conceptually simple, but perhaps tricky at some points; then, we decided to put it in Appendix \[technical lemma\].

\[calculus\] Let $s:[0,a]\times [0,L]\to {{\bf R}}$ be a smooth, non-negative function such that $$s(0,t)=t,\quad s(r,0)=0, \quad s(r,L)=L \quad\text{and}\quad {\dfrac{{\partial}s}{\bdt}}(r,t)\doteq\theta(r,t)>0.$$ Assume that there exist smooth functions $p(r),q(r)$ with $p(r)^2+q(r)^2>0$ such that $$p(r)\cos(\frac{\pi}{L}s(r,t))+q(r)\sin(\frac{\pi}{L}s(r,t))=F(t)$$ where $F(t)$ depends only on $t$. Then $p$ and $q$ are constant and $
{\dfrac{{\partial}s}{\bdr}}=0
$ so that $$s(r,t)=t$$ for all $(r,t)$.

### End of proof of the equality case

Assume that equality holds. Then, if $u$ is an eigenfunction, we know that $u=u(t)$ by the discussion in and $u$ restricts to an eigenfunction on each level circle $\Sigma_r$ for the potential $A=H(t)\,dt$ above (see Fact 3 at the beginning of Section \[equalitycase\] and the second step above).

We assume that $\Phi^A$ is congruent to $\frac 12$ modulo integers. This is the most difficult case; in the other cases the proof is a particular case of this, it is simpler and we omit it.

Recall that each level set $\Sigma_r$ is a circle of length $L$ for all $r$, with metric $g=\theta(r,t)^2\,dt$. As the flux of $A$ is congruent to $\frac 12$ modulo integers, we see that there exist complex-valued functions 