I'm trying to find an orthonormal basis with respect to a weight function $w(x) \equiv 1$ for $L^2$ in terms of the following proposed basis functions: $$ φ_n(x) = D^n \exp(-x^2/2),\quad n=0,1,2,\ldots \tag{1}$$ where $D^n$ is the derivative operator.
Unfortunately, it seems that this cannot be done. Define $$ M_{mn}=\frac{1}{Z_{mn}}\int φ_m(x) φ_n(x) dx $$ where $Z_{mn}=\sqrt{\pi}\ 2^{-(m+n)/2}\ (m+n-1)!!$.
Then, for example, for the first five basis functions we have the following singular matrix: $$ M_{mn}=\left( \begin{array}{cccccc} 1 & 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 & 0 & 1 \\ -1 & 0 & 1 & 0 & -1 & 0 \\ 0 & -1 & 0 & 1 & 0 & -1 \\ 1 & 0 & -1 & 0 & 1 & 0 \\ 0 & 1 & 0 & -1 & 0 & 1 \\ \end{array} \right) $$
As far as my manipulations have been correct it seems to me that Eq. (1) cannot be written in terms of Hermite functions. Question 1: Is this true? Does the singularity of $M_{mn}$ indicate that (linear combinations of) the functions (1) can never form a basis, and thus that they cannot be equivalent to Hermite functions?
I suspect an orthonormal basis in form similar to Eq. (1) might still be possible with a generalization. Rewrite Eq. (1) as $$ φ_n(x)=\sqrt{2π}\ D^n φ(x); \tag{1'} $$ i.e. $φ(x)=\exp(-x^2/2)/\sqrt{2π}$. Now for each $n$ we perform a custom translation and (positive) rescaling of the Gaussian $φ(x)$: $$ Φ_n(x;μ_n,σ_n) = a_n(μ_n,σ_n)\ D^n \left[\frac{1}{σ_n}φ\left(\frac{x-μ_n}{σ_n}\right)\right]. \quad n=0,1,2,\ldots \tag{2} $$ Question 2: For which $(a_n,μ_n,σ_n)$, if any, would Eq. (2) represent an orthonormal basis for $(L^2,w(x)\equiv 1)$? Any pointers would be greatly appreciated.