William R. Green

activity

20122025

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

50 citations · 466 across the 27 of their papers we have counts for

collaborators

30 papers

math.AP2025

The $L^{p}$ -continuity of wave operators for fractional order Schrödinger operators

M. Burak Erdogan, Michael Goldberg, William Green

We consider fractional Schrödinger operators $H = (- Δ)^{α} + V (x)$ in $n$ dimensions with real-valued potential $V$ when $n > 2 α$ , $α > 1$ . We show that the wave operators extend to bounded…

math.AP2025

Dispersive estimates for fractional order Schrödinger operators

M. Burak Erdogan, Michael Goldberg, William Green

We prove dispersive bounds for fractional Schrödinger operators on $R^{n}$ of the form $H = (- Δ)^{α} + V$ with $V$ a real-valued, decaying potential and $α \in / N$ . We deri…

math.AP2024★ 3 cited

$L^{p}$ -continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

M. Burak Erdogan, William R. Green, Kevin LaMaster

We consider the higher order Schrödinger operator $H = (- Δ)^{m} + V (x)$ in $n$ dimensions with real-valued potential $V$ when $n > 4 m$ , $m \in N$ . We adapt our recent results for $m…

math.AP2023★ 2 cited

Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials

M. Burak Erdogan, Michael Goldberg, William R. Green

We prove a family of dispersive estimates for the higher order Schrödinger equation $i u_{t} = (- Δ)^{m} u + V u$ for $m \in N$ with $m > 1$ and $2 m < n < 4 m$ . Here $V$ is a real-valued pote…

math.AP2022★ 12 cited

A note on endpoint $L^{p}$ -continuity of wave operators for classical and higher order Schrödinger operators

M. Burak Erdogan, William R. Green

We consider the higher order Schrödinger operator $H = (- Δ)^{m} + V (x)$ in $n$ dimensions with real-valued potential $V$ when $n > 2 m$ , $m \in N$ . We adapt our recent results for $m…

math.AP2022★ 11 cited

Counterexamples to $L^{p}$ boundedness of wave operators for classical and higher order Schrödinger operators

M. Burak Erdogan, Michael Goldberg, William R. Green

We consider the higher order Schrödinger operator $H = (- Δ)^{m} + V (x)$ in $n$ dimensions with real-valued potential $V$ when $n > 4 m - 1$ , $m \in N$ . We show that for any $\frac{2n}{…

math.AP2021★ 19 cited

The $L^{p}$ -continuity of wave operators for higher order Schrödinger operators

M. Burak Erdogan, William Green

We consider the higher order Schrödinger operator $H = (- Δ)^{m} + V (x)$ in $n$ dimensions with real-valued potential $V$ when $n > 2 m$ , $m \in N$ , $m > 1$ . When $n$ is odd, we prove t…

math.AP2020★ 12 cited

On the one dimensional Dirac equation with potential

Burak Erdogan, William R. Green

We investigate $L^{1} \to L^{\infty}$ dispersive estimates for the one dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural…

math.AP2020★ 15 cited

On the $L^{p}$ boundedness of the Wave Operators for fourth order Schrödinger operators

Michael Goldberg, William R. Green

We consider the fourth order Schrödinger operator $H = Δ^{2} + V (x)$ in three dimensions with real-valued potential $V$ . Let $H_{0} = Δ^{2}$ , if $V$ decays sufficiently and there are no eigenv…

math.AP2020★ 1 cited

Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions

Michael Goldberg, William R. Green

We consider the fourth order Schrödinger operator $H = Δ^{2} + V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution o…

math.AP2019

Strichartz Estimates for the Schrödinger Equation with a Measure-Valued Potential

M. Burak Erdogan, Michael Goldberg, William R. Green

We prove Strichartz estimates for the Schrödinger equation in $R^{n}$ , $n \geq 3$ , with a Hamiltonian $H = - Δ + μ$ . The perturbation $μ$ is a compactly supported measure in $\m…

math.AP2019★ 29 cited

On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances

Burak Erdogan, William R. Green, Ebru Toprak

We study the fourth order Schrödinger operator $H = (- Δ)^{2} + V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and st…

math.AP2018★ 26 cited

On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances

William R. Green, Ebru Toprak

We study the fourth order Schrödinger operator $H = (- Δ)^{2} + V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{- 1}$ decay rate holds in the $L^1\t…

math.AP2018★ 2 cited

The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates

Burak Erdogan, Michael Goldberg, William R. Green

We investigate $L^{1} \to L^{\infty}$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies th…

math.AP2017★ 13 cited

Dispersive estimates for massive Dirac operators in dimension two

M. Burak Erdoğan, William R. Green, Ebru Toprak

We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{- 1}$ decay rate holds in the $L^{1} \to L^{\infty}$ setting if the th…

math.AP2017★ 17 cited

Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions

Burak Erdogan, Michael Goldberg, William R. Green

In this paper we consider Dirac operators in $R^{n}$ , $n \geq 2$ , with a potential $V$ . Under mild decay and continuity assumptions on $V$ and some spectral assumptions on the…

math.AP2017★ 17 cited

On the $L^{p}$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Burak Erdogan, Michael Goldberg, William R. Green

Let $H = - Δ + V$ be a Schrödinger operator on $L^{2} (R^{2})$ with real-valued potential $V$ , and let $H_{0} = - Δ$ . If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-…

math.AP2016★ 21 cited

Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies

Burak Erdogan, William R. Green, Ebru Toprak

We investigate $L^{1} \to L^{\infty}$ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of…

math.AP2016★ 9 cited

On the $L^{p}$ boundedness of wave operators for four-dimensional Schrödinger Operators with a threshold eigenvalue

Michael Goldberg, William R. Green

Let $H = - Δ + V$ be a Schrödinger operator on $L^{2} (R^{4})$ with real-valued potential $V$ , and let $H_{0} = - Δ$ . If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-…

math.AP2016★ 17 cited

The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions

M. Burak Erdogan, William R. Green

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $t^{- 1}$ decay rate as an o…

math.AP2015

Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

William R. Green, Ebru Toprak

We investigate dispersive estimates for the Schrödinger operator $H = - Δ + V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four…

math.AP2015★ 31 cited

The $L^{p}$ boundedness of wave operators for Schrödinger Operators with threshold singularities

Michael Goldberg, William R. Green

Let $H = - Δ + V$ be a Schrödinger operator on $L^{2} (R^{n})$ with real-valued potential $V$ for $n > 4$ and let $H_{0} = - Δ$ . If $V$ decays sufficiently, the wave operators $W_{\pm}=s-…

math.AP2014★ 37 cited

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

Michael Goldberg, William R. Green

We investigate $L^{1} (R^{n}) \to L^{\infty} (R^{n})$ dispersive estimates for the Schrödinger operator $H = - Δ + V$ when there is an eigenvalue at zero energy in even dimensions…

math.AP2014★ 50 cited

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

Michael Goldberg, William R. Green

We investigate $L^{1} (R^{n}) \to L^{\infty} (R^{n})$ dispersive estimates for the Schrödinger operator $H = - Δ + V$ when there is an eigenvalue at zero energy and $n \geq 5$ is od…

The Lp-continuity of wave operators for fractional order Schrödinger operators

Dispersive estimates for fractional order Schrödinger operators

Lp-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials

A note on endpoint Lp-continuity of wave operators for classical and higher order Schrödinger operators

Counterexamples to Lp boundedness of wave operators for classical and higher order Schrödinger operators

The Lp-continuity of wave operators for higher order Schrödinger operators

On the one dimensional Dirac equation with potential

On the Lp boundedness of the Wave Operators for fourth order Schrödinger operators

Time integrable weighted dispersive estimates for the fourth order Schrödinger equation in three dimensions

Strichartz Estimates for the Schrödinger Equation with a Measure-Valued Potential

On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances

On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances

The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates

Dispersive estimates for massive Dirac operators in dimension two

Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions

On the Lp boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies

On the Lp boundedness of wave operators for four-dimensional Schrödinger Operators with a threshold eigenvalue

The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions

Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

The Lp boundedness of wave operators for Schrödinger Operators with threshold singularities

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

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

The $L^{p}$ -continuity of wave operators for fractional order Schrödinger operators

$L^{p}$ -continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

A note on endpoint $L^{p}$ -continuity of wave operators for classical and higher order Schrödinger operators

Counterexamples to $L^{p}$ boundedness of wave operators for classical and higher order Schrödinger operators

The $L^{p}$ -continuity of wave operators for higher order Schrödinger operators

On the $L^{p}$ boundedness of the Wave Operators for fourth order Schrödinger operators

On the $L^{p}$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions

On the $L^{p}$ boundedness of wave operators for four-dimensional Schrödinger Operators with a threshold eigenvalue

The $L^{p}$ boundedness of wave operators for Schrödinger Operators with threshold singularities