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Leonard Gross

Publications and source records attributed to Leonard Gross.

8 recordsLinked to original sources
arXiv2024

Invariance of intrinsic hypercontractivity under perturbation of Schr\"odinger operators

A Schr\"odinger operator that is bounded below and has a unique positive ground state can be transformed into a Dirichlet form operator by the ground state transformation. If the resulting Dirichlet form operator is hypercontractive, Davies and Simon call the Schr\"odinger operator ``intrinsically hypercontractive". I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schr\"odinger operator it remains intrinsically hypercontractive. The proof uses a fortuitous relation between the WKB equation and logarithmic Sobolev inequalities. All bounds are dimension independent. The main theorem will be applied to several examples.

math-ph↗
arXiv2018

Equivalence of helicity and Euclidean self-duality for gauge fields

In the canonical formalism for the free electromagnetic field a solution to Maxwell's equations is customarily identified with its initial gauge potential (in Coulomb gauge) and initial electric field, which together determine a point in phase space. The solutions to Maxwell's equations, all of whose plane waves in their plane wave expansions have positive helicity, thereby determine a subspace of phase space. We will show that this subspace consists of initial gauge potentials which lie in the positive spectral subspace of the operator curl together with initial electric fields conjugate to such potentials. Similarly for negative helicity. Helicity is thereby characterized by the spectral subspaces of curl in configuration space. A gauge potential on three-space has a Poisson extension to a four dimensional Euclidean half space, defined as the solution to the Maxwell-Poisson equation whose initial data is the given gauge potential. We will show that the extension is anti-self dual if and only if the gauge potential lies in the positive spectral subspace of curl. Similarly for self dual extension and negative spectral subspace. Helicity is thereby characterized for a normalizable electromagnetic field by the canonical formalism and (anti-)self duality. For a non-abelian gauge field on Minkowski space a plane wave expansion is not gauge invariant. Nor is the notion of positive spectral subspace of curl. But if one replaces the Maxwell-Poisson equation by the Yang-Mills-Poisson equation then (anti-)self duality on the Euclidean side induces a decomposition of (now non-linear) configuration space similar to that in the electromagnetic case. The strong analogy suggests a gauge invariant definition of helicity for non-abelian gauge fields. We will provide further support for this view.

math-ph↗
arXiv2017

Stability of the Yang-Mills heat equation for finite action

The existence and uniqueness of solutions to the Yang-Mills heat equation over domains in Euclidean three space was proven in a previous paper for initial data lying in the Sobolev space of order one-half, which is the critical Sobolev index for this equation. In the present paper the stability of these solutions will be established. The variational equation, which is only weakly parabolic, and has highly singular coefficients, will be shown to have unique strong solutions up to addition of a vertical solution. Initial data will be taken to be in Sobolev class one-half. The proof relies on an infinitesimal version of the ZDS procedure: one solves first an augmented, strictly parabolic version of the variational equation and then adds to the solution a function which is vertical along the original path. Energy inequalities and Neumann domination techniques will be used to establish apriori initial behavior for solutions.

math.AP↗
arXiv2017

The Yang-Mills heat equation with finite action

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over three dimensional Euclidean space and over a bounded open convex set therein. The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation.

math.AP↗
arXiv2016

Initial behavior of solutions to the Yang-Mills heat equation

We explore the small-time behavior of solutions to the Yang-Mills heat equation with rough initial data. We consider solutions $A(t)$ with initial value $A_0\in H_{1/2}(M)$, where $M$ is a bounded convex region in $\mathbb{R}^3$ or all of $\mathbb{R}^3$. The behavior, as $t\downarrow 0$, of the $L^p(M)$ norms of the time derivatives of $A(t)$ and its curvature $B(t)$ will be determined for $p=2$ and $6$, along with the $H_1(M)$ norm of these derivatives.

math-ph↗
arXiv2014

Neumann domination for the Yang-Mills heat equation

Long time existence and uniqueness of solutions to the Yang-Mills heat equation have been proven over a compact 3-manifold with boundary for initial data of finite energy. In the present paper we improve on previous estimates by using a Neumann domination technique that allows us to get much better pointwise bounds on the magnetic field. As in the earlier work, we focus on Dirichlet, Neumann and Marini boundary conditions. In addition, we show that the Wilson Loop functions, gauge invariantly regularized, converge as the parabolic time goes to infinity.

math-ph↗
arXiv2010

The Yang-Mills heat semigroup on three-manifolds with boundary

Long time existence and uniqueness of solutions to the Yang-Mills heat equation is proven over a compact 3-manifold with smooth boundary. The initial data is taken to be a Lie algebra valued connection form in the Sobolev space $H_1$. Three kinds of boundary conditions are explored, Dirichlet type, Neumann type and Marini boundary conditions. The last is a nonlinear boundary condition, specified by setting the normal component of the curvature to zero on the boundary. The Yang-Mills heat equation is a weakly parabolic non-linear equation. We use a technique of Donaldson and Sadun to convert it to a parabolic equation and then gauge transform the solution of the parabolic equation back to a solution of the original equation. Apriori estimates are developed by first establishing a gauge invariant version of the Gaffney-Friedrichs inequality. A gauge invariant regularization procedure for solutions is also established. Uniqueness holds upon imposition of boundary conditions on only two of the three components of the connection form because of weak parabolicity. This work is motivated by possible applications to quantum field theory.

math.AP↗