## Decay estimates for Schrödinger operators.(English)Zbl 0743.35008

Using the spectral theorem the unitary operator $$e^{itH_ 0}$$ is well defined for $$t\in\mathbb{R}$$, if $$H_ 0=-(\partial/\partial x_ 1)^ 2- \cdots-(\partial/\partial x_ n)^ 2$$ is minus the Laplacian. Given initial data $$f(x)$$ the function $$u(\cdot,t)=e^{itH_ 0}f$$ solves the time dependent Schrödinger equation $$i\partial u/\partial t+H_ 0u=0$$, $$u|_{t=0}=f$$. Since the kernel of $$e^{itH_ 0}$$ is $$(4\pi it)^{n/2}e^{| x-y|^ 2/4it}$$, one sees that the solution is dispersive in the sense that $\| u(\cdot,t)\|_{L^{p'}(\mathbb{R}^ n)}\leq Ct^{-n(1/p-1/2)}\| f\|_{L^ p(\mathbb{R}^ n)}, t>0,\quad\text{ if } 1\leq p\leq 2, \text{ and } 1/p+1/p'=1.$ This paper replaces the free operator $$H_ 0$$ by more general Hamiltonians $$H=- \Delta+V(x)$$ with suitable singularity assumptions on $$V(x)$$. Let $$P_ c$$ denote projection onto the continuous part of the spectrum of $$H$$. The norm result is: Let $$n\geq 3$$. Then if 0 is neither an eigenvalue nor a resonance for $$H$$, $$\| e^{itH}P_ c\psi\|_{L^{p'}(\mathbb{R}^ n)}\leq ct^{-n(1/p-1/2)}\|\psi\|_{L^ P(\mathbb{R}^ n)}$$, $$t>0$$.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35J10 Schrödinger operator, Schrödinger equation 35Q40 PDEs in connection with quantum mechanics

### Keywords:

Schrödinger equation; dispersive
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### References:

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