Gaussian estimates and L p -boundedness of Riesz means

We show that suitable upper estimates of the heat kernel are sufficient to imply the L ^p boundedness of several families of operators associated with the Schr?dinger group in various situations. This generalizes results by Sj?strand and others in the Euclidean case, and by Alexopoulos in the case of Lie groups and Riemannian manifolds. RID="*" ID="*"Research partially supported by the European Commission (European TMR Network "Harmonic Analysis" 1998-2001, Contract ERBFMRX-CT97-0159).     

Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians

The diamagnetic inequality is established for the Schrödinger operator H^(d)₀ in L² d=2,3, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schrödinger operator H^(d)₀–V, using new Hardy type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also proved.Communicated by Bernard Helffersubmitted 02/12/03, accepted 12/03/04     

Partial Gaussian Bounds for Degenerate Differential Operators

Let S be the semigroup on \(L_2({{\bf R}}^d)\) generated by a degenerate elliptic operator, formally equal to \(- \sum \partial_k \, c_{kl} \, \partial_l\), where the coefficients c _kl are real bounded measurable and the matrix C(x)?=?(c _kl (x)) is symmetric and positive semi-definite for all x?∈?R ^d . Let Ω???R ^d be a bounded Lipschitz domain and μ?>?0. Suppose that C(x)?≥?μ I for all x?∈?Ω. We show that the operator P _Ω S _t P _Ω has a kernel satisfying Gaussian bounds and Gaussian Hölder bounds, where P _Ω is the projection of \(L_2({{\bf R}}^d)\) onto L ₂(Ω). Similar results are for the operators u ? χ S _t (χ u), where \(\chi \in C_{\rm b}^\infty({{\bf R}}^d)\) and C(x)?≥?μI for all \(x \in {\mathop{\rm supp}} \chi\).     

Comportement des noyaux de la chaleur des opérateurs de Schrödinger et applications à certaines équations paraboliques semi-linéaires

We consider general Schrödinger operators on domains of Riemannian manifolds with possibly exponential volume growth. We prove sharp large time Gaussian upper bounds. These bounds are then used to prove new Lp-Lp estimates for the corresponding semigroups. Applications to semi-linear parabolic equations are given.     

Remarks on the Cwikel–Lieb–Rozenblum and?Lieb–Thirring Estimates for Schr?dinger Operators on?Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schr?dinger operator −Δ+V on L ²(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schr?dinger operators with complex-valued potentials.     

Semi-groupes sous-Markoviens engendrés par des opérateurs matriciels

We characterize generators of sub-Markovian semigroups onL ^p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A _ij )_1i,jn on (L ^p ()) ⁿ is sub-Markovian if and only if the semigroup generated by the sum of each rowA _i 1+...+A _in (1in), is sub-Markovian. The corresponding result on (C ₀(X)) ⁿ characterizes dissipative operator matrices.

     

Sharp Gaussian bounds and -growth of semigroups associated with elliptic and Schrödinger operators

We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators , we have the -estimate (for large ):

where is the spectral bound of The same estimate holds for elliptic and Schrödinger operators on general domains.

     

Absence de la L∞-contractivité pour les semi-groupes associés aux opérateurs elliptiques complexes sous forme divergence

Soit un opérateur elliptique du second ordre à coefficients complexes sur un ouvert de ^N . On obtient des conditions nécessaires et suffisantes sur les coefficients, pour que le semi-groupe qu'il définit, suivant les conditions au bord considérées, contracte L . On montre en particulier que cette propriété est assez spécifique aux coefficients réels. Abstract. Consider a second order elliptic operator with complex coefficients on an open set of ^N . We obtain necessary and sufficient conditions on the coefficients for the contractivity in L of the semigroup defined under different boundary conditions. In particular, we show that this property is closely related to the fact that the coefficients are actually real valued.