the fundamental identities $$\label{ID}
D^-(\lambda ) \equiv \psi_\lambda (0), \qquad 
D^+(\lambda ) \equiv -\psi'_\lambda (0) ,$$ (also valid for a rescaled potential, i.e., $V(q)= v q^N+ \cdots$).

### The main functional relation [@V0; @V6]

Jointly with the original problem (\[Schr\]), its set of “conjugate" equations is defined by means of complex rotations, as ([@S], ch.2) $$\label{Cnj}
V^{[\ell]}(q) \defi \e^{-\mi\ell\varphi} V(\e^{-\mi\ell\varphi/2}q), \quad
\lambda^{[\ell]} \defi \e^{-\mi\ell\varphi} \lambda, \quad
\varphi \defi {4\pi \over N+2}$$ where $\ell=0,1,\cdots,L-1 \ [{\rm mod}\ L]$ labels the distinct conjugates, of total number $$\label{ELL}
L=N+2 \quad \mbox{in general}, \qquad
L={N \over 2}+1 \quad \mbox{for even polynomial potentials.}$$

The main result to be used throughout is the [*Wronskian identity*]{}, which states a bilinear functional relation between the spectral determinants $D^\pm(\lambda )$ and those of the first conjugate equation, namely $D^{[1]\pm}(\e^{-\mi\varphi} \lambda)$: $$\label{DW}
\e^{+\mi\varphi/4} D^{[1]+}( \e^{-\mi\varphi} \lambda)
D^{[0]-}( \lambda)
-\e^{-\mi\varphi/4} D^{[0]+}( \lambda) 
D^{[1]-}( \e^{-\mi\varphi} \lambda) \equiv 
2 \mi \e^{\mi\varphi\beta_{-1}(0)/2}.$$ It entails an [*exact quantization formula*]{} for the eigenvalues $E_k$, $$\label{EQ}
{2 \over \pi} \arg D^{[1]\pm}(-\e^{-\mi\varphi} E)_{E=E_k} 
- {\varphi \over \pi} \beta_{-1}(0) =
 k+{1 \over 2} \pm {N-2 \over 2(N+2)} 
\qquad \mbox{for } k= \textstyle{0,2,4,\ldots \atop 1,3,5,\ldots}$$ i.e., this condition determines the spectrum $\{E_k\}$ exactly in terms of the spectrum $\{E_k^{[1]}\}$ of 