The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit.As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k.For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k D 0, which yields an asymptotic formula for the reflection coefficient at k D 0 and suggests an annular structure for the solution that may be exploited when k ¤ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as # 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state -one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. c 2000 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. MGELML/3/2/01 2 exists (as far as we believe now) is the free particle (i.e., particle plus field). In the presence of an external potential, however, like the Coulomb potential of a nucleus, a ground state should exist.The difficulty in establishing this ground state comes from the fact that the bottom of the spectrum lies in the continuum (i.e., essential spectrum), not below it, as is the case for the usual Schrödinger equation. We denote the bottom of spectrum of the free-particle Hamiltonian for N particles with appropriate statistics by E 0 (N ). The 'free-particle' Hamiltonian includes the interparticle interaction (e.g., the Coulomb repulsion of electrons) but it does not include the interaction with a fixed external potential, e.g., the interaction with nuclei. When the latter is included we denote the bottom of the spectrum by E V (N ). It is not hard to see in many cases thatis, nevertheless, the bottom of the essential spectrum. The reason is that we can always add arbitrarily many, arbitrarily 'soft' photons that add arbitrarily little energy. It is the soft photon problem that is our primary concern here. The main point of this paper is to show how to overcome this infrared problem and to show, quite generally for a one-particle system, that a ground state exists for all values of the particle mass, the coupling to the field (fine-structure constant α = e 2 / c), the magnetic g-factor, and the ultraviolet cutoff Λ of the electromagnetic field frequencies, provided a bound state exists when the field is turned off. This result implies, in particular, that for a fixed ultraviolet cutoff renormalization of the various physical quantities will not affect the existence of a ground state. Of course, nothing can be said about the limit as the cutoff tends to infinity. We also include a large class of interactions much more general than the usual Coulomb interaction.The model we discuss has been used quite frequently in field theory. In its classical version it was investigated by Kramers [12] who seems to have been the first to point out the possibility of renormalization. The quantized version was investigated by Pauli and Fierz [24] in connection with scattering theory. Most importantly, it was used by Bethe [9] to obtain a suprisingly good value for the Lamb shift.Various restricted versions of the problem have been attacked successfully. In the early seventies Fröhlich investigated the infrared problem in translation invariant models of scalar elect...

We prove a sharp analog of Young's inequality on S N , and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young's inequality on R N to more than three functions, and leads to significant new information about the optimizers and the constants. Math reviews Classification Numbers: 43A15, 52A40, 82C40

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckner's generalization of Onofri's inequality to arbitrary dimensions and determines the cases of equality.