## 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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### References:

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