## 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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### References:

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