11,273 research outputs found
On and Off-diagonal Sturmian operator: dynamic and spectral dimension
We study two versions of quasicrystal model, both subcases of Jacobi
matrices. For Off-diagonal model, we show an upper bound of dynamical exponent
and the norm of the transfer matrix. We apply this result to the Off-diagonal
Fibonacci Hamiltonian and obtain a sub-ballistic bound for coupling large
enough. In diagonal case, we improve previous lower bounds on the fractal
box-counting dimension of the spectrum.Comment: arXiv admin note: text overlap with arXiv:math-ph/0502044 and
arXiv:0807.3024 by other author
Effective constructions in plethysms and Weintraub's conjecture
We give a short proof of Weintraub's conjecture by constructing explicit
highest weight vectors in the symmetric power of an even exterior power
Emergence of complex and spinor wave functions in scale relativity. I. Nature of scale variables
One of the main results of Scale Relativity as regards the foundation of
quantum mechanics is its explanation of the origin of the complex nature of the
wave function. The Scale Relativity theory introduces an explicit dependence of
physical quantities on scale variables, founding itself on the theorem
according to which a continuous and non-differentiable space-time is fractal
(i.e., scale-divergent). In the present paper, the nature of the scale
variables and their relations to resolutions and differential elements are
specified in the non-relativistic case (fractal space). We show that, owing to
the scale-dependence which it induces, non-differentiability involves a
fundamental two-valuedness of the mean derivatives. Since, in the scale
relativity framework, the wave function is a manifestation of the velocity
field of fractal space-time geodesics, the two-valuedness of velocities leads
to write them in terms of complex numbers, and yields therefore the complex
nature of the wave function, from which the usual expression of the
Schr\"odinger equation can be derived.Comment: 36 pages, 5 figures, major changes from the first version, matches
the published versio
Localized and extended states in a disordered trap
We study Anderson localization in a disordered potential combined with an
inhomogeneous trap. We show that the spectrum displays both localized and
extended states, which coexist at intermediate energies. In the region of
coexistence, we find that the extended states result from confinement by the
trap and are weakly affected by the disorder. Conversely, the localized states
correspond to eigenstates of the disordered potential, which are only affected
by the trap via an inhomogeneous energy shift. These results are relevant to
disordered quantum gases and we propose a realistic scheme to observe the
coexistence of localized and extended states in these systems.Comment: Published versio
Time--Splitting Schemes and Measure Source Terms for a Quasilinear Relaxing System
Several singular limits are investigated in the context of a
system arising for instance in the modeling of chromatographic processes. In
particular, we focus on the case where the relaxation term and a
projection operator are concentrated on a discrete lattice by means of Dirac
measures. This formulation allows to study more easily some time-splitting
numerical schemes
Structure Function Scaling of a 2MASS Extinction Map of Taurus
We compute the structure function scaling of a 2MASS extinction map of the
Taurus molecular cloud complex. The scaling exponents of the structure
functions of the extinction map follow the Boldyrev's velocity structure
function scaling of supersonic turbulence. This confirms our previous result
based on a spectral map of 13CO J=1-0 covering the same region and suggests
that supersonic turbulence is important in the fragmentation of this
star--forming cloud.Comment: submitted to Ap
Emergence of complex and spinor wave functions in Scale Relativity. II. Lorentz invariance and bi-spinors
Owing to the non-differentiable nature of the theory of Scale Relativity, the
emergence of complex wave functions, then of spinors and bi-spinors occurs
naturally in its framework. The wave function is here a manifestation of the
velocity field of geodesics of a continuous and non-differentiable (therefore
fractal) space-time. In a first paper (Paper I), we have presented the general
argument which leads to this result using an elaborate and more detailed
derivation than previously displayed. We have therefore been able to show how
the complex wave function emerges naturally from the doubling of the velocity
field and to revisit the derivation of the non relativistic Schr\"odinger
equation of motion. In the present paper (Paper II) we deal with relativistic
motion and detail the natural emergence of the bi-spinors from such first
principles of the theory. Moreover, while Lorentz invariance has been up to now
inferred from mathematical results obtained in stochastic mechanics, we display
here a new and detailed derivation of the way one can obtain a Lorentz
invariant expression for the expectation value of the product of two
independent fractal fluctuation fields in the sole framework of the theory of
Scale Relativity. These new results allow us to enhance the robustness of our
derivation of the two main equations of motion of relativistic quantum
mechanics (the Klein-Gordon and Dirac equations) which we revisit here at
length.Comment: 24 pages, no figure; very minor corrections to fit the published
version: a few typos and a completed referenc
- âŠ