Theory of self-adjoint operators generated by strongly singular, second- order expression of divergence type (Q794258)

Consider the real differential operator in \({\mathbb{R}}^ m: S=-div a(x)\text{grad}+v(x),\) where the matrix \(a(x)>0\) has \(a_{ij}(x)\in L^{\infty}_{loc}\) and the positive (negative) part of v is in \(L^ 1_{loc}\) (resp. in \(L^ p_{loc}\), \(p>m/2)\). The pseudominimal operator, \(A_ 0\), is defined as the restriction of S to functions of compact support, \(f\in W^ 1_ 2\cap L^{\infty},\) such that \(S^ kf\in W^ 1_ 2\cap L^{\infty}\) for all k. \(A_ 0\) is densely defined and symmetric. The note is concerned with the selfadjoint extensions of \(A_ 0\). E.g., in the uniformly elliptic case: a) If \(\sup_{y}\int_{B_ y}(v_-)^ p<\infty\) (where \(B_ y\) is the translation by y of a given ball), \(A_ 0\) is essentially selfadjoint; b) for arbitrary behaviour of \(v_-\) at infinity and arbitrary selfadjoint extension A of \(A_ 0\), local estimates on the integral kernel of \((A-i\tau)^ k\) and on the generalized eigenfunctions of A are obtained.