Michael Loss

Extremals of functionals with competing symmetries

Abstract We present a new method of producing optimizing sequences for highly symmetric functionals. The sequences have good convergence properties built in. We apply the method in different settings to give elementary proofs of some classical inequalities—such as the Hardy-Littlewood-Sobolev and the logarithmic Sobolev inequality—in their sharp form.

Stability of Coulomb systems with magnetic fields. I. The one-electron atom

The ground state energy of an atom in the presence of an external magnetic filedB (with the electron spin-field interaction included) can be arbitrarity negative whenB is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, ∫B2. For a hydrogenic like atom we prove that there is a critical nuclear charge,zc, such that the atom is stable forzzc.

Stability of Coulomb Systems with Magnetic Fields

The ground state energy of an atom in the presence of an external magnetic filedB (with the electron spin-field interaction included) can be arbitrarity negative whenB is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, ∫B2. For a hydrogenic like atom we prove that there is a critical nuclear charge,zc, such that the atom is stable forzzc.

A sharp analog of Young’s inequality on SN and related entropy inequalities

We prove a sharp analog of Young’s inequality on SN, 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 RN to more than three functions, and leads to significant new information about the optimizers and the constants.

Stability of Coulomb systems with magnetic fields. III: Zero energy bound states of the Pauli operator

It is shown that there exist magnetic fields of finite self energy for which the operator σ·(p−A) has a zero energy bound state. This has the consequence that single electron atoms, as treated recently by Fröhlich, Lieb, and Loss [1], collapse when the nuclear charge numberz≧9π2/8α2 (α is the fine structure constant).

Existence of Atoms and Molecules in Non-Relativistic Quantum Electrodynamics

We show that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z, has a ground state when N < Z + 1. The result holds for any value of the fine structure constant α and for any value of the ultraviolet cutoff A on the radiation field. There is no infrared cutoff. The basic mathematical ingredient in our proof is a novel localization of the electromagnetic field in such a way that the errors in the energy are of smaller order than 1/L, where L is the localization radius.

Determination of the spectral gap for Kac's master equation and related stochastic evolution

We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

Stability of Matter in Magnetic Fields

In the presence of arbitrarily large magnetic fields, matter composed of electrons and nuclei was known to be unstable if a or Z is too large. Here we prove that matter is stable if α < 0.06 and Zα 2 < 0.04.

Fluxes, Laplacians, and Kasteleyn’s theorem

The genesis of this paper was an attempt to understand a problem in condensed matter physics related to questions about electron correlations, superconductivity, and electron-magnetic field interactions. The basic idea, which was proposed a few years ago, is that a magnetic field can lower the energy of electrons when the electron density is not small. Certain very specific and very interesting mathematical conjectures about eigenvalues of the Laplacian were made, and the present paper contains a proof of some of them. Furthermore, those conjectures lead to additional natural conjectures about determinants of Laplacians which we both present and prove here. It is not clear whether these determinantal theorems have physical applications but they might, conceivably in the context of quantum field theory. Some, but not all, of the results given here were announced earlier in [LE].

Lieb-Thirring inequalities with improved constants

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schrodinger operators