Abstracts of Talks
The Uncertainty Principle of Fefferman-Phong for Fractional and Logarithmic Order
Yoshinori Morimoto (Kyoto University, Japan)
In the first half of 1980's, the uncertainty principle was discussed by Fefferman-Phong, as the positivity of Schrodinger operators /- \Delta_x+V(x)/. In this talk we extend this principle to the one with the Laplacian /- \Delta_x/ replaced by pseudo-differential operators of fractional and logarithmic order, to apply the study of the smoothness of solutions to the Cauchy problem for some linearized model of spatially inhomogeneous Boltzmann equation of the collision term without angular cut-off.
(The contents of the talk are based on the joint work with R. Alexandre, S. Ukai, C.J.Xu and Tong Yang)
Random Perturbations of Jordan Matrices and the Relevance of Directed Graphs
Brian Davies (Kings College, London)
We determine the spectral asymptotics of a small random perturbation of a standard Jordan block as the size of the block increases. We also study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.
Local Smoothing in the Presence of Lots of Trapping
Maciej Zworski (Berkeley)
The Schrödinger propagator, exp (-it Delta), on Rn is unitary on any Sobolev space so regularity is not improved in progagation. Remarkably, and as has been known for 20 years, the regularity improves when we integrate in time and cut-off in space: more...
The Fefferman-Phong Inequality for Systems
Alberto Parmeggiani (University of Bologna, Italy)
I will review the positive and negative results that I know about the Fefferman-Phong inequality for systems of PDEs.
On the Pseudospectrum of Certain Non-Selfajoint Pseudodifferential Operators
Karel Pravda-Starov (UCLA)
We study the pseudospectral properties of two classes of non-selfadjoint pseudodifferential operators. We first consider the class of differential operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols and explain which are the microlocal properties ruling the spectral stability or instability phenomena appearing under small perturbations for these operators. We then consider a "true" class of pseudodifferential operators and study the phenomena which occur when the Hessian of the principal symbol, in a critical point, defines a non-normal elliptic quadratic differential operator. This second study explains what remains from the spectral instability phenomena appearing for non-normal elliptic quadratic differential operators, when we consider pseudodifferential operators which can be locally "approximated" by such operators.
Approximation and Recovery of Damped Wave Operators
Mark Embree (Rice University)
Damped, vibrating strings provide one of the simplest settings in which to compare results from the spectral theory of non-self-adjoint differential operators to laboratory measurements. In this talk, we shall survey the spectral properties related to a variety of damping mechanisms, along with some allied questions of optimal design (minimizing spectral abscissa, minimizing energy). We shall describe a method for approximating a spatially-dependent viscous damping function from spectral measurements, and then demonstrate the challenges of applying such inverse spectral theory to high-precision data measured on our laboratory's monochord.
This talk describes joint work with Steven J. Cox and Jeffrey Hokanson.
Tunneling and Return to Equilibrium for Kramers-Fokker-Planck Type Operators
Michael Hitrik (UCLA)
For a class of second order supersymmetric differential operators, including the Kramers-Fokker-Planck operator, we determine the semiclassical (here the low temperature) asymptotics for the splitting between the first two eigenvalues, with the first one being 0. Specifically, we consider the case when the exponent of the associated Maxwellian has precisely two local minima. The splitting is then exponentially small and is related to a tunnel effect between the minima. We also show that the rate of return to equilibrium for the associated heat semigroup is dictated by the first non-vanishing eigenvalue. This is joint work with Frédéric Hérau and Johannes Sjöstrand.
The Pseudospectra of Systems
Nils Dencker (Lund University, Sweden)
The pseudospectra (spectral instability) of non-selfadjoint operators is a topic of current interest in applied mathematics. In fact, for non-selfadjoint operators the resolvent could be very large away from the spectrum, making numerical computation of the eigenvalues impossible. This affects the stability of solutions to nonlinear equations and the transition to turbulence for flows.
The occurence of pseudospectra for semiclassical operators is due to the existence of quasimodes, i.e., approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear since the Nirenberg-Treves condition ($\Psi$) is not satisfied for topological reasons, see the paper of Dencker, Sjöstrand and Zworski in CPAM 57:3 (2004).
In this talk we shall explain how these result can be generalized to systems of semiclassical operators, for which new phenomena appear.
Local Solvability Beyond Condition \psi
Ferruccio Colombini (University of Pisa, Italy)
In this lecture we will study the problem of local solvability in some cases when it turns out that condition \psi is not relevant any more: namely, when the regularity of coefficients is low, or when the operator is not of principal type. The results on low regularity of coefficients can be found in a joint work with S.Spagnolo (1989), while those on operators that are not of principal type are contained in a joint work with L.Pernazza and F.Treves (2003) and in a paper in preparation written in collaboration with P.Cordaro and L.Pernazza.
Asypmtotique Spectrale Presque Sûre D'un Opérateur Différentiel Non-Autoadjoint
William Bordeaux-Montrieux (École Polytechnique, France)
Nous considérons un opérateur différentiel simple non-autoadjoint et une perturbation aléatoire. Nous montrons que les grandes valeurs propres se distribuent presque sûrement selon une loi de Weyl dans le pseudospectre.
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