  Relaxation properties of the Smoluchowski diffusion process on a line, in a
force field $ - \nabla U(x) \sim - x^{m-1}$, $m=2n \geq 2$, can be spectrally
quantified in terms of the Schr\"{o}dinger semigroup $\exp (-t\hat{H})$, $t\geq
0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - \Delta +
{\cal{V}}(x)$, where ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$,
for $m>2$ defines a family of higher degree (we call them superharmonic)
double-well systems. $\hat{H}$ has a zero-energy ground state function, which
is a square root $\rho_*^{1/2}(x)$ of the asymptotic stationary probability
density function of the equilibrating diffusion process. Since lowest
eigenvalues of the non-negative operator $\hat{H}$ are expected to determine
relaxation time rates of the process in question, the knowledge of them is
mandatory to justify or refute the spectral relaxation scenario. However, with
a notable exception of $m=2$, $\hat{H}$ refers to Schr\"{o}dinger-type systems
whose analytical and numerically accessible spectral data are either residual
or unavailable at all. With this perspective in mind, we address the spectral
solution problem for $\hat{H}$, provided $U(x)= x^m$ or $m x^m$, for $m>2$. Our
focus is on a detailed analysis of the (conjectured) spectral "closeness" of
$\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta_{\cal{N}}$ in the
interval $[-1,1]$. Intricasies of the $m \to \infty $ behavior are addressed.
