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Michael Goldberg

Publications and source records attributed to Michael Goldberg.

35 records · Page 2Linked to original sources
arXiv2015

The Klein-Gordon Equation on ${\mathbb Z}^2$ and the Quantum Harmonic Lattice

The discrete Klein-Gordon equation on a two-dimensional square lattice satisfies an $\ell^1 \mapsto \ell^\infty$ dispersive bound with polynomial decay rate $|t|^{-3/4}$. We determine the shape of the light cone for any choice of the mass parameter and relative propagation speeds along the two coordinate axes. Fundamental solutions experience the least dispersion along four caustic lines interior to the light cone rather than along its boundary, and decay exponentially to arbitrary order outside the light cone. The overall geometry of the propagation pattern and its associated dispersive bounds are independent of the particular choice of parameters. In particular there is no bifurcation of the number or type of caustics that are present. The discrete Klein-Gordon equation is a classical analogue of the quantum harmonic lattice. In the quantum setting, commutators of time-shifted observables experience the same decay rates as the corresponding Klein-Gordon solutions, which depend in turn on the relative location of the observables' support sets.

math.AP↗
arXiv2015

Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues I: The odd dimensional case

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there is an eigenvalue at zero energy and $n\geq 5$ is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$ for $|t|>1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\qquad\textrm{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form $$ e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, $$ with $A_{-2}$ and $A_{-1}$ finite rank operators mapping $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^\infty$ spaces. The leading order terms $A_{-2}$ and $A_{-1}$ vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t|^{-\frac{n}{2}}A_0$ term also exists as a map from $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$, hence $e^{itH}P_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

math.AP↗
arXiv2015

Dispersive Estimates for higher dimensional Schrödinger Operators with threshold eigenvalues II: The even dimensional case

We investigate $L^1(\mathbb R^n)\to L^\infty(\mathbb R^n)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there is an eigenvalue at zero energy in even dimensions $n\geq 6$. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim |t|^{2-\frac{n}{2}}$ for $|t|>1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{1-\frac{n}{2}},\,\,\,\,\,\text{ for } |t|>1.$$ With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the form \begin{align*} e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}} A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*} with $A_{-2}$ and $A_{-1}$ mapping $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$ while $A_0$ maps weighted $L^1$ spaces to weighted $L^\infty$ spaces. The leading-order terms $A_{-2}$ and $A_{-1}$ are both finite rank, and vanish when certain orthogonality conditions between the potential $V$ and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining $|t|^{-\frac{n}{2}}A_0$ term also exists as a map from $L^1(\mathbb R^n)$ to $L^\infty(\mathbb R^n)$, hence $e^{itH}P_{ac}(H)$ satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

math.AP↗
arXiv2015

The Helmholtz equation with $L^p$ data and Bochner-Riesz multipliers

We prove the existence of $L^2$ solutions to the Helmholtz equation $(-Δ- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-Δ-1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schrödinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the closed subspace of functions whose Fourier transform vanishes on the unit sphere.

math.CA↗
arXiv2013

Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy

We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1/\log t$ for $t>2$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential.

math.AP↗
arXiv2012

Strichartz Estimates and Maximal Operators for the Wave Equation in R^3

We prove sharp Strichartz-type estimates in three dimensions, including some which hold in reverse spacetime norms, for the wave equation with potential. These results are also tied to maximal operator estimates studied by Rogers--Villaroya, of which we prove a sharper version. As a sample application, we use these results to prove the local well-posedness and the global well-posedness for small initial data of semilinear wave equations in R^3 with quintic or higher monomial nonlinearities.

math.AP↗
arXiv2010

Schrödinger dispersive estimates for a scaling-critical class of potentials

We prove a dispersive estimate for the evolution of Schroedinger operators H = -Δ+ V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere on the positive half-line. The proof is an application of a new version of Wiener's L^1 inversion theorem.

math.AP↗
arXiv2008

A dispersive bound for three-dimensional Schroedinger operators with zero energy eigenvalues

We prove a dispersive estimate for the evolution of Schroedinger operators $H = -Δ+ V(x)$ in ${\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\R^3)\cap L^q(\R^3)$, $p < \frac32 < q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\infty)$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable.

math.AP↗
arXiv2008

Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential

We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator $H = (i\nabla + A)^2 + V$ in $\R^n$, $n > 2$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div} A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.

math.AP↗
arXiv2007

Strichartz Estimates for the Schroedinger Equation with Time-Periodic L^{n/2} Potentials

We prove Strichartz estimates for the Schroedinger operator $H = -Δ+ V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies, provided that the associated eigenfunction decays sufficiently rapidly.

math.AP↗
arXiv2007

Strichartz and Smoothing Estimates for Schrödinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions

In this paper we consider magnetic Schrödinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an angular decomposition of the free resolvent, or rather a bound that holds uniformly for all possible angular decompositions. The proof avoids the Fourier transform and instead uses Hörmander's variable coefficient Plancherel theorem for oscillatory integrals.

math.AP↗
arXiv2004

Dispersive Bounds for the three-dimensional Schrodinger equation with almost critical potentials

We prove a dispersive estimate for the time-independent Schrodinger operator H = -Δ+ V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V(x)| < C(1+|x|)^{-2-ε}, is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.

math.AP↗