A remark on papers by Shubin on classical and quantum completeness
نویسندگان
چکیده
Let M be a complete Riemannian manifold and D : C∞ 0 (E) → C∞ 0 (F ) a first order differential operator acting between sections of the hermitian vector bundles E, F . Moreover, let V : C∞(E) → L∞ loc (E) be a self–adjoint zero order differential operator. We give a sufficient condition for the Schrödinger operator H = DD + V to be essentially self–adjoint. This generalizes recent work of I. Oleinik [3, 4, 5], M. Shubin [6, 7], and M. Braverman [1]. We essentially use the method of Shubin. Our presentation shows that there is a close link between Shubin’s self–adjointness condition for the Schrödinger operator and Chernoff’s self–adjointness condition for powers of first order operators. We also discuss nonelliptic operators. However, in this case our result depends on the hyperbolic equation method as well as an additional assumption. We conjecture that the additional assumption turns out to be obsolete in general. For elliptic operators, the paper is completely self–contained.
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تاریخ انتشار 2000