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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

Citations:

Zbl 1080.35102
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