Brownian motion in trapping enclosures: Steep potential wells and false bistability of affiliated Schroedinger type systems.

Langevin and Fokker-Planck equations for the Brownian motion in steep (extremally anharmonic) potential wells of the form $U(x)= x^m/m, m=2n, n>1$ are interpreted as reliable approximations of the reflected Brownian motion in the interval, as the potential steepness grows indefinitely. We investigate a familiar transformation of the involved Fokker-Planck operator to the Hermitian (eventually self-adjoint) Schr\"{o}dinger - type one $-\Delta + {\cal{V}}$, with the two-well (bistable) potential ${\cal{V}}(x)= {\cal{V}}_m(x)= (x^{m-2}/2) [(x^{m}/2) + (1-m)]$. We analyze and resolve somewhat puzzling issue of the absence of negative eigenvalues in such looking-bistable dynamical systems, and that of the existence of spectrally isolated zero-energy ground states, whose squares actually coinicide with Boltzmann-type equilibria $\rho _*(x) \sim \exp (-x^m/m)$ of the related steep-well Langevin (Fokker-Planck) problems. Limits of validity of the spectral "closeness" of $-\Delta + {\cal{V}}$ (with $m$ large) and the Neumann Laplacian $(-\Delta)_{\cal{N}}$ in the interval are established.