Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30We consider an impedance boundary-value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a strip. Pseudo-differential operators are used to deal with this wave diffraction problem. Therefore, single and double layer potentials allow a reformulation of the problem into a system of integral equations. By using operator theoretical methods, the well-posedness of the problem...

We introduce pseudodifferential operators (of infinite order) in the framework of non-quasianalytic classes of Beurling type. We prove that such an operator with (distributional) kernel in a given Beurling class ${}_{\left(\omega \right)}^{\text{'}}$ is pseudo-local and can be locally decomposed, modulo a smoothing operator, as the composition of a pseudodifferential operator of finite order and an ultradifferential operator with constant coefficients in the sense of Komatsu, both operators with kernel in the same class ${}_{\left(\omega \right)}^{\text{'}}$. We also...

We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove...

A necessary and sufficient condition for a bounded operator on $L^2\left(M\right)$, M a Riemannian compact homogeneous space, to be smooth under conjugation by the regular representation is given. It is shown that, if all formal ’Fourier multipliers with variable coefficients’ are bounded, then they are also smooth. In particular, they are smooth if M is a rank-one symmetric space.

Per una classe di operatori pseudodifferenziali a caratteristiche multiple vengono date condizioni necessarie e sufficienti per la validità di stime dal basso «ottimali»

We study one-dimensional Lévy processes with Lévy-Khintchine exponent ψ(ξ²), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition...

We prove a uniform lower bound for the difference λ₂ - λ₁ between the first two eigenvalues of the fractional Schrödinger operator ${(-\Delta )}^{\alpha /2}+V$, α ∈ (1,2), with a symmetric single-well potential V in a bounded interval (a,b), which is related to the Feynman-Kac semigroup of the symmetric α-stable process killed upon leaving (a,b). “Uniform” means that the positive constant $C_{\alpha }$ appearing in our estimate $\lambda ₂-\lambda ₁\ge C_{\alpha }{(b-a)}^{-\alpha }$ is independent of the potential V. In the general case of α ∈ (0,2), we also find a uniform lower bound for...

To every elliptic SG pseudo-differential operator with positive orders, we associate the minimal and maximal operators on $L^p\left(ℝⁿ\right)$, 1 < p < ∞, and prove that they are equal. The domain of the minimal ( = maximal) operator is explicitly computed in terms of a Sobolev space. We prove that an elliptic SG pseudo-differential operator is Fredholm. The essential spectra of elliptic SG pseudo-differential operators with positive orders and bounded SG pseudo-differential operators with orders 0,0 are computed....

We study general continuity properties for an increasing family of Banach spaces $S_w^p$ of classes for pseudo-differential symbols, where $S_w^{\infty }=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\mathrm{Op}\left(S_w^p\right)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\mathrm{Op}\left(S_w^p\right)\mathrm{Op}\left(S_w^r\right)\subset \mathrm{Op}\left(S_w^r\right)$ and $S_w^p·S_w^q\subset S_w^r$, provided $1/p+1/q=1/r$. If instead $1/p+1/q=1+1/r$, then $S_w^{p}w*S_w^q\subset S_w^r$. By modifying the definition of the $S_w^p$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.

Dans cet article on décrit le spectre semi-classique d’un opérateur de Schrödinger sur $ℝ$ avec un potentiel type double puits. La description qu’on donne est celle du spectre autour du maximum local du potentiel. Dans la classification des singularités de l’application moment d’un système intégrable, le double puits représente le cas des singularités non-dégénérées de type hyperbolique.