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Dispersive estimates for matrix and scalar Schrödinger operators in dimension five. (English) Zbl 1373.35266

Summary: We investigate the boundedness of the evolution operators \(e^{itH}\) and \(e^{it\mathcal{H}}\) in the sense of \(L^{1}\to L^{\infty}\) for both the scalar Schrödinger operator \(H=-\Delta+V\) and the non-selfadjoint matrix Schrödinger operator \[ \mathcal H= \bigg[\begin{matrix} -\Delta+\mu-V_1 \quad -V_2\quad \\ \quad V_2 \qquad \Delta-\mu+V_1\end{matrix} \bigg] \] in dimension five. Here \(\mu>0\) and \(V_{1}, V_{2}\) are real-valued decaying potentials. The matrix operator arises when linearizing about a standing wave in certain nonlinear partial differential equations. We apply some natural spectral assumptions on \(\mathcal{H}\), including regularity of the edges of the spectrum \(\pm\mu\).

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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