  Relaxation properties (specifically time-rates) of the Smoluchowski diffusion
process on a line, in a confining potential $ U(x) \sim x^m$, $m=2n \geq 2$,
can be spectrally quantified by means of the affiliated Schr\"{o}dinger
semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled)
motion generator $\hat{H}= - \Delta + {\cal{V}}(x)$ involves a potential
function ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$, which for
$m>2$ has a conspicuous higher degree (superharmonic) double-well form. For
each value of $m>2$, $ \hat{H}$ has the zero-energy ground state eigenfunction
$\rho _*^{1/2}(x)$, where $\rho _*(x) \sim \exp -[U(x)]$ stands for the
Boltzmann equilibrium pdf of the diffusion process. A peculiarity of $\hat{H}$
is that it refers to a family of quasi-exactly solvable Schr\"{o}dinger-type
systems, whose spectral data are either residual or analytically unavailable.
As well, no numerically assisted procedures have been developed to this end.
Except for the ground state zero eigenvalue and incidental trial-error
outcomes, lowest positive energy levels (and energy gaps) of $\hat{H}$ are
unknown. To overcome this obstacle, we develop a computer-assisted procedure to
recover an approximate spectral solution of $\hat{H}$ for $m>2$. This task is
accomplished for the relaxation-relevant low part of the spectrum. By admitting
larger values of $m$ (up to $m=104$), we examine the spectral "closeness" of
$\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta _{\cal{N}}$ in the
interval $[-1,1]$, known to generate the Brownian motion with two-sided
reflection.
