## Commutator methods and a smoothing property of the Schrödinger evolution group.(English)Zbl 0594.35032

Let $$H=-()\Delta +V$$ be a Schrödinger operator in $$L^ 2(R^ n)$$, where V is a real-valued function, which satisfies the condition $$(\partial /\partial x)^{\alpha}V\in L^{\infty}(R^ n)$$ for $$| \alpha | \leq k$$, where k is a given integer.
The main result is the following smoothing property of the unitary evolution group: For $$t\neq 0$$ $$(1+x^ 2)^{-k/2}e^{-itH}(1+x^ 2)^{-k/2}$$ is bounded from $$L^ 2(R^ n)$$ to the Sobolev space $$H^ k(R^ n)$$. The proof uses the boost group $$e^{iax}$$, $$a\in R^ n$$, and commutator methods.
There is a misprint in formula (2.4) in Lemma 2.2: In the integral (t-s) should be replaced by s.

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35B65 Smoothness and regularity of solutions to PDEs 47F05 General theory of partial differential operators
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### References:

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