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Marouane Assal

Publications and source records attributed to Marouane Assal.

9 recordsLinked to original sources
arXiv2024

On the singularities of the spectral shift function for some tight-binding models

We consider perturbed discrete tight-binding models in $\ell^2(\mathbb{Z_h},\mathcal{G})$ describing union of quantum particles with localized interactions, where $\mathbb{Z_h}$ is the 1D lattice $h\mathbb{Z_h}$, $h > 0$, and $\mathcal G$ is a separable Hilbert space. The perturbations play the role of self-adjoint relatively compact (matrix-valued) electric potentials with $\mathcal B(\mathcal G)$-valued coefficients decaying polynomially at infinity. We analyze the Spectral Shift Function (SSF) associated to the pair of the perturbed and the unperturbed operators. On the one hand, we show that the SSF is bounded near the spectral thresholds of the essential spectrum if $\dim(\mathcal G) < +\infty$. On the other hand, if $\dim(\mathcal G) = +\infty$, we show that it may have singularities at some thresholds points $\mu$ of the essential spectrum. In particular, new mechanisms allowing the SSF to have singularities at the thresholds are exhibited, based on the degeneracy of the spectrum of the unperturbed operator. Moreover, we give the main terms of the asymptotic behaviors of the SSF near $\mu$ described in terms of some explicit effective Berezin-Toeplitz type operators. These results are completed by Levinson type formulas and examples of eigenvalues asymptotics for power-like and exponential decay potentials.

math.SP↗
arXiv2024

Transition of the semiclassical resonance widths across a tangential crossing energy-level

We consider a 1D $2\times 2$ matrix-valued operator \eqref{System0} with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order $n$, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order $2n$. We compute the transfer matrix at this point between the incoming and outgoing microlocal solutions and apply it to the semiclassical distribution of resonances at the energy crossing level. It is described in terms of a generalized Airy function. This result generalizes \cite{FMW1} to the tangential crossing and \cite{AFH1} to the crossing at a turning point.

math-ph↗
arXiv2022

Semiclassical resonance asymptotics for systems with degenerate crossings of classical trajectories

This paper is concerned with the asymptotics of resonances in the semiclassical limit $h\to 0^+$ for $2\times 2$ matrix Schrödinger operators in one dimension. We study the case where the two underlying classical Hamiltonian trajectories cross tangentially in the phase space. In the setting that one of the classical trajectories is a simple closed curve whereas the other one is non-trapping, we show that the imaginary part of the resonances is of order $h^{(m_0+3)/(m_0+1)}$, where $m_0$ is the maximal contact order of the crossings. This principal order comes from the subprincipal terms of the transfer matrix at crossing points which describe the propagation of microlocal solutions from one trajectory to the other. In addition, we compute explicitly the leading coefficient of the resonance widths in terms of the probability amplitudes associated with all the \textit{generalized classical trajectories} escaping to infinity from the closed trajectory.

math-ph↗
arXiv2022

Clusters of resonances for a non-selfadjoint multichannel discrete Schr\"odinger operator

We study the distribution of resonances for discrete Hamiltonians of the form $H_0+V$ near the thresholds of the spectrum of $H_0$. Here, the unperturbed operator $H_0$ is a multichannel Laplace type operator on $\ell^2(\mathbb Z; \mathbb C^N) \cong \ell^2(\mathbb Z)\otimes \mathbb C^N$ and $V$ is a non-selfadjoint compact perturbation. We compute the exact number of resonances and give a precise description on their location in clusters around some special points in the complex plane.

math-ph↗
arXiv2022

Metastable diffusions with degenerate drifts

We study the spectrum of the semiclassical Witten Laplacian $Δ_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $Δ_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.

math.AP↗
arXiv2019

Eigenvalue splitting for a system of Schrödinger operators with an energy-level crossing

We study the asymptotic distribution of the eigenvalues of a one-dimensional two-by-two semiclassical system of coupled Schrödinger operators in the presence of two potential wells and with an energy-level crossing. We provide Bohr-Sommerfeld quantization condition for the eigenvalues of the system on any energy-interval above the crossing and give precise asymptotics in the semiclassical limit $h\to 0^+$. In particular, in the symmetric case, the eigenvalue splitting occurs and we prove that the splitting is of polynomial order $h^{\frac32}$ and that the main term in the asymptotics is governed by the area of the intersection of the two classically allowed domains.

math-ph↗
arXiv2017

Semiclassical Trace Formula and Spectral Shift Function for Systems via a Stationary Approach

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint Schrödinger operators with matrix-valued potentials. We give Weyl type semiclassical asymptotics with sharp remainder estimate for the spectral shift function, and, under the existence of a scalar escape function, a full asymptotic expansion in the strong sense for its derivative. A time-independent approach enables us to treat certain potentials with energy-level crossings.

math-ph↗
arXiv2016

Long time semiclassical Egorov theorem for $\hbar$-pseudodifferential systems

In the Heisenberg picture, we study the semiclassical time evolution of a bounded quantum observable $Q^w(x,\hbar D_x)$ associated to a $(m\times m)$ matrix-valued symbol $Q$ generated by a semiclassical matrix-valued Hamiltonian $H\sim H_0+\hbar H_1$. Under a non-crossing assumption on the eigenvalues of the principal symbol $H_0$ that ensures the existence of almost invariant subspaces of $L^{2}(\mathbb R^n)\otimes \mathbb C^m$, and for a class of observables that are semiclassically block-diagonal with respect to the projections onto these almost invariants subspaces, we establish a long time matrix-valued version for the semiclassical Egorov theorem valid in a large time interval of Ehrenfest type $T(\hbar)\simeq log(\hbar^{-1})$.

math-ph↗