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 ⟨χr_′^ξ| is the linear functional that corresponds to any ket |ψ⟩, the value ⟨r^′|ψ⟩. In condensed writing ⟨χr_′^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 ⟨χr_′^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 χr_′^ξ(r) proceed as nascent delta function, turning into a real delta function in the limit where ξ→0. In this regard, χr_′^ξ(r) acts as a limit of an integral operator kernel in a convolution integration procedure.