages, 8 figure
Thermodynamic limits of sperm swimming precision
Sperm swimming is crucial to fertilise the egg, in nature and in assisted
reproductive technologies. Modelling the sperm dynamics involves elasticity,
hydrodynamics, internal active forces, and out-of-equilibrium noise. Here we
demonstrate experimentally the relevance of energy dissipation for sperm
beating fluctuations. For each motile cell, we reconstruct the time-evolution
of the two main tail's spatial modes, which together trace a noisy limit cycle
characterised by a maximum level of precision . Our results indicate
, remarkably close to the estimated precision of a
dynein molecular motor actuating the flagellum, which is bounded by its energy
dissipation rate according to the Thermodynamic Uncertainty Relation. Further
experiments under oxygen deprivation show that decays with energy
consumption, as it occurs for a single molecular motor. Both observations can
be explained by conjecturing a high level of coordination among the
conformational changes of dynein motors. This conjecture is supported by a
theoretical model for the beating of an ideal flagellum actuated by a
collection of motors, including a motor-motor nearest neighbour coupling of
strength : when is small the precision of a large flagellum is much
higher than the single motor one. On the contrary, when is large the two
become comparable.Comment: Main Text with Appendices (14 pages, 9 figures) plus Supplementary
Information, Accepted for Publication in PRX-Lif
- …