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Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue. (English) Zbl 1079.81021

Summary: Let \(H = -\Delta+V(x)\) be a three-dimensional Schrödinger operator. We study the time decay in \(L^{p}\) spaces of scattering solutions \(e^{-itH}P_c u\), where \(P_c\) is the orthogonal projection onto the continuous spectral subspace of \(L^2(\mathbb{R}^3)\) for \(H\). Under suitable decay assumptions on \(V(x)\) it is shown that they satisfy the so-called \(L^p\)-\(L^q\) estimates \(\| e^{-itH} P_cu \|_p \leq (4\pi| t|)^{-3(1/2-1/p)} \| u\|_q\) for all \(1 \leq q \leq 2 \leq p \leq \infty\) with \(1/p +1/q =1\) if \(H\) has no threshold resonance and eigenvalue; and for all \(3/2 < q \leq 2 \leq p < 3\) if otherwise.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
35P15 Estimates of eigenvalues in context of PDEs
35Q40 PDEs in connection with quantum mechanics
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