A Floquet Operator with Pure Point Spectrum and Energy Instability

Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi- Periodic Perturbations

We prove that the $1-d$ quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper \cite{EV} in the general quasi-periodic setting. The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation.

Weakly regular Floquet Hamiltonians with pure point spectrum

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: i…

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

We prove that the Floquet spectrum of a class of time-periodic Schroedinger equations under a a mildly nonlinear resonant forcing is purely absolutely continuous.

On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

By a straightforward generalisation, we extend the work of Combescure from rank-1 to rank-N perturbations. The requirement for the Floquet operator to be pure point is established and compared to that in Combescure. The result matches that in McCaw. The method here is an alternative to that work. We show that if the condition for the Floquet operator to be pure point is relaxed, then in the case of the delta-kicked Harmonic oscillator, a singularly continuous component of the Floquet operator spectrum exists. We also provide an in depth discussion of the conjecture presented in Combescure of t…

Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues

We prove a localization theorem for continuous ergodic Schrödinger operators $ H_ω:= H_0 + V_ω$, where the random potential $ V_ω$ is a nonnegative Anderson-type perturbation of the periodic operator $ H_0$. We consider a lower spectral band edge of $ σ(H_0) $, say $ E= 0 $, at a gap which is preserved by the perturbation $ V_ω$. Assuming that all Floquet eigenvalues of $ H_0$, which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval $ I $ containing 0 such that $ H_ω$ has only pure point spectrum in $ I $ for almost all $ …

柔性壁附近两层振荡流的 Floquet 稳定性分析

Floquet stability analysis of a two-layer oscillatory flow near a flexible wall

本文对柔性壁分隔的两层平行振荡流进行线性 Floquet 稳定性分析，揭示临界条件及 Reynolds 数、通道宽度和壁面刚度对扰动波长与失稳的影响。