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, $m \in \mathbb{N}^n$ 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 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 non-normalizable, 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: $$\dim \text{Ann}(I) = 2,~~~~~~~\dim \text{Ann}(I)\cap \mathcal H =0.$$