C e n t r e  d e  r e c h e r c h e s  m a t h é m a t i q u e s

John Toth

Professor Toth received his Ph.D. in 1993 at M.I.T. under the supervision of Victor Guillemin. He was a Benjamin Pierce Assistant professor at Harvard from 1993 to 1995, when he took up an appointment at McGill University in Montréal. Since his arrival at McGill he has been a mainstay of the analysis group, and has contributed actively to their graduate programme with several quite wonderful advanced graduate courses. He was an invited member of the Fields Institute for the fall of 1997 for their year on Microlocal Analysis.

Professor Toth's work so far has centred on the theme of quantization and the asymptotic behaviour of quantum systems, in particular quantum integrable systems, as one goes to the classical limit of h --> 0. There has, of course been much work in this area, concentrating mostly on eigenvalue asymptotics and the corresponding approximate eigenfunctions (le. quasimodes). One of the original features of his approach is that he obtains sharp asymptotic bounds for the actual eigenfunctions. This allows him, for example, to prove various examples of eigenfunction accumulation ("scarring"), for which there had previously been only computational evidence, as well as to show, in a precise, quantitative way, how the eigenfunctions and the corresponding classical dynamics interrelate.

The theme of quantum integrable systems is already present in his thesis work, in which he studied the quantizations of geodesic flow on the ellipsoid and of the classical Neumann oscillator. He shows that these systems are quantum integrable and moreover, they admit a separation of variables. reducing the problem to studying solutions of a single "generalized Lamé" ordinary differential equation. A careful use of the properties of the solutions to such equations, and methods such as the micro-local wave-averaging ansatz and the WKB approximation allow him to analyse the spectra of these systems, tying them to a third system, that of the rigid body in a vacuum.

It is in the papers [4], [5] that his study of eigenfunction behaviour begins in earnest. The paper [4] shows how one can obtain a sequence of eigenfunctions which accumulate along a hyperbolic geodesic for the 3-dimensional rigid body, obtaining explicit bounds for the eigenfunctions near the geodesic. He moreover shows that these modes detect not only the geodesic, but also conjugate points. This was one of the first examples of scarring to be established on a rigorous level. In [8], using a microlocal quantum Birkhoff normal form, this scarring phenomenon was shown to hold for a large number of integrable systems in arbitrary dimension. Scarring is conjectured not to exist in the ergodic case (Sarnak has shown that this is indeed the case for some arithmetic quotients), so that in the very least, this work indicates a strong qualitative difference in the quantum setting between integrable systems and ergodic ones. One can also look for concentration phenomena in cases when the bicharacteristic flow of the classical system accumulates on a more general (singular) variety and this is the subject of [7].

In [5], the eigenfunctions of a general quantum integrable system are examined, as one tends to the classical limit, using the FBI transform. From an L2 estimate for the FBI transform of the eigenfunctions, he obtains pointwise bounds on the norms of the eigenfunctions associated with a fixed energy level set. These bounds are a considerable sharpening of the universal (Seeger-Sogge) bounds. He also obtains exponential decay estimates for the eigenfunctions of a wide variety of quantum integrable systems in arbitrary dimension. The techniques of the paper lead naturally to a microlocal study of tunnelling, and this is the topic of [11].

In another vein, he has successfully applied microlocalisation techniques to questions involving the generalised Toeplitz operators of Boutet de Monvel and Guillemin. These arise in the quantlzation of such natural varieties as coadjoint orbits. A first paper [6] studies such operators on the torus, and again obtains bounds on the eigenfunctions. In [10], these are extended to arbitrary compact analytic manifolds.

Professor Toth is one of the leading young microlocal analysts in the world. His work combines a beautiful geometric insight into the nature of the eigenfunctions he is studying and detailed knowledge of the special functions involved in particular cases, both of these complementing a powerful analytic technique. His work builds on that of some of the leaders in our discipline, and improves it considerably in the particular cases, mostly integrable systems, which he considers. It is a great pleasure to have him as a colleague and to watch his rapidly developing understanding of these quite intricate questions.

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