Related papers: Semiclassical functional calculus for $h$-dependen…

We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also…

Let $T\colon H^1({\mathbb R})\to H^1({\mathbb R})$ be a bounded Fourier multiplier on the analytic Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ and let $m\in L^\infty({\mathbb R}_+)$ be its symbol, that is,…

In the present paper we consider the functional dissipativity of the Dirichlet problem for systems of partial differential operators of the form $\partial_{h} ({\mathop{\mathscr A}\nolimits}^{hk}(x)\partial_{k})$ (${\mathop{\mathscr…

Given a complex, separable Hilbert space $\cH$, we consider differential expressions of the type $\tau = - (d^2/dx^2) + V(x)$, with $x \in (a,\infty)$ or $x \in \bbR$. Here $V$ denotes a bounded operator-valued potential $V(\cdot) \in…

Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic…

Let $(M,g_0)$ be a compact Riemmanian manifold of dimension $n$. Let $P_0 (\h) := -\h^2\Delta_{g}+V$ be the semiclassical Schr\"{o}dinger operator for $\h \in (0,\h_0]$, and let $E$ be a regular value of its principal symbol…

We consider the Dirichlet realization of the operator $-h^2\Delta+iV$ in the semi-classical limit $h\to0$, where $V$ is a smooth real potential with no critical points. For a one dimensional setting, we obtain the complete asymptotic…

The essential support of the symbol of a semiclassical pseudodifferentail operator is characterized by semiclassical wavefront sets of distributions. The proof employs a coherent state whose center in phase space is dependent on Planck's…

Let $(\varphi_t)_{t\geq 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $\Omega = h(\mathbb{D})$. We study the point spectrum $\sigma_p(\Delta\mid_{H^p})$ of $\Delta$, the…

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey-$s$ regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is…

Using the Davies-Helffer-Sj\"ostrand functional calculus based on almost analytic extensions, we address the following problem: Given self-adjoint operators $S_j$, $j=1,2$, in $\mathcal{H}$, and functions $f$ in an appropriate class, for…

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the…

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a…

In this paper we extensively investigate the class of conditionally positive definite operators, namely operators generating conditionally positive definite sequences. This class itself contains subnormal operators, $2$- and $3$-isometries…

For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} \Lambda_{\varepsilon}^{S}(T) := \sigma_S (T) \bigcup \left \{ q…

We study the set ${\cal C}$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space ${\cal H}$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted…

We consider an operator function (F(\lambda)) for (\lambda\in(\sigma,\tau)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal…

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…

Motivated by the study of layer potentials on manifolds with straight conical or cylindrical ends, we introduce and study two classes (or calculi) of pseudodifferential operators defined on manifolds with cylindrical ends: the class of…

We describe some semiclassical spectral properties of Harper-like operators, i.e. of one-dimensional quantum Hamiltonians periodic in both momentum and position. The spectral region corresponding to the separatrices of the classical…