arXiv subjects

Maher Zerzeri

Publications and source records attributed to Maher Zerzeri.

16 recordsLinked to original sources

arXiv2026

Stationary phase with Cauchy singularity. A critical point of signature $(+,-)$

Asymptotic expressions for an integral appearing in the solution of a d-bar problem are presented. The integral is a solid Cauchy transform of a function with a rapidly oscillating phase with a small parameter $h$, $0<h\ll 1$. Whereas standard steepest descent approaches can be applied to the case where the stationary points of the phase $ω_{k}$, $k=1,\ldots, N$ are far from the singularity $ζ$ of the integrand, a polarization approach is proposed for the case that $|ζ-ω_{k}|<\mathcal{O}(\sqrt{h})$ for some $k$. In this case the problem is studied in $\mathbb{C}^{2}$ ($\widetildeω:=\overlineω$ is treated as an independent variable) on steepest descent contours. An application of Stokes' theorem allows for a decomposition of the integral into three terms for which asymptotic expressions in terms of special functions are given.

math.AP↗

arXiv2026

Higher-Order Approximation of Coherent State Dynamics in Self-Interacting Quantum Field Theories

We study the propagation of coherent states in self-interacting bosonic quantum field theories in the semi-classical (mean-field) regime. Relying on Hepp's method and a detailed analysis of the associated classical and quantum field dynamics, non-linear and linear respectively, we construct an asymptotic expansion of arbitrary order for the quantum evolution of coherent states. The results are first established for the spatially cutoff $P(ϕ)_2$ model, under standard assumptions ensuring essential self-adjointness of the Hamiltonian and well-posedness of the classical flow, and are then extended to a class of non-polynomial analytic interactions. This work refines and generalizes earlier results, which identified only the leading-order term of the expansion.

math-ph↗

arXiv2025

Counter-examples to the fractal Weyl law for semiclassical resonances

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.

math.AP↗

arXiv2021

Landau-Zener Formula in a "Non-Adiabatic" regime for avoided crossings

We study a two-level transition probability for a finite number of avoided crossings with a small interaction. Landau-Zener formula, which gives the transition probability for one avoided crossing as $e^{-π\frac{\varepsilon^{2}}{h}}$, implies that the parameter $h$ and the interaction $\varepsilon$ play an opposite role when both tend to $0$. The exact WKB method produces a generalization of that formula under the optimal regime $\frac{h}{\varepsilon^2}$ tends to~0. In this paper, we investigate the case $\frac{\varepsilon^2}{h}$ tends to 0, called "non-adiabatic" regime. This is done by reducing the associated Hamiltonian to a microlocal branching model which gives us the asymptotic expansions of the local transfer matrices.

math-ph↗

arXiv2021

Resonances over a potential well in an island

In this paper we study the distribution of scattering resonances for a multidimensional semi-classical Schrödinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.

math.AP↗

arXiv2020

Semiclassical Gevrey operators in the complex domain

We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is $\leq 2$.

math.AP↗

arXiv2020

Semiclassical Gevrey operators and magnetic translations

We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas of the time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is $\geq 2$.

math.AP↗

arXiv2020

An example of resonance instability

We construct a semiclassical Schrödinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.

math.SP↗

arXiv2017

Propagation des singularités et résonances

In the framework of semiclassical resonances, we make more precise the link between polynomial estimates of the extension of the resolvent and propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.

math.AP↗

arXiv2016

Barrier-top resonances for non globally analytic potentials

We give the semiclassical asymptotic of barrier-top resonances for Schrödinger operators on ${\mathbb R}^{n}$, $n \geq 1$, whose potential is $C^{\infty}$ everywhere and analytic at infinity. In the globally analytic setting, this has already been obtained. Our proof is based on a propagation of singularities theorem at a hyperbolic fixed point that we establish here. This last result refines a theorem of the same authors, and its proof follows another approach.

math.AP↗

arXiv2016

Resonances for homoclinic trapped sets

We study semiclassical resonances generated by homoclinic trapped sets. First, under some general assumptions, we prove that there is no resonance in a region below the real axis. Then, we obtain a quantization rule and the asymptotic expansion of the resonances when there is a finite number of homoclinic trajectories. The same kind of results is proved for homoclinic sets of maximal dimension. Next, we generalize to the case of homoclinic/heteroclinic trajectories and we study the three bump case. In all these settings, the resonances may either accumulate on curves or form clouds. We also describe the corresponding resonant states.

math.AP↗

arXiv2014

Rate of decay of some Petrowsky-like dissipative systems

In this paper, we show that the fastest decay rate for some Petrowsky-like dissipative systems is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the corresponding operator satisfied some spectral gap condition. We give also some applications to illustrate our setting.

math.AP↗

arXiv2012

On the classical limit of self-interacting quantum field Hamiltonians with cutoffs

We study, using Hepp's method, the propagation of coherent states for a general class of self interacting bosonic quantum field theories with spatial cutoffs. This includes models with non-polynomial interactions in the field variables. We show indeed that the time evolution of coherent states, in the classical limit, is well approximated by time-dependent affine Bogoliubov unitary transformations. Our analysis relies on a non-polynomial Wick quantization and a specific hypercontractive estimate.

math-ph↗

arXiv2011

Spectral shift function for slowly varying perturbation of periodic Schroedinger operators

In this paper we study the asymptotic expansion of the spectral shift function for the slowly varying perturbations of periodic Schrödinger operators. We give a weak and pointwise asymptotics expansions in powers of $h$ of the derivative of the spectral shift function corresponding to the pair $\big(P(h)=P_0+ϕ(hx),P_0=-Δ+V(x)\big),$ where $ϕ(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-δ})$ for some $δ>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$. To prove the pointwise asymptotic expansion of the spectral shift function, we establish a limiting absorption Theorem for $P(h)$.

math.SP↗

arXiv2011

Spectral shift function for perturbed periodic Schroedinger operators. The large-coupling constant limit case

In the large coupling constant limit, we obtain an asymptotic expansion in powers of $μ^{-\frac{1}δ}$ of the derivative of the spectral shift function corresponding to the pair $\big(P_μ=P_0+μW(x),P_0=-Δ+V(x)\big),$ where $W(x)$ is positive, $W(x)\sim w_0(\frac{x}{|x|})|x|^{-δ}$ near infinity for some $δ>n$ and $w_0\in {\mathcal C}^\infty(\mathbb S^{n-1};\,\mathbb R_+).$ Here $\mathbb S^{n-1}$ is the unite sphere of the space $\mathbb R^n$ and $μ$ is a large parameter. The potential $V$ is real-valued, smooth and periodic with respect to a lattice $Γ$ in ${\mathbb R}^n$.

math.SP↗

arXiv2009

Spectral projection, residue of the scattering amplitude, and Schrodinger group expansion for barrier-top resonances

We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrodinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrodinger group in terms of these resonances.

math.AP↗