[Paper Review] Decay Estimates and Strichartz Estimates of Fourth-order Schrödinger Operator

Hongliang Feng, Avy Soffer+1 more|arXiv (Cornell University)|Mar 1, 2017

Advanced Mathematical Physics Problems39 references3 citations

TL;DR

This paper establishes decay and Strichartz estimates for the fourth-order Schrödinger operator $H = (-\Delta)^2 + V$ in $\mathbb{R}^d$ for $d=3$ and $d\geq 5$, using resolvent asymptotics and conjugate operator methods. It derives Jensen-Kato-type decay estimates, local decay estimates, $L^1 \to L^\infty$ decay for $d=3$, and endpoint global Strichartz estimates for $d\geq 5$, under suitable spectral and potential decay assumptions.

ABSTRACT

We study time decay estimates of the fourth-order Schrödinger operator $H=(-Δ)^{2}+V(x)$ in $\mathbb{R}^{d}$ for $d=3$ and $d\geq5$. We analyze the low energy and high energy behaviour of resolvent $R(H; z)$, and then derive the Jensen-Kato dispersion decay estimate and local decay estimate for $e^{-itH}P_{ac}$ under suitable spectrum assumptions of $H$. Based on Jensen-Kato decay estimate and local decay estimate, we obtain the $L^1 ightarrow L^{\infty}$ estimate of $e^{-itH}P_{ac}$ in $3$-dimension by Ginibre argument, and also establish the endpoint global Strichartz estimates of $e^{-itH}P_{ac}$ for $d\geq5$. Furthermore, using the local decay estimate and the Georgescu-Larenas-Soffer conjugate operator method, we prove the Jensen-Kato type decay estimates for some functions of $H$.

Motivation & Objective

• To establish time decay estimates for the fourth-order Schrödinger operator $H = (-\Delta)^2 + V$ in $\mathbb{R}^d$ for $d=3$ and $d\geq 5$.
• To analyze the low- and high-energy behavior of the resolvent $R(H;z)$ near $z=0$ and at infinity.
• To derive Jensen-Kato-type dispersion decay and local decay estimates for $e^{-itH}P_{ac}$ under spectral assumptions.
• To prove $L^1 \to L^\infty$ decay estimates for $d=3$ using the Ginibre argument and endpoint Strichartz estimates for $d\geq 5$.
• To extend decay estimates to functions of $H$ using the Georgescu-Larenas-Soffer conjugate operator method, overcoming degeneracy at $\xi=0$.

Proposed method

• Derive asymptotic expansions of the free resolvent $R(H_0;z)$ and the perturbed resolvent $R(H;z)$ near $z=0$ using spectral and potential decay assumptions.
• Establish the limiting absorption principle for $R(H;z)$ to analyze spectral properties at the threshold energy.
• Use the Ginibre argument to derive $L^1 \to L^\infty$ decay estimates for $d=3$ based on $L^p$-boundedness of $P_{ac}$ and Jensen-Kato decay.
• Prove endpoint global Strichartz estimates for $d\geq 5$ using the combination of Jensen-Kato and local decay estimates.
• Apply the Georgescu-Larenas-Soffer conjugate operator method to derive Jensen-Kato-type decay estimates starting from local decay estimates.
• Use Fredholm theory and Hilbert-Schmidt norm estimates to prove invertibility of operators like $QM_0Q$ and construct asymptotic expansions of $M(\mu)^{-1}$ near $\mu=0$.

Experimental results

Research questions

• RQ1 How do the resolvent $R(H;z)$ and its asymptotic behavior at $z=0$ and infinity affect decay properties of $e^{-itH}P_{ac}$?
• RQ2 Can Jensen-Kato-type and local decay estimates be established for $H = (-\Delta)^2 + V$ despite the degeneracy of $(-\Delta)^2$ at $\xi=0$?
• RQ3 What $L^1 \to L^\infty$ decay rate is achievable for $e^{-itH}P_{ac}$ in $d=3$ under the Ginibre argument?
• RQ4 What endpoint Strichartz estimates hold for $e^{-itH}P_{ac}$ in $d\geq 5$, and how are they derived from decay estimates?
• RQ5 Can the conjugate operator method yield Jensen-Kato-type decay estimates for functions of $H$ starting only from local decay?

Key findings

• For $d=3$, the paper establishes an $L^1 \to L^\infty$ decay estimate for $e^{-itH}P_{ac}$ with rate $\langle t \rangle^{-3/2}$, derived via the Ginibre argument.
• For $d\geq 5$, the paper proves endpoint global Strichartz estimates for $e^{-itH}P_{ac}$, extending the classical Strichartz theory to fourth-order operators.
• The Jensen-Kato-type decay estimate $\|\langle x\rangle^{-\sigma}e^{-itH}P_{ac}\langle x\rangle^{-\sigma}\|_{L^2 \to L^2} \leq C\langle t \rangle^{-3/2}$ is established for $d=3$ under spectral assumptions.
• The asymptotic expansion of $M(\mu)^{-1}$ near $\mu=0$ is derived, showing that the inverse exists and is meromorphic, with the process terminating in finite steps due to the growth condition $\limsup_{\iota \downarrow 0} \|\iota^4 M(\iota)^{-1}\| < \infty$.
• The invertibility of $QM_0Q$ in $L^2(\mathbb{R}^3)$ is shown to be equivalent to zero being a regular spectral point of $H$, which ensures the validity of the decay estimates.
• The conjugate operator method allows derivation of Jensen-Kato-type decay estimates from local decay estimates alone, providing a new route that avoids the non-degeneracy assumptions of Murata.