Sandro Graffi

Time Quasi-Periodic Unbounded Perturbations of Schrödinger Operators and KAM Methods

Abstract: We 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+εP(ωt) for ε 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 Kuksins estimate of solutions of homological equations with non-constant coefficients.

Classical limit of the Quantized Hyperbolic Toral Automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N→∞,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.

Spectra of PT-symmetric operators and perturbation theory

A criterion is formulated for existence and another for the non-existence of complex eigenvalues for a class of non-self-adjoint operators in Hilbert space invariant under a particular discrete symmetry. Applications to the PT-symmetric Schrodinger operators are discussed.

The Schrödinger equation and canonical perturbation theory

LetT0(ħ, ω)+εV be the Schrödinger operator corresponding to the classical HamiltonianH0(ω)+εV, whereH0(ω) is thed-dimensional harmonic oscillator with non-resonant frequencies ω=(ω1, ... , ωd) and the potentialV(q1, ... ,qd) is an entire function of order (d+1)−1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT0(ħ, ω) admits a convergent expansion in powers of ħ of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues λn(ħ, ε) ofT0+εV such thatnħ coincides with a KAM torus are given, up to order ε∞, by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in ħ.

MEAN-FIELD APPROXIMATION OF QUANTUM SYSTEMS AND CLASSICAL LIMIT

We prove that, for a smooth two-body potentials, the quantum mean-field approximation to the nonlinear Schroedinger equation of the Hartree type is stable at the classical limit h \to 0, yielding the classical Vlasov equation.

symmetric non-self-adjoint operators, diagonalizable and non-diagonalizable, with a real discrete spectrum

Consider in L(R), d ≥ 1, the operator family H(g) := H0 + igW . H0 = a∗1a1 + . . .+ a∗ d ad + d/2 is the quantum harmonic oscillator with rational frequencies , W a P symmetric bounded potential, and g a real coupling constant. We show that if |g| < ρ, ρ being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator H(g) = a∗ 1 a1 + a ∗ 2 a2 + iga ∗ 2 a1 has real discrete spectrum but is not diagonalizable.Consider in , the operator family . H0 = a*1a1 + + a*dad + d/2 is the quantum harmonic oscillator with rational frequencies, W is a symmetric bounded potential, and g is a real coupling constant. We show that if |g| < ?, ? being an explicitly determined constant, the spectrum of H(g) is real and discrete. Moreover we show that the operator has a real discrete spectrum but is not diagonalizable.

Thermodynamical limit for correlated Gaussian random energy models

Abstract: Let {EΣ(N)}ΣΣN be a family of |ΣN|=2N centered unit Gaussian random variables defined by the covariance matrix CN of elements cN(Σ,τ):=Av(EΣ(N)Eτ(N)) and the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N=N1+N2, and all pairs (Σ,τ)ΣN×ΣN: where πk(Σ),k=1,2 are the projections of ΣΣN into ΣNk. The condition is explicitly verified for the Sherrington-Kirkpatrick, the even p-spin, the Derrida REM and the Derrida-Gardner GREM models.

The mathematical aspects of quantum maps

to Dynamical Systems.- Number Theoretic Background.- Mathematical Aspects of Quantum Maps.- Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems.- From Normal to Anomalous Deterministic Diffusion.

Absolute Continuity of the Floquet Spectrum for a Nonlinearly Forced Harmonic Oscillator

Abstract: We prove that the Floquet spectrum of the time periodic Schrödinger equation corresponding to a mildly nonlinear resonant forcing, is purely absolutely continuous for μ suitably small.

Quantization of the classical Lie algorithm in the Bargmann representation

Abstract For any polynomial perturbation of a d -dimensional system of non-resonant harmonic oscillators it is proved, writing the classical Hamiltonian in complex coordinates and the corresponding Schrodinger operator in the Bargmann representation, that the classical Lie algorithm generating the canonical perturbation expansion can be “exactly” quantized to yield the Rayleigh-Schrodinger series near any unperturbed bound state.