arXiv subjects

Mouez Dimassi

Publications and source records attributed to Mouez Dimassi.

11 recordsLinked to original sources

arXiv2024

Spectral analysis and best decay rate of the wave propagator on the tadpole graph

We consider the damped wave semigroup on the tadpole graph ${\mathcal R}$. We first give a meticulous spectral analysis, followed by a judicious decomposition of the resolvent's kernel. As a consequence, and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of the energy space $\mathcal{H}$, we establish the exponential decay of the corresponding energy, with the optimal decay rate dictated by the spectral abscissa of the relevant operator.

math.AP↗

arXiv2024

A Gaussian Beam Construction of de Haas-vanAlfven Resonances

In this article we consider a Bloch electron in a crystal lattice subject to slowy varying external magnetic fields. We offer an explanation of de Haas-van Alfven oscillations in terms of energy levels of approximate eigenfunctions for the magnetic Schrodinger operator by using a gaussian beam construction for a small enough magnetic field.

math-ph↗

arXiv2020

Absence of embedded eigenvalues for Hamiltonian with crossed magnetic and electric fields

In the presence of the homogeneous electric field ${\bf E}$ and the homogeneous perpendicular magnetic field ${\bf B}$, the classical trajectory of a quantum particle on ${\mathbb R}^2$ moves with drift velocity $\alpha$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.

math.SP↗

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 Schr\"odinger 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↗

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↗

arXiv2013

Upper bound for the counting function of interior transmission eigenvalues

For the complex interior transmission eigenvalues (ITE) we study for small $\theta > 0$ the counting function $$N(\theta, r) = #\{\lambda \in \C:\: \lambda \: {\rm is} \: {\rm (ITE)},\: |\lambda| \leq r, \: 0 \leq \arg \lambda \leq \theta\}.$$ We obtain for fixed $\theta > 0$ an upper bound $N(\theta, r) \leq C r^{n/2}, \: r \geq r(\theta).$

math.SP↗

arXiv2013

Trace asymptotics formula for the Schr\"odinger operators with constant magnetic fields

In this paper, we consider the 2D- Schr\"odinger operator with constant magnetic field $H(V)=(D_x-By)^2+D_y^2+V_h(x,y)$, where $V$ tends to zero at infinity and $h$ is a small positive parameter. We will be concerned with two cases: the semi-classical limit regime $V_h(x,y)=V(h x,h y)$, and the large coupling constant limit case $V_h(x,y)=h^{-\delta} V(x,y)$. We obtain a complete asymptotic expansion in powers of $h^2$ of ${\rm tr}(\Phi(H(V),h))$, where $\Phi(\cdot,h)\in C^\infty_0(\mathbb R;\mathbb R)$. We also give a Weyl type asymptotics formula with optimal remainder estimate of the counting function of eigenvalues of $H(V)$.

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 Schr\"odinger 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 $\big(P(h)=P_0+\phi(hx),P_0=-\Delta+V(x)\big),$ where $\phi(x)\in {\mathcal C}^\infty(\mathbb R^n,\mathbb R)$ is a decreasing function, ${\mathcal O}(|x|^{-\delta})$ for some $\delta>n$ and $h$ is a small positive parameter. Here the potential $V$ is real, smooth and periodic with respect to a lattice $\Gamma$ 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 $\mu^{-\frac{1}{\delta}}$ of the derivative of the spectral shift function corresponding to the pair $\big(P_\mu=P_0+\mu W(x),P_0=-\Delta+V(x)\big),$ where $W(x)$ is positive, $W(x)\sim w_0(\frac{x}{|x|})|x|^{-\delta}$ near infinity for some $\delta>n$ 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 $\mu$ is a large parameter. The potential $V$ is real-valued, smooth and periodic with respect to a lattice $\Gamma$ in ${\mathbb R}^n$.

math.SP↗

arXiv2010

Spectral problems for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y), \: B > 0, \epsilon > 0$. We prove that the operator $H(B, \epsilon)$ has at most a finite number of embedded eigenvalues on $\R$ which is a step to the proof of the conjecture of absence of embedded eigenvalues of $H$ in $\R.$ Applying the formula for $\xi'(\lambda, B, \epsilon)$, we obtain a semiclassical asymptotics of the spectral shift function related to the operators $H_0(h) = (hD_x - By)^2 + h^2D_y^2 + \epsilon x$ and $H(h) = H_0(h) + V(x,y).$

math-ph↗

arXiv2009

Spectral shift function for operators with crossed magnetic and electric fields

We obtain a representation formula for the derivative of the spectral shift function $\xi(\lambda; B, \epsilon)$ related to the operators $H_0(B,\epsilon) = (D_x - By)^2 + D_y^2 + \epsilon x$ and $H(B, \epsilon) = H_0(B, \epsilon) + V(x,y), \: B > 0, \epsilon > 0$. We establish a limiting absorption principle for $H(B, \epsilon)$ and an estimate ${\mathcal O}(\epsilon^{n-2})$ for $\xi'(\lambda; B, \epsilon)$, provided $\lambda \notin \sigma(Q)$, where $Q = (D_x - By)^2 + D_y^2 + V(x,y).$

math-ph↗