Related papers: KAM for high-dimensional nonlinear quantum harmoni…
Li\'enard-type nonlinear one-dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schr\"odinger equation is exactly solved in momuntum space…
Solving a quadratic nonlinear system of equations (QNSE) is a fundamental, but important, task in nonlinear science. We propose an efficient quantum algorithm for solving $n$-dimensional QNSE. Our algorithm embeds QNSE into a…
The problem of quantum harmonic oscillator with "regular+random" square frequency, subjected to "regular+random external force, is considered in framework of representation of the wave function by complex-valued random process. Average…
Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous…
It is proved that the KAM tori (thus quasi-periodic solutions) are long time stable for nonlinear Schrodinger equation.
From the beginning of KAM theory, it was realized that its applicability to realistic problems depended on developing quantitative estimates on the sizes of the perturbations allowed. In this paper we present results on the existence of…
The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is…
In this paper, we consider a classical Hamiltonian normal form with degeneracy in normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy…
Multiparametric quantum $gl(2)$ algebras are presented according to a classification based on their corresponding Lie bialgebra structures. From them, the non-relativistic limit leading to quantum harmonic oscillator algebras is implemented…
Quantum harmonic oscillators linearly coupled through coordinates and momenta, represented by the Hamiltonian $ {\hat H}=\sum^2_{i=1}\left( \frac{ {\hat p}^{2}_i}{2 m_i } + \frac{m_i \omega^2_i}{2} x^2_i\right) +{\hat H}_{int} $, where the…
Nonlinear time-dependent partial differential equations are essential in modeling complex phenomena across diverse fields, yet they pose significant challenges due to their computational complexity, especially in higher dimensions. This…
For nonautonomous Hamiltonian systems and their quantisations we discuss properties of the quantised systems, related to those of the corresponding classical systems, described by the KAM-related theories: the proper KAM, the averaging…
We consider a set of N linearly coupled harmonic oscillators and show that the diagonalization of this problem can be put in geometrical terms. The matrix techniques developed here allowed for solutions in both the classical and quantum…
The quantum mechanical equivalent of parametric resonance is studied. A simple model of a periodically kicked harmonic oscillator is introduced which can be solved exactly. Classically stable and unstable regions in parameter space are…
Quantum computing offers a promising avenue for advancing computational methods in science and engineering. In this work, we introduce the quantum asymptotic numerical method (qANM), a framework for solving nonlinear problems using quantum…
We are dealing in this work with such formal and conceptual extensions of nonrelativistic quantum mechanics (QM) which contain QM with its standard formalism and interpretation as a subtheory. QM is here primarily equivalently reformulated…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…
We show the existence of a continuum of Hamiltonian structures for the two-dimensional isotropic harmonic oscillator. In particular, a continuum of Hamiltonian structures having noncommutative coordinates is presented. A study of the…
We investigate the connection between the linear harmonic oscillator equation and some classes of second order nonlinear ordinary differential equations of Li\'enard and generalized Li\'enard type, which physically describe important…
This paper is devoted to find the exact solution of the harmonic oscillator in a position-dependent 4-dimensional noncommutative phase space. The noncommutative phase space that we consider is described by the commutation relations between…