Consider the inverse spectral problem in the case of one dimensional Schrodinger equation with a periodic potential. It is well known how to restore the potential from a total set of spectral data (boundaries of zones=periodic eigenvalues and normalization constants). Obviously, the quasi-momentum (a Floquet exponent) should also be expressed in terms of spectral data. I'm searching for the analytical expression of the quasi-momentum, at least in the simplest case of the single-zone potential where
spectral data consist from three eigenvalues E0, E1, E2.
In the case of single-zone potential I have found the following answer (see book by J. Moser, Integrable systems and spectral theory, 1999, Ijevsk).
We define Floquet multiplier as follows $\psi_E(x+1)=\exp(w(E))\psi_E(x)$. In the case of single-zone potential we can express $w(E)$ through the conformal mapping (The Schwarz-Christoffel Transformation)
$w(E)=\int\limits_{E_0}^{E}(E-E')[-(E-E_0)(E-E_1)(E-E_2)]^{-1/2}$,
where $E'$ is defined by the spectrum $spec=\{E_0, E_1, E_2\}$. One of the ways to find $E'$ is to impose compatibility condition
$w(E_1)=w(E_2)$.
Now I have the next question. How I can effectively solve the following equation
$w(E|E_0, E_1, E_2)=w(E|E'_0, E'_1, E'_2)$
?
Is there any solution at all?
The Schwarz-Christoffel Transformation (nice explanation can be found here http://mathfaculty.fullerton.edu/mathews//c2003/SchwarzChristoffelMod.html) maps upper half-plane $Im E>0$ to a polygon in the complex $w$-plane.
The boundary of the polygon can be defined as follows. In the example above we first have half-line $(-\infty, w(E_0)=0$. At $w(E_0)=0$ we apply counter-clockwise rotation by $\pi/2$ and plot vertical interval $(0,w(E_1))$. At $w(E_1)$ we apply one more counter-clockwise rotation $\pi/2$ and plot horizontal interval $(w(E_1),w(E'))$ (Note, that $E'$ belongs to the forbidden band $E_1
One more comment. It happens that if we fix the period of the potential, it gives additional restrictions for a possible choice of spectral data. I.e. E_0, E_1, E_2 can not be choosen arbitrary for the potential with a period L. I write this comment since my first impression from Moser's book was opposite.
If we choose arbitrary E_0, E_1, E_2, then the period L is determined by
$w(E_1)=\int\limits_{E_0}^{E}(E-E')[-(E-E_0)(E-E_1)(E-E_2)]^{-1/2}$
as follows L=Pi / Im[w(E_1)]
This can be found in @article{marchenko1975characterization,
title={A characterization of the spectrum of Hill's operator},
author={Marchenko, Vladimir Alexandrovich and Ostrovskii, Iosif Vladimirovich},
journal={Matematicheskii Sbornik},
volume={139},
number={4},
pages={540--606},
year={1975},
publisher={Russian Academy of Sciences, Branch of Mathematical Sciences}
}
after Eq. (5.23) (in a slightly modified form).
If I understand correctly, single-zone potential is defined uniquely by E_0, E_1, E_2 up to a translation.
Single zone potential $V(x|E0,E1,E2)$ almost coincides with the $(E2-E1)[Cos(2 Pi x/L)+E0]$ (see attachment)
If we work with 1-periodic potential, possible values of $E1$ and $E2$ are given by the curve, plotted in the figure (see attachment). Without loss of generality, I choose $E0=0$. $E1$ has the maximal value $E1(max)=Pi^2$.
Another interesting thing is that the limit E1->E0 of the single-zone potential corresponds to the single-soliton potential. See gif-file attached as an example
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