Considering singular Sturm--Liouville differential expressions of the type \[ \tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \; \alpha \mathbb{R}, \] we employ some Sturm comparison-type results in spirit Kurss to derive criteria for $\tau_{\alpha}$ be limit point and circle case at $x=0$. More precisely, if $\alpha \mathbb{R}$ $0 0$ such that $0<x$ sufficiently small, \begi...

We consider perturbed quadharmonic operators, ? 4 + V , acting on sections of a Hermitian vector bundle over complete Riemannian manifold, with the potential satisfying bound from below by non-positive function depending distance point. Under bounded geometry assumption and underlying we give sufficient condition for essential self-adjointness such operators. then apply this to prove separation...