The linear response function χ(r,r′): another perspective.

In this paper, we propose a conceptual approach to assign a "mathematical meaning" to the non-local function χ (r , r ′) . Mathematical evaluation of this kernel remains difficult since it is a function depending on six Cartesian coordinates. The idea behind this approach is to look for a limit process in order to explore mathematically this non-local function. According to our approach, the bra ⟨ χ r ′ ξ | is the linear functional that corresponds to any ket | ψ ⟩ , the value ⟨ r ′ | ψ ⟩ . In condensed writing ⟨ χ r ′ ξ | ⟨ r | ψ ⟩ = ⟨ r ′ | ψ ⟩ , and this is achieved by exploiting the sifting property of the delta function that gives it the sense of a measure, i.e. measuring the value of ψ (r) at the point r ′ . It is worth noting that ⟨ χ r ′ ξ | is not an operator in the sense that when it is applied on a ket, it produces a number ψ (r = r ′) and not a ket. The quantity χ r ′ ξ (r) proceed as nascent delta function, turning into a real delta function in the limit where ξ → 0 . In this regard, χ r ′ ξ (r) acts as a limit of an integral operator kernel in a convolution integration procedure. [ABSTRACT FROM AUTHOR]