Dispersive decay estimates for Dirac equations with a domain wall (Q6641706)

The authors consider the initial value problem: \(i\partial _{t}\alpha =D(\tau )\alpha \), \(\alpha (0,x)=\alpha _{0}(x)\in L^{2}(\mathbb{R};\mathbb{C }^{2})\), where \(\alpha (t,x):\mathbb{R}_{t}\times \mathbb{R}_{x}\rightarrow \mathbb{C}^{2}\) is a complex vector amplitude, and \(D\) the Dirac Hamiltonian defined for each \(\tau \in \mathbb{R}/2\pi Z\), through \(D(\tau )=i\sigma _{3}\partial _{x}+\sigma _{1}1_{(-\infty ,0)}(x)+\sigma _{\ast }(\tau )1_{[0,\infty )}(x)\), \(\sigma _{j}\) denoting the standard Pauli matrix, \( \sigma _{\ast }\tau )=\left( \begin{array}{cc} 0 & e^{-i\tau } \\\Ne^{i\tau } & 0 \end{array} \right) \), and \(1_{S}(x)\) the indicator function of a set \(S\subseteq \mathbb{ R}\). The purpose of the paper is to describe the time-dynamics, as the dislocation parameter \(\tau \) varies, in particular the dispersive decay of solutions as \(t\rightarrow \infty \), for initial data projected onto the continuous spectral part of \(D(\tau )\). The authors recall spectral properties of the Hamiltonian \(D(\tau )\) and the main result of the paper proves that for any \(\epsilon >0\), there exists \(C_{\epsilon }>0\), which is independent of \(\tau \in \lbrack 0,2\pi ]\), such that \(\left\Vert \left\langle x\right\rangle ^{-2}e^{iD(\tau )t}P_{ac}(D(\tau ))\left\langle D(\tau )\right\rangle ^{-3/2-\epsilon }\left\langle x\right\rangle ^{-2}\right\Vert _{L^{1}\rightarrow L^{\infty }}\leq C_{\epsilon }\frac{1}{ \left\langle t\right\rangle ^{1/2}}\frac{1}{1+sin^{2}(\tau /2)t}\), where \( P_{ac}(D(\tau ))\) is the projection operator onto the continuous spectral subspace of \(D(\tau )\). For the proof, the authors introduce the resolvent kernel \(\mathfrak{R}(\omega ,\tau )=(D(\tau )-\omega )^{-1}\). They present an explicit expression as an oscillatory integral of \(\alpha (x,t)=e^{-iD(\tau )t}P(D(\tau ))\left\langle D(\tau )\right\rangle ^{3/2-\varepsilon }\alpha _{0}\), where \(P\) is the projection onto the positive part of the continuous spectrum \(P(D(\tau ))=P_{ac}(D(\tau )1_{[1,\infty )})\). They divide the oscillatory integral into high-energy and low-energy components, and they prove that the high-energy component is bounded from above uniformly in \(\tau \) using a stationary phase argument, and that the low-energy component is bounded from above in two steps: first a uniform \(t^{-1/2}\) bound is proved, then a \(t^{-3/2}\) decay rate for nonresonant \(\tau \). The authors also prove alternative variations of time-decay upper bounds, according to the assumptions dealing with the spatial localization of the initial conditions.