1,721,027 research outputs found
Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods
No full textWe eliminate by KAM methods the time dependence in a class of linear differential equations in l2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H0 + ∈ P (ωt) for ∈ small. Here H0 is the one-dimensional Schrödinger operator p2 + V, V(x) ∼ |x|α, α > 2 for |x| → ∞, the time quasi-periodic perturbation P may grow as |x|β, β < (α - 2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients
Resonant harmonic oscillators and eigenvalue multiplicity
No full textExplicit formulas are worked out for the eigenvalue multiplicity of a systemof independent quantum harmonic oscillators in the general case of 1\leq s\leqn-1 resonance relations among the frequencies\om_1,\ldots,\om_n. As a particular case we prove that, even though the quantumnumbers are always less than the degrees of freedom, the \ev s are in generalintrinsically degenerate only in the completely resonant case
Normal forms and quantization formulae
No full textWe consider the Schrödinger operator Q = -ħ2 Δ+V in Rn, where V (x) → +∞ as \x | → +∞, is Gevrey of order l and has a unique non-degenerate minimum. A quantization formula up to an error of order e-c|lnħ|-a is obtained for all eigenvalues of Q lying in any interval [0, | \lnħ-b], with a > 1 and 0 0. For eigenvalues in [O, ħδ], 0 < δ < 1, the error is of order e-cħl|l. The proof is based upon uniform Nekhoroshev estimates on the quantum normal form constructed quantizing the Lie transformation
Long time semiclassical approximation of quantum flows: a proof of the Ehrenfest time
No full textLet H be a holomorphic Hamiltonian of quadratic growth on R2n, b a holomorphic exponentially localized observable, H, B the corresponding operators on L2(Rn) generated by Weyl quantization, and U(t) = exp iHt/ħ. It is proved that the L2 norm of the difference between the Heisenberg observable Bt = U(t)BU(-t) and its semiclassical approximation of order N - 1 is majorized by KN N(6n+1)N ħ-4/9(-ħlogħ)N for t ∈ [0, Tn(ħ)], where Tn(ħ) := -2logħ/[α(6n + 3)(N - 1)] and α := ∥Hess(x,ξ)H∥. Choosing a suitable N(ħ) the error is majorized by Cħlog| log ħ|, 0 ≤ t ≤ | log ħ|/log | log ħ| (here K and C are explicit constants independent of N, ħ)
Recenti ricerche su una categoria di demodulatori di frequenza a correlazione ampiezza fase
No full textAl demodulatore classico ideale di frequenza si può imputare una perdita di informazione in quanto non tiene alcun conto dell'ampiezza istantanea, la quale è statisticamente dipendente dalla deviazione di fase prodotta dal rumore. Di qui l'idea di studiare una categoria di demodulatori FM, costituiti da due demodulatori di tipo classico, uno di frequenza ed uno di ampiezza, e da un ulteriore circuito capace di effettuare un opportuno "confronto" fra le deviazioni di frequenza e di ampiezza prodotte dal rumore, allo scopo di pervenire ad un miglior rapporto segnale/rumore in uscita.La ricerca ha dato origine a risultati di qualche interesse, che talora si inquadrano in un contesto più generale e che, in parte, sono già stati oggetto di pubblicazioni [6,7]. La presente comunicazione riassume i risultati fino ad oggi ottenuti ed accenna ad alcune prospettive di ulteriori sviluppi
Ground states for a class of deterministic spin models with glassy behaviour
No full textWe consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with \ha\ , where is the discrete sine Fourier transform. The ground state found by these authors for odd and prime is shown to become asymptotically dege\-ne\-ra\-te when is a product of odd primes, and to disappear for even. This last result is based on the explicit construction of a set of eigenvectors for , obtained through its formal identity with the imaginary part of the propagator of the quantized unit symplectic matrix over the -torus.
Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator
No full textWe construct a family of global Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. This construction is based on the geometric approach to the corresponding HamiltonJacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics
Quality & Quantity: Introduction
No full textThis special issue of Quality and Quantity stems from the workshop “How can
mathematics contribute to social science” organized by Nicola Bellomo, Pierluigi
Contucci and Sandro Graffi at the Department of Mathematics of the University of
Bologna in March 2006. The purpose of the meeting was to bring together expertises
from the two fields aiming at each other from the methodological perspective within
the social science investigations.
This collection, which follows the invitation of the Editor Vittorio Capecchi, has
a similar intention: to allow the interested reader to have a series of papers in which
problems coming from Social Sciences are studied and developed using methods and
models originated in the mathematical-physics investigations. It is our belief that the
two communities have a lot to learn from each other and our hope is to stimulate
further studies by suggesting some bridge between them
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