This article is concerned with estimations from below for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Concentrating on the case of odd ℓ, it continues the work done in part I [21] which dealt with even ℓ.

This paper is devoted to the spectral analysis of a non elliptic operator $A$, deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator $A$ has been derived, we determine its continuous spectrum. Then, we show that $A$ is unbounded from below and that it has a sequence of negative eigenvalues tending to $-\infty$. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some conditions...

This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some...

Asymptotics with sharp remainder estimates are recovered for number $𝐍\left(\tau \right)$ of eigenvalues of operator $A(x,D)-tW(x,x)$ crossing level $E$ as $t$ runs from $0$ to $\tau$, $\tau \to \infty$. Here $A$ is periodic matrix operator, matrix $W$ is positive, periodic with respect to first copy of $x$ and decaying as second copy of $x$ goes to infinity, $E$ either belongs to a spectral gap of $A$ or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.

Il est bien connu que les fréquences propres associées à un d'Alembertien amorti sont confinées dans une bande parallèle à l'axe réel. Nous rappelons l'asymptotique de Weyl pour la distribution des parties réelles des fréquences propres, nous montrons que «presque toutes» les fréquences propres appartiennent à une bande déterminée par la limite de Birkhoff du coefficient d'amortissement. Nous montrons aussi que certaines moyennes des parties imaginaires convergent vers la moyenne du coefficient...

This article studies the asymptotic behavior of the number $N\left(\lambda \right)$ of the negative eigenvalues $\<-\lambda$ as $\lambda \to +0$ of the two dimensional Pauli operators with electric potential $V\left(x\right)$ decaying at $\infty$ and with nonconstant magnetic field $b\left(x\right)$, which is assumed to be bounded or to decay at $\infty$. In particular, it is shown that $N\left(\lambda \right)=(1/2\pi ){\int }_{V\left(x\right)\>\lambda }b\left(x\right)dx(1+o\left(1\right))$, when $V\left(x\right)$ decays faster than $b\left(x\right)$ under some additional conditions.

In this paper we consider the $\varphi$-Laplacian problem with Dirichlet boundary condition, $$-\mathrm{div}\left(\varphi \left(\right|\nabla u\left|\right)\frac{\nabla u}{\left|\nabla u\right|}\right){=\lambda g(·)\varphi \left(u\right)\text{in}\Omega ,\lambda \in ℝ\text{and}u|}_{\partial \Omega }=0.$$ The term $\varphi$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty }\left(\Omega \right)$ and $\Omega \subseteq {ℝ}^N$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both a minimizer...