The linear response function \(\chi (\textbf{r}, \textbf{r}^{'})\): another perspective

In this paper, we propose a conceptual approach to assign a “mathematical meaning” to the non-local function \(\chi (\textbf{r}, \mathbf{r'})\) $\chi (\mathbf{r},{\mathbf{r}}^{\prime })$ . 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 \(\langle \chi ^{\xi }_{r'} \vert \) $⟨{\chi }_{r^{\prime }}^{\xi }|$ is the linear functional that corresponds to any ket \(\vert \psi \rangle \) $|\psi ⟩$ , the value \(\langle \textbf{r}' \vert \psi \rangle \) $⟨{\mathbf{r}}^{\prime }|\psi ⟩$ . In condensed writing \(\langle \chi ^{\xi }_{r'} \vert \, \langle \textbf{r} \vert \psi \rangle = \langle \textbf{r}' \vert \psi \rangle \) $⟨{\chi }_{r^{\prime }}^{\xi }|⟨\mathbf{r}|\psi ⟩=⟨{\mathbf{r}}^{\prime }|\psi ⟩$ , 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 \(\psi (\textbf{r})\) $\psi (\mathbf{r})$ at the point \(\textbf{r}'\) ${\mathbf{r}}^{\prime }$ . It is worth noting that \(\langle \chi ^{\xi }_{r'} \vert \) $⟨{\chi }_{r^{\prime }}^{\xi }|$ is not an operator in the sense that when it is applied on a ket, it produces a number \(\psi (\textbf{r} = \textbf{r}')\) $\psi (\mathbf{r}={\mathbf{r}}^{\prime })$ and not a ket. The quantity \(\chi ^{\xi }_{r'} (\textbf{r})\) ${\chi }_{r^{\prime }}^{\xi }(\mathbf{r})$ proceed as nascent delta function, turning into a real delta function in the limit where \(\xi \rightarrow 0\) $\xi \to 0$ . In this regard, \(\chi ^{\xi }_{r'} (\textbf{r})\) ${\chi }_{r^{\prime }}^{\xi }(\mathbf{r})$ acts as a limit of an integral operator kernel in a convolution integration procedure.