Related papers: 1d Quantum Harmonic Oscillator Perturbed by a Pote…

We prove that there is an invariant torus with given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for…

We consider a Hamiltonian $H=H^{0}(p)+\kappa H^{1}(p,q,t)$, $(p,q)\in {\mathbb{R}}^{n} \times {\mathbb{T}}^n$, $t\in{\mathbb{R}}$ where $\kappa \in {\mathbb{R}}$ is a small perturbation parameter and $p$, $q$ are the action and angle…

The use of fractional momentum operators and fractionary kinetic energy used to model linear damping in dissipative systems such as resistive circuits and a spring-mass ensambles was extended to a quantum mechanical formalism. Three…

We consider time dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$: $$ {\rm i} \partial_t u=\frac12(-\partial_{x_1}^2-\partial_{x_2}^2 + x_1^2+x_2^2)u +V(t, x, D)u, \qquad \ x \in \mathbb{R}^2, $$ where $V(t, x, D)$ is a…

We study the canonical quantization of the damped harmonic oscillator by resorting to the realization of the q-deformation of the Weyl-Heisenberg algebra (q-WH) in terms of finite difference operators. We relate the damped oscillator…

Let $J_\sigma$ be the Dunkl harmonic oscillator on ${\mathbb{R}}$ ($\sigma>-\frac{1}{2}$). For $0<u<1$ and $\xi>0$, it is proved that, if $\sigma>u-\frac{1}{2}$, then the operator $U=J_\sigma+\xi|x|^{-2u}$, with appropriate domain, is…

The first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak local energy decay estimate holds and proves that solutions to the linear wave equation with variable coefficients in $\mathbb R^{1+3}$,…

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential $V(r)=(a^2\omega^2/4)sinh(r/a)^2$ where $a$ is the curvature radius and $r$ is the geodesic distance from a fixed center. Thus the potential is…

Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the…

We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible…

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a view point of an index theorem by using an explicit matrix representation. For a positive deformation parameter $q$ or…

The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems, respectively. In this paper, we build on these techniques to develop an…

We consider the scenario of a fluctuating spacetime due to a deformed commutation relation with a fluctuating deformation parameter, or to a fluctuating metric tensor. By computing the resulting dynamics and averaging over these…

We solve the generalized relativistic harmonic oscillator in 1+1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state…

Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $\sigma $ on $H$. Let $\kappa$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(\kappa^{1/2}…

In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad…

Quantum systems lose coherence upon interaction with the environment and tend towards classical states. Quantum coherence is known to exponentially decay in time so that macroscopic quantum superpositions are generally unsustainable. In…

We prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear (also called strongly nonlinear) autonomous Hamiltonian differentiable perturbations of the mKdV equation. The proof is…

Through a set of generators that preserves the hermiticity and trace of density matrices, we analyze the damping of harmonic oscillator in open quantum systems into four modes, distinguished by their specific effects on the covariance…

We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to…