Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRigorous estimates for the relegation algorithm
63
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We revisit the relegation algorithm by Deprit et al. (Celest. Mech. Dyn. Astron. 79:157-182, 2001) in the light of the rigorous Nekhoroshevs like theory. This relatively recent algorithm is nowadays widely used for implementing closed form analytic perturbation theories, as it generalises the classical Birkhoff normalisation algorithm. The algorithm, here briefly explained by means of Lie transformations, has been so far introduced and used in a formal way, i.e. without providing any rigorous convergence or asymptotic estimates. The overall aim of this paper is to find such quantitative estimates and to show how the results about stability over exponentially long times can be recovered in a simple and effective way, at least in the non-resonant case.
rate research
Read More
We reconsider the Schroder-Siegel problem of conjugating an analytic map in $mathbb{C}$ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension $n>1$. Assuming a condition which is equivalent to Brunos one on the eigenvalues $lambda_1,ldots,lambda_n$ of the linear part we show that the convergence radius $rho$ of the conjugating transformation satisfies $ln rho(lambda )geq -CGamma(lambda)+C$ with $Gamma(lambda)$ characterizing the eigenvalues $lambda$, a constant $C$ not depending on $lambda$ and $C=1$. This improves the previous results for $n>1$, where the known proofs give $C=2$. We also recall that $C=1$ is known to be the optimal value for $n=1$.
We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + i b)tau + (c + i d) g}$ with two complex parameters and H{o}lder functions $f,: g$ generalizing the case $Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].
We consider the linear and nonlinear Schrodinger equation for a Bose-Einstein condensate in a harmonic trap with $cal {PT}$-symmetric double-delta function loss and gain terms. We verify that the conditions for the applicability of a recent proposition by Mityagin and Siegl on singular perturbations of harmonic oscillator type self-adjoint operators are fulfilled. In both the linear and nonlinear case we calculate numerically the shifts of the unperturbed levels with quantum numbers $n$ of up to 89 in dependence on the strength of the non-Hermiticity and compare with rigorous estimates derived by those authors. We confirm that the predicted $1/n^{1/2}$ estimate provides a valid upper bound on the the shrink rate of the numerical eigenvalues. Moreover, we find that a more recent estimate of $log(n)/n^{3/2}$ is in excellent agreement with the numerical results. With nonlinearity the shrink rates are found to be smaller than without nonlinearity, and the rigorous estimates, derived only for the linear case, are no longer applicable.
We study the hyperboloidal initial value problem for the one-dimensional wave equation perturbed by a smooth potential. We show that the evolution decomposes into a finite-dimensional spectral part and an infinite-dimensional radiation part. For the radiation part we prove a set of Strichartz estimates. As an application we study the long-time asymptotics of Yang-Mills fields on a wormhole spacetime.
We study the survival probability associated with a semi-classical matrix Shrodinger operator that models the predissociation of a general molecule in the Born-Oppenheimer approximation. We show that it is given by its usual time-dependent exponential contribution, up to a reminder term that is exponentially small (in the semiclassical parameter) with arbitrarily large rate of decay. The result applies in any dimension, and in presence of a number of resonances that may tend to infinity as the semiclassical parameter tends to 0.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
