Rose–Hulman Institute of Technology

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20002026

most citedChimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

801 citations

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Counting papers with at most 25 authors, so collaboration author lists do not stand in for the people here.

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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.AP2024★ 1 cited

Inverse Iteration for the Laplace Eigenvalue Problem with Robin and Mixed Boundary Conditions

Benjamin Lyons, Emily Ruttenberg, Nicholas Zitzelberger

We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove conv…

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.AP2024★ 1 cited

The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions

William R. Green, Connor Lane, Benjamin Lyons +2

We investigate dispersive estimates for the massless three dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $\langle t\rangl…

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★ 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.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★ 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★ 11 cited

Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations

Kyle M. Claassen, Mathew A. Johnson

We consider the existence and stability of real-valued, spatially antiperiodic standing wave solutions to a family of nonlinear Schrödinger equations with fractional dispersion and…

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★ 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…

math.AP2013★ 45 cited

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

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

We investigate $L^{1} (R^{4}) \to L^{\infty} (R^{4})$ dispersive estimates for the Schrödinger operator $H = - Δ + V$ when there are obstructions, a resonance or an eigenvalue, at…

math.AP2013★ 12 cited

Time decay estimates for the wave equation with potential in dimension two

William R. Green

We study the wave equation with potential $u_{tt} - Δ u + V u = 0$ in two spatial dimensions, with $V$ a real-valued, decaying potential. With $H = - Δ + V$ , we study a variety of mapping estim…

math.AP2012★ 30 cited

A weighted dispersive estimate for Schrödinger operators in dimension two

M. Burak Erdoğan, William R. Green

Let $H = - Δ + V$ , where $V$ is a real valued potential on $R^{2}$ satisfying $|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H = - Δ + V$ , then $$ \…

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

Inverse Iteration for the Laplace Eigenvalue Problem with Robin and Mixed Boundary Conditions

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

The Massless Dirac Equation in Three Dimensions: Dispersive estimates and zero energy obstructions

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 Lp boundedness of the Wave Operators for fourth order Schrödinger operators

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

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

Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schrödinger Equations

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

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

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

Time decay estimates for the wave equation with potential in dimension two

A weighted dispersive estimate for Schrödinger operators in dimension two

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