We build a new estimate relative with Hermite functions based upon oscillatory integrals and Langer's turning point theory. From it we show that the equation $$ i \partial_t u =-\partial_x^2 u+x^2 u+ε\langle x\rangle^μ W(νx,ωt)u,\quad u=u(t,x),~x\in\mathbb R,~ 0\leq μ<\frac13,$$ can be reduced in $\mathcal H^1(\mathbb R)$ to an autonomous system for most values of the frequency vector $ω$ and $ν$, where $W(φ, θ)$ is a smooth map from $ \mathbb T^d\times \mathbb T^n$ to $\mathbb R$ and odd in $φ$.
We prove reducibility of a transport equation on the $d$-dimensional torus $T^d$ with a time quasi-periodic unbounded perturbation. As far as we know this is the first example of a reducibility result for an equation in more than one dimensions with unbounded perturbations. Furthermore the unperturbed problem has eigenvalues whose differences are dense on the real axis.
In this paper we study reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows to deal with perturbations of order strictly larger than the ones considered in all the previous papers.
We build a new estimate for the normalized eigenfunctions of the operator $-\partial_{xx}+\mathcal V(x)$ based on the oscillatory integrals and Langer's turning point method, where $\mathcal V(x)\sim |x|^{2\ell}$ at infinity with $\ell>1$. From it and an improved reducibility theorem we show that the equation \[\textstyle {\rm i}\partial_t ψ=-\partial_x^2 ψ+\mathcal V(x) ψ+ε\langle x\rangle^μ W(νx,ωt)ψ,\quad ψ=ψ(t,x),~x\in\mathbb R, ~μ<\min\left\{\ell-\frac23,\frac{\sqrt{4\ell^2-2\ell+1}-1}2\right\},\] can be reduced in $L^2(\mathbb R)$ to an autonomous system for most values of the frequency …
We prove the reducibility of quantum harmonic oscillators in $\mathbb R^d$ perturbed by a quasi-periodic in time potential $V(x,ωt)$ with $\mathit{logarithmic~decay}$. By a new estimate built for solving the homological equation we improve the reducibility result by Grébert-Paturel(Annales de la Faculté des sciences de Toulouse : Mathématiques. $\mathbf{28}$, 2019).
We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb R$ perturbed by a quasi-periodic in time potential $V(x,ωt)$ under the following conditions, namely there is a $C>0$ such that \begin{equation*} |V(x,θ)|\le C,\quad|x\partial_xV(x,θ)|\le C,\quad\forall~(x,θ)\in\mathbb R\times\mathbb T_σ^n. \end{equation*} The corresponding perturbation matrix $(P_i^j(θ))$ is proved to satisfy $(1+|i-j|)| P_i^j(θ)|\le C$ and $\sqrt{ij}|P_{i+1}^{j+1}(θ)-P_i^j(θ)|\le C$ for any $θ\in\mathbb T_σ^n$ and $i,j\geq 1$. A new reducibility theorem is set up under this kind of decay in the perturba…
We investigate the large $n$ asymptotics of the $n$-th eigenvalue for a class of unbounded self-adjoint operators defined by infinite Jacobi matrices with discrete spectrum. In the case of the quantum Rabi model we obtain the first three terms of the asymptotics which determine the parameters of the model. This paper is based on our previous paper [5] that it completes and improves.
The Floquet-Magnus expansion of unbounded operators
Institute of Theoretical Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学理论物理研究所）；School of Mathematical Sciences, Lancaster University（兰卡斯特大学数学科学学院）
We compute the resolvent of the anti-commutator operator $XP+PX$ and of the quantum harmonic oscillator Hamiltonian operator $\frac{1}{2}(X^2+P^2)$. Using Stone's formula for finding the spectral resolution of an, either bounded or unbounded, self-adjoint operator on a Hilbert space, we also compute their Vacuum Characteristic Function (Quantum Fourier Transform). We also show how Stone's formula is applied to the computation of the Vacuum Characteristic Function of finite dimensional quantum observables. The method is proposed as an analytical alternative to the algebraic (or Heisenberg) appr…
