Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem

Authors : Hany ATİA, Hala H EMAM

Pages : 1321-1332

Doi:10.15672/hujms.1021920

View : 24 | Download : 8

Publication Date : 2022-10-01

Article Type : Research Paper

Abstract :Consider the tri-harmonic differential expression $L_{V}^{\nabla}u=\leftinsert ignore into journalissuearticles values(\nabla^{+}\nabla\right);^{3}u+Vu$, on sections of a hermitian vector bundle over a complete Riemannian manifold $\leftinsert ignore into journalissuearticles values(M,g\right);$ with metric $g$, where $\nabla$ is a metric covariant derivative on bundle E over complete Riemannian manifold, $\nabla^{+}$ is the formal adjoint of $\nabla$ and $V$ is a self adjoint bundle on $E$. We will give conditions for $L_{V}^{\nabla}$ to be essential self-adjoint in $L^{2}\leftinsert ignore into journalissuearticles values(E\right);.$ Additionally, we provide sufficient conditions for $L_{V}^{\nabla}$ to be separated in $L^{2}\leftinsert ignore into journalissuearticles values( E\right);$. According to Everitt and Giertz, the differential operator $L_{V}^{\nabla}$ is said to be separated in $L^{2}\leftinsert ignore into journalissuearticles values( E\right); $ if for all $u$ $\in L^{2}\leftinsert ignore into journalissuearticles values( E\right);$ such that $L_{V}^{\nabla}u\in L^{2}\leftinsert ignore into journalissuearticles values( E\right); $, we have $Vu\in L^{2}\leftinsert ignore into journalissuearticles values( E\right);$.
Keywords : essential self adjoint, separation problem, Riemannian manifold, covariant tri harmonic