Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues
Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues
复制标题
流形、本质自伴性和特征值缺失的薛定谔算子
DOI：
10.1007/bf02921722
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
N. Garofalo
中科院分区：
文献类型：
--
作者：
H. Donnelly;N. Garofalo
设M是一个完备的非紧黎曼流形。若Δ表示M的拉普拉斯算子，则有相应的薛定谔算子− Δ +V.给出了V的条件，保证了− Δ + V的本质自伴性.特别地，若V ∈ Qα，loc（Mn），局部Stummel类，且V ≥ − c在紧集外，则− Δ +维斯在C0∞（Mn）上本质自伴.此外，还证明了在点处强奇异的势的本质自伴性。我们还研究了−Δ +维斯的本征值的缺失。这依赖于Rellich型恒等式。强奇异势的结果利用了经典的不确定性原理的推广，在Rn，黎曼流形的极点。
Suppose thatMnis a complete, noncompact, Riemannian manifold. If Δ denotes the Laplace operator ofM, one has associated Schrödinger operators − Δ +V.Conditions onVare formulated, which ensures the essential self-adjointness of − Δ +V. In particular, ifV∈ Qα,loc(Mn), the local Stummel class, andV ≥ − coutside of a compact set, then − Δ +Vis essentially self-adjoint on C0∞(Mn). In addition, essential self-adjointness is proved for potentials which are strongly singular at a point. The absence of eigenvalues of −Δ +Vis also studied. This relies upon Rellich-type identities. The results on strongly singular potentials make use of a generalization of the classical uncertainty principle, inRn, to Riemannian manifolds with a pole.