In QM we use position eigenstates $|x\rangle$ represented by Dirac delta function $\langle x' | x \rangle= \delta (x-x')$. They are non-physical because this distribution is not square integrable. I was thus wondering if it was possible to define a distribution $\chi (x)$ such that $$\int |\chi(x)|^2 = 1$$ and that $\chi (x) = 0 $ for $x\neq 0$.
I imagine that even if such a distribution exists, its utilisation is inconvenient, but still I was curious about this.
Thank you for your help!