Self-adjointness of Schrödinger-type operators with singular potentials on manifolds of bounded geometry. (English) Zbl 1037.58013

Summary: We consider the Schrödinger type differential expression \[ H_V=\nabla^*\nabla+V, \] where \(\nabla\) is a \(C^{\infty}\)-bounded Hermitian connection on a Hermitian vector bundle \(E\) of bounded geometry over a manifold of bounded geometry \((M,g)\) with metric \(g\) and positive \(C^{\infty}\)-bounded measure \(d\mu\), and \(V=V_1+V_2\), where \(0\leq V_1\in L_{\text{loc}}^1(\text{End\,} E)\) and \(0\geq V_2\in L_{\text{loc}}^1(\text{End\,} E)\) are linear self-adjoint bundle endomorphisms. We give a sufficient condition for self-adjointness of the operator \(S\) in \(L^2(E)\) defined by \(Su=H_Vu\) for all \(u\in\text{Dom}(S)=\{u\in W^{1,2}(E): \int\langle V_1u,u\rangle\,d\mu<+ \infty\) and \(H_Vu\in L^2(E)\}\). The proof follows the scheme of Kato, but it requires the use of more general version of Kato’s inequality for Bochner Laplacian operator as well as a result on the positivity of \(u\in L^2(M)\) satisfying the equation \((\Delta_M+b)u=\nu\), where \(\Delta_M\) is the scalar Laplacian on \(M\), \(b>0\) is a constant and \(\nu\geq 0\) is a positive distribution on \(M\).


58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P05 General topics in linear spectral theory for PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47F05 General theory of partial differential operators
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