Spectral surfaces for families of Schrödinger operators

Let $A$ be a self-adjoint operator with the simple discrete spectrum $a_0<a_1<\dots<a_n<\dots$,
$$ A\varphi_n=a_n\varphi_n,\quad n\in\mathbb Z;\quad a_n\to\infty, $$
and $B$ be an operator subordinate to $A$. Then there arises the spectral surface
$$ S=\{(z,E)\in\mathbb C^2|\, (Az+B)f=Ef \text{ for some } f\neq 0\text{ in }H\}. $$
In which disk $|z|<R_n$ is the branch $E_n(z)$, $E_n(0)=a_n$ well-defined? Is the surface $S$ reducible?
We will discuss these and related questions in the case of
(a) Schrödinger–Hill operator $ Ly=-y''+v(x)y,\quad 0\leqslant x\leqslant 2\pi, \quad v(x+2\pi)=v(x) $ and
(b) (an)harmonic operator and its perturbations $ My=(-y''+q(x)y)+zw(x)y,\quad x\in\mathbb R. $
The works by C. Bender and T. Wu, and A. Eremenko and A. Gabrielov were the starting point for us.
The lecture is based on (joint) results of the speaker, Pl. Djakov, J. Adduci, P. Siegl, J. Viola.