相关论文: The Weyl symbol of Schr\"odinger semigroups

In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl…

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl $h-$pseudodifferential operators, with constants independent of…

The construction, in [AJN], of a pseudodifferential calculus analogous to the Weyl calculus, in an infinite dimensional setting, required the introduction of convenient classes of symbols. In this article, we proceed with the study of these…

We establish a criterion for the validity of the classical (non-semiclassical) Weyl law for Schr\"odinger operators $ H=\Delta+V $ on complete Riemannian manifolds. In contrast to existing results, our approach does not rely on standard…

We consider the long time semiclassical evolution for the linear Schr\"odinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to $\hbar^{-2+\epsilon},\ \epsilon>0$, the symbol of a propagated…

The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

We study Schr\"odinger operators on $\mathbb{R}^2$ $$ H = \left(-\frac{\partial^2}{\partial x_1^2}\right)^{\alpha/2} + \left(-\frac{\partial^2}{\partial x_2^2}\right)^{\alpha/2} + V, $$ for $\alpha \in (0,2)$ and some sufficiently regular,…

In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in $\mathbb R^n$ under the influence of a variable magnetic field $B$. It incorporates phase factors defined…

We derive a representation formula for the Weyl solution to the Schr\"odinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with…

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…

We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we…

Let $ H:=-\tfrac12\Delta+V$ be a one-dimensional continuum Schr\"odinger operator. Consider ${\hat H}:= H+\xi$, where $\xi$ is a translation invariant Gaussian noise. Under some assumptions on $\xi$, we prove that if $V$ is locally…

We analyze semi-classical Schr\"odinger operators with potentials of class $C^{1,1/2}$ and establish commutator estimates for the associated projection operators in Schatten norms. These are then applied to prove quantitative versions of…

In this work, we use semigroup integral to evaluate zeta-function regularized determinants. This is especially powerful for non--positive operators such as the Dirac operator. In order to understand fully the quantum effective action one…

We continue the study of the perturbation problem discussed in \cite{CP3} and get rid of the 'slow variation' assumption by considering symbols of the form $a\big(x+\delta\,F(x),\xi\big)$ with $a$ a real H\"{o}rmander symbol of class…

An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary…

We investigate the group $\mathcal{H}_\mathbb{C}$ of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space $H$. Irreducible representations of the Weyl--Schr{\"o}dinger type on the space $L^2_\chi$…

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion…

The Weyl modules are the standard modules for the Schur algebra. Their duals (the costandard modules) have well-known constructions as quotients of exterior powers and as submodules of symmetric powers. This paper presents analogous…