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We study a Schr\"odinger-like equation for the anharmonic potential $x^{2 \alpha}+\ell(\ell+1) x^{-2}-E$ when the anharmonicity $\alpha$ goes to $+\infty$. When $E$ and $\ell$ vary in bounded domains, we show that the spectral determinant for the central connection problem converges to a special function written in terms of a Bessel function of order $\ell+\frac{1}{2}$ and its zeros converge to the zeros of that Bessel function. We then study the regime in which $E$ and $\ell$ grow large as well, scaling as $E\sim \alpha^2 \varepsilon^2$ and $\ell\sim \alpha p$. When $\varepsilon$ is greater than $1$ we show that the spectral determinant for the central connection problem is a rapidly oscillating function whose zeros tend to be distributed according to the continuous density law $\frac{2p}{\pi}\frac{\sqrt{\varepsilon^2-1}}{\varepsilon}$. When $\varepsilon$ is close to $1$ we show that the spectral determinant converges to a function expressed in terms of the Airy function $\operatorname{Ai}(-)$ and its zeros converge to the zeros of that function. This work is motivated by and has applications to the ODE/IM correspondence for the quantum KdV model.

We consider a 3-dimensional Pfaffian system, whose z-component is a differential system with irregular singularity at infinity and Fuchsian at zero. In the first part of the paper, we prove that its Frobenius integrability is equivalent to the sixth Painlev\'e equation PVI. The coefficients of the system will be explicitly written in terms of the solutions of PVI. In this way, we remake a result of [44, 61]. We then express in terms of the Stokes matrices of the 3x3 irregular system the monodromy invariants p_{jk}=Tr(M_jM_k) of the 2-dimensional isomonodromic Fuchsian system with four singularities, traditionally associated to PVI [23, 55] and used to solve the non-linear connection problem. Several years after [44, 61], the authors of [14] showed that the computation of the monodromy data of a class of irregular systems may be facilitated in case of coalescing eigenvalues. This coalescence corresponds to the critical points (fixed singularities) of PVI. In the second part of the paper, we classify the branches of PVI transcendents holomorphic at a critical point such that the analyticity and semisimplicity properties described in [14] are satisfied, and we compute the associated Stokes matrices and the invariants p_{jk}. Finally, we compute the monodromy data parametrizing the chamber of a 3-dim Dubrovin-Frobenius manifold associated with a transcendent holomorphic at x=0., Comment: 69 pages, 1 table, 1 figure