Self-adjoint abstract differential operators

Abstract

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H₁ whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A≥E, B(x)≥0, and the operator B(x) is bounded for all x. An operator L₀ is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) εH₁ according to the formula: L₀y=−y″ + Q(x)y (−∞<x<∞). It is proved that the closure of the operator L₀ is a self-adjoint operator in H₁ under the given assumptions.