the first conjugate potential $V^{[1]}$, (of which the left-hand-side $D^{[1]\pm}(-\e^{-\mi\varphi} E)$ are functionals). (Eq.(\[EQ\]) together with all its conjugates appear to form a determined system for the resolution of all the spectra $\{E_k^{[\ell]}\}$ at once.)

Here we will often invoke the specific results relating to the [*homogeneous*]{} potentials $$\label{Sch}
\hat H_N \defi -{\d^2 \over \d q^2}+q^N, \quad q\in[0,+\infty) \qquad 
N\ge 1 \mbox{ integer.}$$ The corresponding formulae are collated in greater detail separately in Appendix, and also in Table 1 below for $N=1,\ 4$.

Statement of the problem
------------------------

The previous exact analysis of Sec. 1.2 is controlled in an essential manner by the degree $N$ of the potential, quite sensibly since $q^N$ defines the most singular interaction term. Still, there are interesting transitional situations where this parameter $N$ can diverge or behave discontinuously while the quantum problem itself has a well-defined limit (example: the term $q^N$ is multiplied by a coupling constant $g \to 0$). Earlier studies of such problems show those limits to be very singular, and non-uniform over the energy range. An unexplored challenge is then to control the limiting behaviour of spectral functions, which are symmetric functions involving all eigenvalues simultaneously. The earlier analyses, which essentially work 