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).
\[ \mathcal{H} = \left[\begin{matrix}{-\Delta + \mu - V_1}& -V_2\\V_2 & {\Delta - \mu + V_1} \end{matrix} \right], \]
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 |
Citations:
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