Alberto Parmeggiani

Spectral theory of non-commutative harmonic oscillators : an introduction

The Harmonic Oscillator.- The Weyl-Hormander Calculus.- The Spectral Counting Function N(?) and the Behavior of the Eigenvalues: Part 1.- The Heat-Semigroup, Functional Calculus and Kernels.- The Spectral Counting Function N(?) and the Behavior of the Eigenvalues: Part 2.- The Spectral Zeta Function.- Some Properties of the Eigenvalues of .- Some Tools from the Semiclassical Calculus.- On Operators Induced by General Finite-Rank Orthogonal Projections.- Energy-Levels, Dynamics, and the Maslov Index.- Localization and Multiplicity of a Self-Adjoint Elliptic 2x2 Positive NCHO in .

A Class of Counterexamples to the Fefferman–Phong Inequality for Systems

Abstract We give here a class of counterexamples to the Fefferman–Phong inequality for systems of pseudodifferential operators, which contains Brummelhuis’ one as a particular case. The main ingredient in the proof is the use of “localized operators” associated with the system, and Hörmanders example of a positive-semidefinite matrix whose Weyl quantization is not nonnegative. For the considered class, in the “isotropic” case, the Sharp Gårding inequality cannot be improved.

A Generalization of Hörmander's Inequality-I

We give sufficient conditions to generalize Hörmanders inequality to the case of operators with multiple characteristics of order higher than two

Lower Bounds for Pseudodifferential Operators

We start off by fixing some notation (see Sjostrand [6]). Let X be an open subset of R n (more generally, X can be a C ∞ n-dimensional manifold without boundary) and let ∑ ⊂ T * (X\0 ≃. X × (R n \{0}) be a C∞ conic sub-manifold. With µ∈ R and h ∈ Z + = {0, 1, 2,…}, we denote by N µ,h (X, ∑) the set of all classical symbols of order µ, p(x,ξ) ∼ ∑ j ≥0 p µ-j (x, ξ), such that for any j ≥ 0 one has

On the Cauchy Problem for Hyperbolic Operators with Double Characteristics

\left| {{{p}_{{\mu - j}}}\left( {x,\xi } \right)} \right|{\underset{\raise0.3em\hbox{

Lower Bound Estimates Without Transversal Ellipticity

\smash{\scriptscriptstyle\thicksim}

A Note on Kohn’s and Christ’s Examples

}}{ < }}{{\left| \xi \right|}^{{\mu - j}}}dis{{t}_{\Sigma }}{{\left( {x,\xi } \right)}^{{{{{\left( {h - 2j} \right)}}_{ + }}}}}

A necessary and sufficient condition for a lower bound for fourth-order pseudodifferential operators

where t + =max{t, 0} and dist∑(x,ξ) denotes the distance of x,ξ/∣ξ∣) to{ ( y,η) ∈ ∑;∣η∣ =1}. OPNμ,h (X, ∑) will then denote the corresponding