The continuous spectra of elliptic differential operators in unbounded domains
We study some spectral properties of the differential operator \[ Lu=-div A(du+iub)-iA(b,du+iub)+qu= \] \[ -\frac{1}{\sqrt{\det g}}\sum^{n}_{j,k=1}(\frac{\partial}{\partial x_ j}+ib_ j)\sqrt{\det g}a^{jk}(\frac{\partial}{\partial x_ k}+ib_ k)u+qu \] in a domain \(\Omega\) of a non-compact oriented Riemannian \(C^{\infty}\) n-manifold \({\mathcal M}\) with metric \(g=\sum g_{jk}dx_ j\otimes dx_ k\) (n\(\geq 2)\). Here, \(A=\sum a^{jk}(\partial /\partial x_ j)\otimes \partial /\partial x_ k\) is a real symmetric (2,0) tensor field, \(b=\sum b_ jdx_ j\) a real differential 1-form, q a scalar function, \(i=\sqrt{-1}\). Let H be a self-adjoint realization in \(L^ 2(\Omega)\) of L. Our main concerns in this paper are: (i) What conditions exclude positive eigenvalues of H? (ii) Is the continuous spectrum of H absolutely continuous? (iii) Does the essential spectrum of H fill up the positive real axis?
oriented Riemannian \(C^{\infty }\) n-manifold
scientific article; zbMATH DE number 4071508