The AHP Prize 2015 was awarded to Ira Herbst and Juliane Rama for the paper

The article deals with the following physically important stability question: Suppose the atomic or molecular Schroedinger operator has a resonance (defined in the usual way as a pole of the analytic continuation of resolvent matrix elements). Does a perturbed Schroedinger operator with the addition of a small constant electric field have a nearby resonance?While the analysis presented in the award-winning paper is restricted to toy models of Schroedinger operators (Friedrichs Hamiltonians and rank-one perturbations of free 1D Hamiltonians), the conclusions of these investigations are somewhat astonishing and cast some doubts on the interpretation of resonance poles and their relations with dynamically metastable states. In the case of perturbations with a DC electric field, the authors prove that resonances of the perturbed operator actually approach the real axis when the perturbation is turned off, showing that any resonance of the original operator is in fact unstable under such perturbations.On the other hand, the resonances appear to be stable under an AC electric field perturbation. Some interesting numerical calculations pertaining to the Friedrichs model are also presented.

2014The AHP Prize 2014 was awarded to David Damanik, Jake Fillman and Anton Gorodetski for the paper

The article describes a large class of potentials for which the one-dimensional Schrödinger operator on the continuum has a Cantor spectrum with zero Lebesgue measure: these are the first examples known so far in the literature. In addition, there is a subclass of such potentials for which the authors obtain a formula for the local Hausdorff dimension of the spectrum. They prove that the latter converges to 1 in the limit of small coupling or in the high energy limit, and to zero in the infinite coupling limit. They also show the existence, in some cases, of ''pseudo-bands'' where the local Hausdorff dimension is equal to 1. The method of study uses a suspension construction permitting to exploit the enormous quantity of results obtained for the discrete version of the Schrödinger operator in one-dimension since the early eighties. The paper is masterly written and is easy to read even for non experts.

2013The AHP Prize 2013 was awarded to Dean Baskin for the paper

This paper proves local-in-time Strichartz estimates for Klein-Gordon PDEs on a wide class of spacetimes which are asymptotically De Sitter. Strichartz estimates relate norms of the solution fields and their time derivatives over bounded time ranges to (spatial) Sobolev norms of the initial data, or, alternatively, to norms of source fields for inhomogeneous versions of the Klein-Gordon PDEs. Such estimates provide crucial information regarding the dispersive character of solutions of the Klein-Gordon equation on these spacetimes. As a very informative application, the author uses the obtained bounds to prove a global existence theorem for small data for semi-linear Klein-Gordon-type equations on asymptotically De Sitter spacetimes. The small data global existence application shows that despite the local-in-time character of the estimates proven, and despite their loss of derivatives, this sort of result could be very useful in future analyses of long-time behavior of solutions in general relativity, a notoriously difficult problem.

2012The AHP Prize 2012 was awarded to Semyon Dyatlov for the paper

Relatively isolated physical black holes in the universe are expected to settle, locally, into spacetimes modeled by the Kerr or Kerr-de Sitter solutions of the Einstein equations. The analysis of distant observations of such black holes is likely to involve the resonant modes of these solutions; hence the mathematical study of these modes is physically important. One of the most direct approaches to the study of the resonant modes of a given spacetime is via the analysis of its quasi-normal modes (QNMs). The paper uses techniques from microlocal analysis to study the QNMs of the Kerr-de Sitter solutions. The author obtains strong results characterizing the asymptotics of the QNMs, and also proves that the quasi-normal modes collectively serve as an appropriate and useful basis in terms of which solutions of the wave equation on Kerr-de Sitter spacetime backgrounds can be expanded.

2011The AHP Prize 2011 was awarded to László Erdős and Antti Knowles for the paper

The paper studies the quantum evolution for random Hamiltonians given by band matrices H_xy with a quite general type of even distribution of independent matrix elements. It established the diffusive behavior of the evolution for times smaller than O(W^d/3), where W is the band width and d the space dimension. The proof is based on the expansion in "nonbacktracking powers" of the Hamiltonian given by Chebyshev polynomials of H with corrections, due to the fluctuations of the absolute values of the matrix elements, that the authors control through a sophisticated graphical classification scheme. The result, providing also a lower bound on the localization length of the eigenfunctions and an upper bound on the largest eigenvalue of the Hamiltonian, in an important contribution to the rigorous (de)localization theory for random Schrödinger operators.

2010The AHP Prize 2010 was awarded to J.-M. Barbaroux, T. Chen, V. Vougalter and S. Vugalter for the paper

The paper is devoted to the nonrelativistic QED Hamiltonian (sometimes called the Pauli-Fierz model) that attempts to capture Quantum Electrodynamics in the low energy regime. Charged particles, treated nonrelativistically, are minimally coupled to relativistic quantized photons. In this way one obtains a well-defined self-adjoint Hamiltonian depending on the fine structure constant α.A formal (due to the presence of continuous spectrum) perturbative treatment of such a model leads to expressions for the ground state in terms of powers of α with logarithmic corrections. It was an observation of Hainzl and Hainzl and Seiringer that the results from the perturbation theory can be turned into rigorous upper and lower bounds (by adjusting constants). There has been a string of papers using this method. The prize-winning article is its high point, determining an exact expression for the hydrogen atom binding energy up to order α⁵ log α⁻¹, with rigorous bounds for the o(α⁵ log α⁻¹) reminder. This required fifty pages of hard estimates and a number of ingenious innovations.

2009The AHP Prize 2009 was awarded to D. Dolgopyat and B. Fayad for the paper

This paper concerns outer billiards, a dynamical system similar to the conventional (inner) billiards. Their study was put forward by J. Moser in the 1970s and provides an interesting example of an area preserving two- dimensional mapping with an explicit geometrical description. In particular, Moser posed the problem whether the orbits of the outer billiards can escape to infinity. The motivation for this question was that if the boundary of the outer billiard table is strictly convex and sufficiently smooth, then KAM-type arguments prove that all orbits stay bounded.Surprisingly enough, the methods of this paper are essentially of the KAM type, although the authors prove an “anti-KAM” kind of result; these methods are likely to be applicable to similar problems, and their theorem opens the door for further study. To summarize, this paper solves an old problem in an unexpected way, and its method can certainly be applicable to a bunch of new models, too.

2008The AHP Prize 2008 was awarded jointly toP. Bálint and I. P. Tóth for the paper

Billiards with some hyperbolicity have played a key role in the development of dynamical systems, since they represent a highly nontrivial natural example of chaotic dynamics. The nominated paper is very well-written and accessible to non billiard experts. It settles a long standing conjecture (modulo an additional assumption which is most likely generic) and clarifies our understanding of ergodicity and mixing properties of billiards.and toL. Parnovski for the paper

The Bethe-Sommerfeld conjecture concerns a basic property of an operator with wide application in physics and was considered a challenging problem in spectral theory in the last decades. In the case of rational lattices and in all dimensions the proof has achieved by Skriganov (1984) and Scrikanov & Sobolev (2006). The definitive result has been obtained by Parnowski in this paper, which proves the conjecture for any periodicity lattice, in all dimensions greater than two and with an arbitrary smooth potential.

2007The AHP Prize 2007 is attributed to Fabien Vignes-Tourneret for the paper "Renormalization of the Orientable Non-commutative Gross–Neveu Model"This paper introduces new methods in noncommutative field theory and solves several non-trivial and difficult mathematical issues. It opens a new category of quantum field theories to renormalization, namely non-commutative Fermionic quantum field theories. This is an important step for physics as well as for mathematics, as the condensed matter version of such theories, although still to be developed, should be the relevant framework for a future deeper understanding of the physics of the quantum Hall effect.

2006For 2006, the AHP Prize laureates are Giuseppe Benfatto, Alessandro Giuliani and Vieri Mastropietro for their paper entitled "Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures” in which they prove that the weak coupling 2D Hubbard model away from half filling is a Landau Fermi liquid up to exponentially small temperatures.On the picture: Vincent Rivasseau, Chief Editor of the Journal Annales Henri Poincaré, poses with the winners of the AHP Prize during the award ceremony at the Annual Meeting of the Swiss Physical Society (March 27 2008, Genève). From left to right: Marc Herbstritt (Birkhäuser Verlag), Vieri Mastropietro (University Roma "Tor Vergata", Italy, AHP Prize Winner 2006), Alessandro Giuliani (University Roma Tre, Italy, AHP Prize Winner 2006), Giuseppe Benfatto (University Roma "Tor Vergata", Italy, AHP Prize Winner 2006), Alexander Sobolev (University College London, U.K., AHP Prize Winner 2005) and Vincent Rivasseau (Chief Editor Annales Henri Poincaré).

2005For 2005, Alexander V. Sobolev receives the AHP Prize for his paper entitled "Integrated Density of States for the Periodic Schrödinger Operator in Dimension Two" in which he provides a rigorous and insightful investigation on the high energy asymptotics of the density of states for the Schrödinger operator L2.

2004 AHP Prize for Nandor Simanyi.

2003 AHP Prize for Alessandro Pizzo.

2002 AHP Prize for Lorenzo Bertini, Stella Brassesco, Paolo Buttà, and Errico Presutti.

2001 AHP Prize for Galina Perelman.

In 2000, two AHP Prizes were exceptionally awarded to Michael T. Anderson and to Gueorgui Popov.