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Hart F. Smith

Publications and source records attributed to Hart F. Smith.

14 recordsLinked to original sources
arXiv2026

Lindblad evolution with subelliptic diffusion

We consider classical/quantum correspondence in Lindblad evolution with jump operators for which the corresponding Fokker--Planck equation is subelliptic. This allows us to consider the physical model proposed by Zurek and Paz, and to extend some of the recent mathematical results of Hernandez, Ranard and Riedel, Galkowski and Zworski, and Li, where the diffusion term in the Fokker-Planck equation was assumed elliptic. We consider the case where the jump operators $\ell_j$ in the Lindbladian are linear functions of $x$, and place an assumption which implies that the H\"ormander condition holds for the resulting Fokker-Planck equation. By constructing a suitable parametrix for this equation we show that the semiclassical derivative estimates established for elliptic diffusion also hold in the subelliptic case, with global bounds in $L^p$ for all $1\le p\le \infty$.

math.AP↗
arXiv2021

On the trace of the wave group and regularity of potentials

For the wave equation $\partial_t^2-Δ+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $t\rightarrow 0$ for the relative trace of the wave group.

math.AP↗
arXiv2018

On the trace of Schrödinger heat kernels and regularity of potentials

For the Schrödinger operator $-Δ_\rm{g}+V$ on a complete Riemannian manifold with real valued potential $V$ of compact support, we establish a sharp equivalence between Sobolev regularity of $V$ and the existence of finite-order asymptotic expansions as $t\rightarrow 0$ of the relative trace of the Schrödinger heat kernel. As an application, we generalize a result of Sà Barreto and Zworski, concerning the existence of resonances on compact metric perturbations of Euclidean space, to the case of bounded measurable potentials.

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arXiv2018

Dispersive estimates for the wave equation on Riemannian manifolds of bounded curvature

We establish space-time dispersive estimates for solutions to the wave equation on compact Riemannian manifolds with bounded sectional curvature, with the same exponents as for $C^\infty$ metrics. The estimates are for bounded time intervals, so by finite propagation velocity the results apply also on non-compact manifolds under appropriate uniform conditions. We assume a priori that in local coordinates the metric tensor components satisfy ${\rm g}_{ij}\in W^{1,p}$ for some $p>d$, which ensures that the curvature tensor is well defined in the weak sense, but this can be relaxed to any assumption that suffices for the local harmonic coordinate calculations in the paper.

math.AP↗
arXiv2014

Heat traces and existence of scattering resonances for bounded potentials

We show that, in odd dimensions, any real valued, bounded potential of compact support has at least one scattering resonance. For dimensions three and higher this was previously known only for sufficiently smooth potentials. The proof is based on an inverse result, which states that the trace of the associated heat kernel has an appropriate asymptotic expansion if and only if the potential is smooth.

math.AP↗
arXiv2012

Pointwise bounds on quasimodes of semiclassical Schrodinger operators in dimension two

We prove optimal pointwise bounds on quasimodes of semiclassical Schrodinger operators with arbitrary smooth real potentials in dimension two. This end-point estimate was left open in the general study of semiclassical Lp bounds conducted by Koch-Tataru-Zworski. However, we show that their results imply the two dimensional end-point estimate by scaling and localization.

math.AP↗
arXiv2011

Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds $(Ω,\g)$ with boundary, for both the compact case and the case that $Ω$ is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents $(p,q)$ for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $L^4_tL^\infty_x$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary.

math.AP↗
arXiv2011

Strichartz estimates for Dirichlet-wave equations in two dimensions with applications

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors \cite{SmSo00}, Burq \cite{B} and Metcalfe \cite{M} showing that local Strichartz estimates for obstactles imply global ones require that the Sobolev index, $γ$, equal 1/2 when $n=2$. We overcome this difficulty by interpolating between energy estimates ($γ=0$) and ones for $γ=\frac12$ that are generalizations of Minkowski space estimates of Fang and the third author \cite{FaWa2}, \cite{FaWa}, the second author \cite{So08} and Sterbenz \cite{St05}.

math.AP↗
arXiv2009

On Abstract Strichartz Estimates and the Strauss Conjecture for Nontrapping Obstacles

The purpose of this paper is to show how local energy decay estimates for certain linear wave equations involving compact perturbations of the standard Laplacian lead to optimal global existence theorems for the corresponding small amplitude nonlinear wave equations with power nonlinearities. To achieve this goal, at least for spatial dimensions $n=3$ and 4, we shall show how the aforementioned linear decay estimates can be combined with "abstract Strichartz" estimates for the free wave equation to prove corresponding estimates for the perturbed wave equation when $n\ge3$. As we shall see, we are only partially successful in the latter endeavor when the dimension is equal to two, and therefore, at present, our applications to nonlinear wave equations in this case are limited.

math.AP↗
arXiv2009

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.

math.AP↗
arXiv2007

Subcritical Lp bounds on spectral clusters for Lipschitz metrics

We establish asymptotic bounds on the L^p norms of spectrally localized functions in the case of two-dimensional Dirichlet forms with coefficients of Lipschitz regularity. These bounds are new for the range p>6. A key step in the proof is bounding the rate at which energy spreads for solutions to hyperbolic equations with Lipschitz coefficients.

math.AP↗
arXiv2006

On the $L^p$ norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain $L^8$ norm bounds for spectral clusters associated to elliptic second order operators on two-dimensional manifolds with boundary. The result leads to optimal $L^q$ bounds, in the range $2\le q\le\infty$, for $L^2$-normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp $L^q$ estimates in higher dimensions for a range of exponents $\bar{q}_n\le q\le \infty$.

math.AP↗