The mathematical theory of resonances whose widths are exponentially small, II (original) (raw)
Approximation of the Width of Resonances
Applied Mathematical Sciences, 2011
We study the Schrodinger operator, with a many-body potential of Coulomb type when h tends to 0. We consider the case where two electronic levels cross, and that the resonances appear (due to the well created by the second electron level). We show that these resonances ...
Partial widths of resonances by analytic continuation from real eigenvalues
Chemical Physics Letters, 1990
An analytic continuation method for obtaining resonance widths from real eigenvalues in stabilization graphs has been extended to the case of multiple decay channels. An approximate procedure for decoupling the exit channels permits calculation of partial widths in reasonable agreement with exact values, especially for narrow resonances. Results are reported for a simple model scattering system introduced by Fels and Hazi.
Extended Fundamental Model of Resonance
2003
Fundamental models are the simplest, one degree of freedom Hamiltonians that serve as a tool to understand the qualitative effects of various resonances. A new, extended fundamental model (EFM) is proposed in order to improve the classical, Andoyer type, second fundamental model (SFM). The EFM Hamiltonian differs from the SFM by the addition of a term with the third power of momentum; it depends on two free parameters. The new model is studied for the case of a firstorder resonance, where up to five critical points can be present. Similarly, to the respective SFM, it admits only the saddle-node bifurcations of critical points, but its advantage lies in the capability of generating the separatrix bifurcations, known also as saddle connections. The reduction of parameters for the EFM has been performed in a way that allows the use of the model in the case of the so-called abnormal resonance.
Sharp exponential bounds on resonances states and width of resonances
Advances in Applied Mathematics, 1988
We prove sharp exponential bounds on resonance states for the shape (a-decay) and Stark resonances. We use these bounds to estimate the width of these resonances. The bound on width of Stark resonances gives a partial generalization of the classical Oppenheimer formula derived originally for the hydrogen atom. o 198x Academic Press. Inc.
6 the Width of Resonances for Slowly Varying Perturbations of One-Dimensional Periodic
2015
In this talk, we report on results about the width of the resonances for a slowly varying perturbation of a periodic operator. The study takes place in dimension one. The perturbation is assumed to be analytic and local in the sense that it tends to a constant at +∞ and at −∞; these constants may differ. Modulo an assumption on the relative position of the range of the local perturbation with respect to the spectrum of the background periodic operator, we show that the width of the resonances is essentially given by a tunneling effect in a suitable phase space. Résumé. Dans cet exposé, nous décrirons le calcul de la largeur des résonances de perturbations lentes d'opérateurs de Schrödinger périodiques. Cetteétude est unidimensionnelle. Les perturbations lentes considérées sont analytiques et locales au sens où elles tendent vers une constante en +∞ et en −∞ ; ces deux constantes peuvent toutefoisêtre différentes. Sous des hypothèses adéquates sur la position relative de l'image de la perturbation locale par rapport au spectre de l'opérateur de Schrödinger périodique, nous démontrons que la largeur des résonances est donnée par un effet tunnel dans un espace de phase adéquat.
A Rational Approach to the Resonance Region
Nuclear Physics B - Proceedings Supplements, 2009
Resonance Saturation in QCD can be understood in the large-Nc limit from the mathematical theory of Padé Approximants to meromorphic functions.
Journal of Physics A: Mathematical and Theoretical
We consider the large L limit of one dimensional Schrödinger operators H L = −d 2 /dx 2 + V 1 (x) + V 2,L (x) in two cases: when V 2,L (x) = V 2 (x − L) and when V 2,L (x) = e −cL δ(x − L). This is motivated by some recent work of Herbst and Mavi where V 2,L is replaced by a Dirichlet boundary condition at L. The Hamiltonian H L converges to H = −d 2 /dx 2 + V 1 (x) as L → ∞ in the strong resolvent sense (and even in the norm resolvent sense for our second case). However, most of the resonances of H L do not converge to those of H. Instead, they crowd together and converge onto a horizontal line: the real axis in our first case and the line Im(k) = −c/2 in our second case. In the region below the horizontal line resonances of H L converge to the reflectionless points of H and to those of −d 2 /dx 2 + V 2 (x). It is only in the region between the real axis and the horizontal line (empty in our first case) that resonances of H L converge to resonances of H. Although the resonances of H may not be close to any resonance of H L we show that they still influence the time evolution under H L for a long time when L is large.
Theory of parametric resonance: modern results
2003 IEEE International Workshop on Workload Characterization (IEEE Cat. No.03EX775), 2003
Linear dynatnical systems with many degrecs of freedom with periodic coefficients also depending on constmt parameters arc considcred. Stability of the trivial solution is studied with the use of the Floqiet theory. First arid second order derivativcs of the Floqqet matrix with rcspect to parameters are derivcd in terms of matriciants of the main and adjoint problems and derivatives of the system matrix. This allows to find &e derivatives of simple multipliers, responsible for svability of the system, with respect to parameters and predict llicir bchavior with a change of parameters. lliis shown how to use this information in gradipt procedures for stabilization or destabilization of the system. As a numerical example, the system described by Carsson-Cambi equation is considered. Then, strong and week interactions of multipliers on the complex plane are studied, and geometric interpretation o f thesc intcractions is giveo. As application or the dcvcloped theory the resonance domains for Hill's equation with damping are studied. It is shown that they represunt halves of cones in the three-parameter space. Then, parametric resonance of a pendulum with damping &d vibrating suspension point following arbitrary periodic law is considered, and the parametric resoniince domains are Sound. Another important application of damped Hill's equation is connected with the study:of stability of periodic motions in non-linear dynamical systems. I t is shown how to find stable and unstable regimes for harmonically excited Duffing's equation. Then, linear vihrational systems with periodic coefficients depending on three independent parametdrs: frequency and amplitude of pcriodic excitation, and damping parameter are considered with the assumption that the last two quantities arc small. Instability of $he trivial solution of thc system (parametric resonance);is studied. For arbitrary matrix of periodic excitation 4id positive definitc damping matrix general expressions for domains of the main (simple) and coinbinatibn resonances arc dcrived. Two important specific casesiof excitatioii matrix are studied: a symmetric matrix and a stationary matrix multiplied by a scalar perioclic function. I t is shown that in both cases the resonancc domains arc halves of cones in the thrce-dimensional space with the boundary surface coefficients depcnding only on the cigenfrequencies, eigenmodes and system matrices. The obtained relations allow to analyze influence of growing eigenfrequencies and resonance number on resonance domains. Two mechanical problems arc considered and solved Bolotin's problem of dynamic stability of a beam loadcd by periodic bending moments, and parametric resonance of a nonuniform column loaded by periodic longitudinal force. The lecture is a review of the recent results on parametric resonance obtained by the author with We consider a system of linear difrerential equations where G = G(1,p) is a real square matrix of dimension rn , which is smoothly depending on a vector of real parameters p = (pI,pz, ...p,,) and is a continuous periodic function of the time G(t,p) = c(t + T,p), T being a period. We denote linear independent solutions of system (I) as x l ( t ) , x 2 ( t ) , ..., x , ( t ) and Sorm out of them a fundamental matrix X(t) = [xl(t),X*(t) ,..., X,(t)].