at fixed $N$, need further development to handle such non-uniform regimes.

Here we will begin to gather some insight about this issue by examining three model problems at varying depths.

Exercise I: infinite square well as $N\to +\infty$ limit
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The infinite square-well potential over the interval $[-1,+1]$ can be realized as the $N\to +\infty$ limit of the homogeneous potential $V(q) \equiv |q|^N$. The solutions of the Schrödinger equation with this potential indeed approach those of the infinite square well, but the limiting behaviours are interestingly singular, and non-uniform with respect to the quantum number $k$ [@FB].

The $N = +\infty$ problem
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The limiting Schrödinger operator $\hat H_\infty$ is given by $V(q) \equiv 0$ in $[-1,+1]$ with Dirichlet boundary conditions at $q=\pm 1$. At infinite $N$, the spectrum of $|q|^N$ becomes explicit again, $\{E_k=(k+1)^2 {\pi^2 / 4}\}$, of growth order $\mu_\infty = {1 \over 2}$. As with finite $N$, the operator $\hat H_\infty$ splits into $\hat H_\infty^+$ (over the even eigenfunctions, labeled by even $k$) and $\hat H_\infty^-$ (over the odd eigenfunctions, labeled by odd $k$).

Here the immediately explicit spectral functions are the [*Fredholm*]{} determinants $\Delta^\pm$ of eq.(\[DFr\]) (they reduce to standard Weierstrass products), $$\label{Frinf}
\Delta_\infty^+(\lambda) = \cos 