On an inclusion of the essential spectrum of Laplacians under non-compact change of metric (Q2880666)

Let \((M,g)\) be a connected smooth Riemannian manifold without boundary. One says that the Laplacian \(\Delta_g\) is \textit{essentially self-adjoint} if it has a unique self-adjoint extension \(\bar\Delta_g\).NEWLINENEWLINEThe author shows: \medbreak\noindent Theorem 1: Let \(g\) and \(g^\prime\) be Riemannian metrics such that \(g=g^\prime\) outside a subset \(K\) of \(M\). If \(\Delta_g\) is essentially self-adjoint in \(L^2(g)\) and if the Cauchy boundary of \(K\) with respect to \(g^\prime\) is almost polar, then \(\Delta_{g^\prime}\) is essentially self-adjoint in \(L^2(g^\prime)\). If additionally there is a function \(\chi\) on \(M\) satisfying \(\text{grad}_g\chi\in L^\infty\), \(\Delta_g\chi\in L^\infty\), and \(\chi|_K=1\), and if the inclusion \(H_0^1(N,g)\subset L^2(N,g)\) is compact for some \(N\) containing an \(\varepsilon\) neighborhood of the support of \(\chi\), then the essential spectrum of \(\bar\Delta_g\) is contained in the essential spectrum of \(\bar\Delta_{g^\prime}\). \medbreak \textit{K. Furutani}'s stability result [Proc. Japan Acad., Ser. A 56, 425--428 (1980; Zbl 0483.58019)] then follows as a special case: \medbreak\noindent Theorem 2: Let \(g\) and \(g^\prime\) be Riemannian metrics on \(M\) such that \(g=g^\prime\) outside a compact subset \(K\) of \(M\). If \(\Delta_g\) is essentially self-adjoint in \(L^2(g)\), then \(\Delta_{g^\prime}\) is essentially self-adjoint in \(L^2(g^\prime)\) and the two operators \(\bar\Delta_g\) and \(\bar\Delta_{g^\prime}\) have the same essential spectrum. \medbreak Theorem 1 extends Theorem 2 in a non-trivial fashion as the set \(K\) in Theorem 1 may have infinite volume measured with respect to the metric \(g^\prime\) and its completion may involve a singular set such as the fractal set to which the metric is not extendable.