## Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three. II.(English)Zbl 1146.35324

Summary: We investigate boundedness of the evolution $$e^{it\mathcal H}$$ in the sense of $$L^2(\mathbb R^3)\rightarrow L^{2}(\mathbb R^3)$$ as well as $$L ^{1}(\mathbb R^3)\rightarrow L^\infty(\mathbb R^3)$$ for the non-selfadjoint operator
$\mathcal{H} = \left[\begin{matrix}{-\Delta + \mu - V_1}& -V_2\\V_2 & {\Delta - \mu + V_1} \end{matrix} \right],$
where $$\mu >0$$ and $$V _{1}, V_{2}$$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave, and the aforementioned bounds are needed in the study of nonlinear asymptotic stability of such standing waves. We derive our results under some natural spectral assumptions (corresponding to a ground state soliton of NLS), but without imposing any restrictions on the edges $$\pm \mu$$ of the essential spectrum. Our goal is to develop an “axiomatic approach,” which frees the linear theory from any nonlinear context in which it may have arisen.
For part I see Dyn. Partial Differ. Equ. 1, No. 4, 359–379 (2004; Zbl 1080.35102).

### MSC:

 35B45 A priori estimates in context of PDEs 47D06 One-parameter semigroups and linear evolution equations 35Q40 PDEs in connection with quantum mechanics 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Zbl 1080.35102
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### References:

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