Consider quantum harmonic oscillator, perturbed by an even almost-periodic complex-valued potential with bounded derivative and primitive. Suppose that we know the first correction to the spectral asymptotics $\{Δμ_n\}_{n=0}^\infty$ ($Δμ_n=μ_n-μ_n^0+o(n^{-1/4})$, where $μ_n^0$ and $μ_n$ is the spectrum of the unperturbed and the perturbed operators, respectively). We obtain the formula that recovers the frequencies and the Fourier coefficients of the perturbation.
We study the asymptotic behavior of the spectrum of a quantum system which is a perturbation of a spherically symmetric anharmonic oscillator in dimension 2. We prove that a large part of its eigenvalues can be obtained by Bohr-Sommerfeld quantization rule applied to the normal form Hamiltonian and also admit an asymptotic expansion at infinity. The proof is based on the generalization to the present context of the normal form approach developed in [BLM20b] (see also [PS10]) for the particular case of $T^d$.
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).
In this work, concentration properties of quasimodes for perturbed semiclassical harmonic oscillators are studied. The starting point of this research comes from the fact that, in the presence of resonances between frequencies of the harmonic oscillator and for a generic bounded perturbation, the set of semiclassical measures for quasimodes of sufficiently small width is strictly smaller than the corresponding set of semiclassical measures for the unperturbed system. In this work we precise the description of this set taking into account the Diophantine properties of the vector of frequencies,…
In this work we address the problem of the quantization of a simple harmonic oscillator that is perturbed by a time dependent force. The approach consists of removing the perturbation by a canonical change of coordinates. Since the quantization procedure uses the classical Hamiltonian formalism as staring point, the change of variables is carried out using canonical transformations, and to transform between the quantized systems the canonical transformation is implemented as a unitary transformation mapping the states of the perturbed and unperturbed system onto each other.
We consider a boundary value problem for a general second order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric conditions on the shapes of the cavities and no conditions on their distribution in the domain. On the boundaries of the cavities a nonlinear Robin condition is imposed. The sizes of the cavities and the minimal distance between them are supposed to satisfy a certain simple condition ensuring that under the homogenization the cavities disappear and we obtain a…
We consider a boundary value problem for a general second order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization we obtain a similar problem in a non-perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of th…
