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

We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials U(x) ∼ x m , m = 2n ⩾ 2. This is paralleled by a transformation of each mth diffusion generator L = DΔ + b(x)∇, and likewise the related Fokker-Planck operator L* = DΔ − ∇[b(x) ⋅], into the affiliated Schrödinger one Ĥ = − D Δ + V ( x ) . Upon a proper adjustment of operator domains, the dynamics is set by semigroups exp(tL), exp(tL*) and exp(−tĤ), with t ⩾ 0. The Feynman-Kac integral kernel of exp(−tĤ) is the major building block of the relaxation process transition probability density, from which L and L* actually follow. The spectral 'closeness' of the pertinent Ĥ and the Neumann Laplacian − Δ N in the interval is analyzed for m even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the m-family of operators Ĥ with a priori chosen V ( x ) ∼ x m , when Ĥ becomes spectrally 'close' to the Dirichlet Laplacian − Δ D for large m. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for Ĥ with a bistable-looking potential V ( x ) = a x 2 m − 2 − b x m − 2 , a , b , > 0 , m > 2 has been addressed.