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We investigate signatures of convergence for a sequence of diffusionprocesses on a line, in conservative force fields stemming from superharmonicpotentials $U(x)\sim x^m$, $m=2n \geq 2$. This is paralleled by atransformation of each $m$-th diffusion generator $L = D\Delta + b(x)\nabla $,and likewise the related Fokker-Planck operator $L^*= D\Delta - \nabla [b(x)\,\cdot]$, into the affiliated Schr\"{o}dinger one $\hat{H}= - D\Delta +{\cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics isset by semigroups $\exp(tL)$, $\exp(tL_*)$ and $\exp(-t\hat{H})$, with $t \geq0$. The Feynman-Kac integral kernel of $\exp(-t\hat{H})$ is the major buildingblock of the relaxation process transition probability density, from which $L$and $L^*$ actually follow. The spectral "closeness" of the pertinent $\hat{H}$and the Neumann Laplacian $-\Delta_{\cal{N}}$ in the interval is analyzed for$m$ even and large. As a byproduct of the discussion, we give a detaileddescription of an analogous affinity, in terms of the $m$-family of operators$\hat{H}$ with a priori chosen ${\cal{V}}(x) \sim x^m$, when $ \hat{H}$ becomesspectrally "close" to the Dirichlet Laplacian $-\Delta_{\cal{D}}$ for large$m$. For completness, a somewhat puzzling issue of the absence of negativeeigenvalues for $\hat{H}$ with a bistable-looking potential ${\cal{V}}(x)=ax^{2m-2} - bx^{m-2}, a, b, >0, m>2$ has been addressed.

Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials