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A dispersive bound for three-dimensional Schrödinger operators with zero energy eigenvalues. (English) Zbl 1223.35265

Consider \(V\in L^p(\mathbb{R}^3)\cap L^q(\mathbb{R}^3)\) for exponents \(p<\frac32<q\). Here all function spaces are complex and over \(\mathbb{R}^3\). Denote by \(H:=-\Delta+V\) the corresponding Schrödinger operator, which may not be symmetric. A resonance \(\Psi\) of \(H\) is a distributional solution of \(H\Psi=\lambda^2\Psi\), for some \(\lambda\in\mathbb{R}\), such that \(\Psi\in L^3_{\text{weak}}\smallsetminus L^2\). Denote by \(X_1\) the set of \(\Psi\in L^2\) that are solutions of \(H\Psi=0\), that is, the zero energy eigenfunctions. Define inductively, if \(X_k\subseteq L^1\), the space \[ X_{k+1}:=\{\Psi\in L^3_{\text{weak}}\mid H\Psi\in X_k\}. \] The main result is a bound on the \(L^\infty\)-norm of the time evolution of initial values \(f\in L^1\) away from the generalized eigenspaces of \(H\), in terms of the \(L^1\)-norm of \(f\). Suppose that \(H\) has no resonances, that \(X_k\subseteq L^1\) for each \(k\in\mathbb{N}\), and that \(\bigcup_{k\in\mathbb{N}}X_k\) is finite dimensional. Setting \(P\) to the sum of all spectral projections to generalized eigenspaces of eigenvalues of \(H\), there is \(C>0\) such that \[ \|e^{-itH}(I-P)f\|_\infty\leq C|t|^{-3/2}\|f\|_1 \] for all \(f\in L^1\) and \[ \|e^{-itH}(I-P)f\|_2\leq C\|f\|_2 \] for all \(f\in L^2\).

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81U30 Dispersion theory, dispersion relations arising in quantum theory
35J10 Schrödinger operator, Schrödinger equation
47D08 Schrödinger and Feynman-Kac semigroups
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