I want to apply the theory of $D$-modules to solve operator equations of several variables in the *Bargmann space*
$$\mathcal H :=\bigg\{\psi \in  \mathcal O^\text{an}_{\mathbb{C}^n}\,\,\bigg|\,\,\,\int_{\mathbb{C}}d^{2n}x\,|\psi(x)|^2e^{-\|x\|^2}<\infty\bigg\}.$$
I can find a lot of references on $D$-modules that give good algorithms that solve these types of equations. However, I can't find good references that consider convergence properties of these solutions.

I had a naive idea about how to quickly compute the dimension of the solution space $\text{Ann}(I)$ associated with a given holonomic ideal, and this relies on the correspondence between holonomic $D$-modules and local systems on $\mathbb{C}^n$. Indeed, to compute the behavior of such a local system, we would look at the limiting behavior of the connection form at large $|x|\to \infty$:
$$A_j(x)dx^j \underset{|x|\to\infty}{\sim} \sum_{|\alpha| = m}x^\alpha A_j^{(\alpha)}dx^j.$$
Here, $m :=\deg A$ gives the leading order contribution to the connection form. By analyzing the spectral properties of the $A^{(\alpha)}_j$, this would then determine how many solutions are normalizable or not (suppose there are no singular points, for simplicity). However, this analysis fails for some simple examples. My question is, *when this analysis fails, why does it fail?*:

*Test case: Kummer's equation (WORKS)*

Let $n=1$, and consider the principal ideal in the Weyl algebra generated by $$l:=x\partial^2-(x-b)\partial -a.$$
We can always solve the ideal $I$ by constructing a local system, in this case we get the following flat connection:
\begin{align}
A(x)dx &=  \frac{1}{x}\begin{pmatrix}0&a\\ a&x+b\end{pmatrix}dx\underset{|x|\to \infty}{\sim}\begin{pmatrix}0&0\\ 0&1\end{pmatrix}dx
\end{align}
In this case, $A^{(\alpha)}$ has two eigenspaces, one with zero eigenvalue, and one with eigenvalue one. So we would expect the following asymptotics:
\begin{align}
\psi_1(x) &\underset{|x|\to\infty}{\sim} e^x \cdot(\text{lower-order terms})\\
\psi_2(x) &\underset{|x|\to\infty}{\sim} e^0 \cdot(\text{lower-order terms})
\end{align}
In either case, both solutions (if analytic), would lie in the Bargmann space $\mathcal H$. Indeed, the solutions to the holonomic ideal are explicitly
\begin{align}
\psi_1(x) = \,_1F_1(a;b;x),~~~~~~~\psi_2(x) = \,U(a;b;x)
\end{align}
where here, $U$ denotes *Tricomi*'s confluent hypergeometric function. At large $|x|\to\infty$, Kummer's hypergeometric function indeed grows like $x^{a-b}e^x$, whereas Tricomi's function grows like $x^{-a}$ (which is indeed in the Bargmann space, when analytic), confirming our intuition from the local systems perspective.


*Test case: Hermite's equation (DOESN'T WORK)*

Let $n=1$, and consider the principal ideal in the Weyl algebra generated by $$\partial^2+x\partial +a.$$
In this case we get the following flat connection:
\begin{align}
A(x)dx &= \begin{pmatrix}0&-a\\ 1&-x\end{pmatrix}dx\underset{|x|\to \infty}{\sim}-x\bigg(\begin{pmatrix}0&0\\ 0&1\end{pmatrix} + O\bigg(\frac{1}{x}\bigg)\bigg)dx
\end{align}
In this case, $A^{(\alpha)}$ has two eigenspaces, one with zero eigenvalue, and one with eigenvalue $x$. So we would expect the following asymptotics:
\begin{align}
\psi_1(x) &\underset{|x|\to\infty}{\sim} e^\frac{-x^2}{2} \cdot(\text{lower-order terms})\\
\psi_2(x) &\underset{|x|\to\infty}{\sim} e^0 \cdot(\text{lower-order terms})
\end{align}
In this case, one of the solutions is unnormalizable, whereas the other one should be normalizable. So the solution space should be one-dimensional. In summary, based on the theory of $D$-modules, we have two analytic solutions (as $\text{Sing}(I) = 0$), and we should always lose one solution upon passing to the Bargmann space:
$$\dim \text{Ann}(I) = 2,~~~~~~~\dim \text{Ann}(I)\cap \mathcal H =1.$$


However, the solutions to the holonomic ideal are explicitly
\begin{align}
\psi_1(x) = \,_1F_1(a/4;1/2;x^2),~~~~~~~\psi_2(x) =x\,_1F_1(a/4+1/2;3/2;x^2)
\end{align}
In fact, both functions have the asymptotics of the type $O(e^{x^2})$ and are unnormalizable, violating our intuition which was based on the mapping to a local system. So in fact we lose *both* solutions:
$$\dim \text{Ann}(I) = 2,~~~~~~~\dim \text{Ann}(I)\cap \mathcal H =0.$$