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

Jensen, Arne

Math. Z. 191, 53-59 (1986).

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

Keywords:

Schrödinger operator; smoothing property; unitary evolution group; Sobolev space; boost group; commutator methods

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