Classically, Hardy’s inequality enables to estimate the spectral gap of a one-dimensional diffusion up to a factor belonging to $[1,4]$. The goal of this paper is to better understand the latter factor, at least in a symmetric setting. In particular, we will give an asymptotical criterion implying that its value is exactly 4. The underlying argument is based on a semi-explicit functional for the spectral gap, which is monotone in some rearrangement of the data. To find it will resort to some regularity...

We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on $0$-forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of ${\Delta }_{f,h}^{\left(0\right)}$ and solves efficiently the question of weakly resonant wells.