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Michael Loss

Publications and source records attributed to Michael Loss.

At least 73 records · Page 4Linked to original sources
arXiv2007

Determination of the spectral gap in the Kac model for physical momentum and energy conserving collisions

The Kac model describes the local evolution of a gas of $N$ particles with three dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve momentum as well as energy. The state space of this walk is a sphere of dimension $3N - 4$. The Kac conjecture concerns the spectral gap in the one step transition operator $Q$ for this walk. In this paper, we compute the exact spectral gap. As in previous work by Carlen, Carvalho and Loss where a lower bound on the spectral gap was proved, we use a method that relates the spectral properties of $Q$ to the spectral properties of a simpler operator $P$, which is simply an average of certain non commuting projections. The new feature is that we show how to use a knowledge of certain eigenfunctions and eigenvalues of $P$ to determine spectral properties of $Q$, instead of simply using the spectral gap for $P$ to bound the spectral gap for $Q$, inductively in $N$, as in previous work. The methods developed here can be applied to many other high--dimensional stochastic process, as we shall explain. We also use some deep results on Jacobi polynomials to obtain the required spectral information on $P$, and we show how the identity through which Jacobi polynomials enter our problem may be used to obtain new bounds on Jacobi polynomials.

math-ph↗
arXiv2006

Relativistic hydrogenic atoms in strong magnetic fields

In the Dirac operator framework we characterize and estimate the ground state energy of relativistic hydrogenic atoms in a constant magnetic field and describe the asymptotic regime corresponding to a large field strength using relativistic Landau levels. We also define and estimate a critical magnetic field beyond which stability is lost.

math.AP↗
arXiv2006

Lowest energy states in nonrelativistic QED: atoms and ions in motion

Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and $N$ electrons coupled to the radiation field. Since the total momentum $P$ is conserved, the Hamiltonian $H$ admits a fiber decomposition with respect to $P$ with fiber Hamiltonian $H(P)$. A stable atom, resp. ion, means that the fiber Hamiltonian $H(P)$ has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for $H(P)$ under (i) an explicit bound on $P$, (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons.

math-ph↗
arXiv2005

Inequalities for means of chords, with application to isoperimetric problems

We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $L^2(\mathbb{R}^2)$ with an attractive interaction supported on a closed curve $Γ$, formally given by $-Δ-αδ(x-Γ)$; we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread $Γ$ in $\mathbb{R}^3$, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems reduce to purely geometric questions about inequalities for mean values of chords of $Γ$. We prove an isoperimetric theorem for $p$-means of chords of curves when $p \leq 2$, which implies in particular that the global extrema for the physical problems are always attained when $Γ$ is a circle. The article finishes with a discussion of the $p$--means of chords when $p > 2$.

math-ph↗
arXiv2005

An Inequality of Hadamard Type for Permanents

Let F be an N x N complex matrix whose jth column is the vector f_j in C^N. Let |f_j|^2 denote the sum of the absolute squares of the entries of f_j. Hadamard's inequality for determinants states that |\det(F)| <= \prod_{j=1}^N|f_j|. Here we prove a sharp upper bound on the permanent of F, which is |perm(F)| <= N!N^{-N/2} \prod_{j=1}^N|f_j|, and we determine all of the cases of equality.

math.CA↗
arXiv2005

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

We prove a Lieb-Thirring type inequality for potentials such that the associated Schrödinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.

math-ph↗
arXiv2004

A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities

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.FA↗
arXiv2004

The thermodynamic limit for matter interacting with Coulomb forces and with the quantized electromagnetic field: I. The lower bound

The proof of the existence of the thermodynamic limit for electrons and nuclei interacting via the Coulomb potential, in the framework of non-relativistic quantum mechanics, was accomplished decades ago. This result did not take account of interactions caused by magnetic fields, however, (the spin-spin interaction, in particular) or of the quantized nature of the electromagnetic field. Recent progress has made it possible to undertake such a proof in the context of non-relativistic QED. This paper contains one part of such a proof by giving a lower bound to the free energy which is proportional to the number of particles and which takes account of the fact that the field, unlike the particles, is never confined to a finite volume. In the earlier proof the lower bound was a `two line' corollary of the `stability of matter'. In QED the proof is much more complicated.

math-ph↗
arXiv2004

Connection between the Lieb--Thirring conjecture for Schroedinger operators and an isoperimetric problem for ovals on the plane

To determine the sharp constants for the one dimensional Lieb--Thirring inequalities with exponent gamma in (1/2,3/2) is still an open problem. According to a conjecture by Lieb and Thirring the sharp constant for these exponents should be attained by potentials having only one bound state. Here we exhibit a connection between the Lieb--Thirring conjecture for gamma=1 and an isporimetric inequality for ovals in the plane.

math-ph↗
arXiv2004

A Note on Polarization Vectors in Quantum Electrodynamics

A photon of momentum k can have only two polarization states, not three. Equivalently, one can say that the magnetic vector potential A must be divergence free in the Coulomb gauge. These facts are normally taken into account in QED by introducing two polarization vectors epsilon_λ(k) with lambda in {1,2}, which are orthogonal to the wave-vector k. These vectors must be very discontinuous functions of k and, consequently, their Fourier transforms have bad decay properties. Since these vectors have no physical significance there must be a way to eliminate them and their bad decay properties from the theory. We propose such a way here.

math-ph↗
arXiv2003

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 alpha and for any value of the ultraviolet cutoff Lambda on the radiation field. There is no infrared cutoff. The basic mathematical ingredient in our proof is a novel way of localizing 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.

math-ph↗
arXiv2002

A Bound on Binding Energies and Mass Renormalization in Models of Quantum Electrodynamics

We study three well known models of matter coupled to the ultraviolet cutoff, quantized radiation field and to the Coulomb potential of arbitrarily many nuclei. Two are nonrelativistic: the first uses the kinetic energy (p+eA(x))^2 and the second uses the Pauli-Fierz energy (p+eA(x))^2 +eσ\cdot B(x). The third, no-pair model, is relativistic and replaces the kinetic energy with the Dirac operator D(A), but restricted to its positive spectral subspace, which is the ``electron subspace''. In each case we are able to give an upper bound to the {\it binding} energy -- as distinct from the less difficult ground state energy. This implies, for the first time we believe, an estimate, albeit a crude one, of the mass renormalization in these theories.

math-ph↗
arXiv2002

Stability of a Model of Relativistic Quantum Electrodynamics

The relativistic ``no pair'' model of quantum electrodynamics uses the Dirac operator, D(A), for the electron dynamics together with the usual self-energy of the quantized ultraviolet cutoff electromagnetic field A -- in the Coulomb gauge. There are no positrons because the electron wave functions are constrained to lie in the positive spectral subspace of some Dirac operator, D, but the model is defined for any number, N, of electrons, and hence describes a true many-body system. In addition to the electrons there are a number, K, of fixed nuclei with charges \leq Z. If the fields are not quantized but are classical, it was shown earlier that such a model is always unstable (the ground state energy E=-\infty) if one uses the customary D(0) to define the electron space, but is stable (E > - const.(N+K)) if one uses D(A) itself (provided the fine structure constant αand Z are not too large). This result is extended to quantized fields here, and stability is proved for α=1/137 and Z \leq 42. This formulation of QED is somewhat unusual because it means that the electron Hilbert space is inextricably linked to the photon Fock space. But such a linkage appears to better describe the real world of photons and electrons.

math-ph↗
arXiv2001

Determination of the Spectral Gap for Kac's Master Equation and Related Stochastic Evolutions

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.

math-ph↗
arXiv2001

Ground States in Non-relativistic Quantum Electrodynamics

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 {\it 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.

math-ph↗
arXiv1999

A simple proof of a theorem of Laptev and Weidl

A new and elementary proof of a recent result of Laptev and Weidl is given. It is a sharp Lieb-Thirring inequality for one dimensional Schroedinger operators with matrix valued potentials.

math-ph↗