arXiv subjects

William R. Green

Publications and source records attributed to William R. Green.

30 records · Page 2Linked to original sources

arXiv2016

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

Let $H=-Δ+V$ be a Schrödinger operator on $L^2(\mathbb R^4)$ with real-valued potential $V$, and let $H_0=-Δ$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^4)$ for all $1\leq p\leq \infty$ if zero is not an eigenvalue or resonance, and on $\frac43<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p(\mathbb R^4)$ for $1\leq p\leq \frac43$ by direct examination of the integral kernel of the leading terms. Furthermore, if $\int_{\mathbb R^4} xV(x) ψ(x) \, dx=0$ for all zero energy eigenfunctions $ψ$, then the wave operators are bounded on $L^p$ for $1 \leq p<\infty$.

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arXiv2016

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

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 operator from the Hardy space $H^1$ to $BMO$, the space of functions of bounded mean oscillation. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold s-wave resonance does not destroy the $t^{-1}$ decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap.

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arXiv2016

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

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 spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion $$ e^{itH}χ(H) P_{ac}(H)=O(1/(\log t)) A_0+ O(1/t)A_1+O((t\log t)^{-1})A_2+ O(t^{-1}(\log t)^{-2})A_3. $$ Here $A_0,A_1:L^1(\mathbb R^n)\to L^\infty (\mathbb R^n)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

math.AP↗

arXiv2016

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

Let $H=-Δ+V$ be a Schrödinger operator on $L^2(\mathbb R^n)$ with real-valued potential $V$ for $n > 4$ and let $H_0=-Δ$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^n)$ for all $1\leq p\leq \infty$ if zero is not an eigenvalue, and on $1<p<\frac{n}{2}$ if zero is an eigenvalue. We show that these wave operators are also bounded on $L^1(\mathbb R^n)$ by direct examination of the integral kernel of the leading term. Furthermore, if $\int_{\mathbb R^n} V(x) ϕ(x) \, dx=0$ for all eigenfunctions $ϕ$, then the wave operators are $L^p$ bounded for $1\leq p<n$. If, in addition $\int_{\mathbb R^n} xV(x) ϕ(x) \, dx=0$, then the wave operators are bounded for $1\leq p<\infty$.

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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.

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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↗

arXiv2014

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

We study the wave equation with potential $u_{tt}-Δu+Vu=0$ in two spatial dimensions, with $V$ a real-valued, decaying potential. With $H=-Δ+V$, we study a variety of mapping estimates of the solution operators, $\cos(t\sqrt{H})$ and $\frac{\sin(t\sqrt{H})}{\sqrt{H}}$ under the assumption that zero is a regular point of the spectrum of $H$. We prove a dispersive estimate with a time decay rate of $|t|^{-\frac{1}{2}}$, a polynomially weighted dispersive estimate which attains a faster decay rate of $|t|^{-1}(\log |t|)^{-2}$ for $|t|>2$. Finally, we prove dispersive estimates if zero is not a regular point of the spectrum of $H$.

math.AP↗

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.

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arXiv2013

Exponential Decay for dispersion managed solitons for general dispersion profiles

We show that any weak solution of the Gabitov-Turitsyn equation describing dispersion managed solitons together with its Fourier transform decay exponentially. This strong regularity result extends a recent result of Erdogan, Hundertmark and Lee in two directions, to arbitrary non-negative average dispersion and, more importantly, to rather general dispersion profiles, which cover most, if not all, physically relevant cases.

math.AP↗

arXiv2012

Dispersive estimates for matrix Schrödinger operators in dimension two

We consider the non-selfadjoint operator [\cH = [{array}{cc} -Δ+ μ-V_1 & -V_2 V_2 & Δ- μ+ V_1 {array}]] where $μ>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ dispersive decay estimates for the evolution $e^{it\cH}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R^2)},\,\,\,\,\,\,\,\, |t| >2, $$ with $w(x)=\log^2(2+|x|)$.

math.AP↗

arXiv2012

A weighted dispersive estimate for Schrödinger operators in dimension two

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 $$ \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)}, |t| >2, $$ with $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights.

math.AP↗

arXiv2012

Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{-1}, \text{for} |t|>1.$$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.

math.AP↗