# Mc’Kean’s transformation for 3-rd order operators

Andrey Badanin  and  Evgeny L. Korotyaev  Saint-Petersburg State University,
Universitetskaya nab. 7/9, St. Petersburg, 199034 Russia,
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider a non-self-adjoint third order operator
$(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy$ with 1-periodic coefficients
$p,q$. This operator is the L-operator in the Lax pair for the good Boussinesq
equation on the circle. In 1981, McKean introduced a transformation that
reduces the spectral problem for this operator to a spectral problem for the
Hill operator with a potential that depends analytically on the energy. In the
present paper we are studying this transformation.

###### Key words and phrases:

3-order operator, eigenvalues, asymptotics

###### 1991 Mathematics Subject Classification:

47E05, 34L20, 34L40

##  1\. Introduction and main results

###  1.1. Introduction

We consider a non-self-adjoint operator ${\mathcal{L}}={\mathcal{L}}(\psi)$
acting on $L^{2}({\mathbb{R}})$ and given by

${\mathcal{L}}y=(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy,$ (1.1)

where the 1-periodic coefficients $p,q$ satisfy

$\psi=(p,q)\in{\mathcal{H}}\oplus{\mathcal{H}},$

the Hilbert space
${\mathcal{H}}=L_{\mathbb{R}}^{2}({\mathbb{T}}),{\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}$,
equipped with the norm $\|f\|^{2}=\int_{0}^{1}|f(x)|^{2}dx$. Let
${\mathcal{H}}_{\mathbb{C}}$ be the complexification of the real space
${\mathcal{H}}$. The operator is defined on the domain

$\displaystyle\mathop{\mathrm{Dom}}\nolimits({\mathcal{L}})=\Big{\\{}y\in
L^{2}({\mathbb{R}}):(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy\in
L^{2}({\mathbb{R}}),y^{\prime\prime},(y^{\prime\prime}+py)^{\prime}\in
L^{1}({\mathbb{R}})\Big{\\}}.$ (1.2)

The operator ${\mathcal{L}}$ is the $L$-operator in the Lax pair for the good
Boussinesq equation on the circle. The inverse problem for this operator was
studied by McKean’s [McK81]. The solution of the inverse problem uses a
transformation that reduces the operator ${\mathcal{L}}$ to the Schrodinger
operator with a periodic energy-dependent potential. Studying this
transformation is the main goal of the present work.

Recall that in order to recover the potential of the Schrödinger operator with
a periodic potential, three sequences are required:

1) Sequence of eigenvalues of a 2-periodic problem.

2) Sequence of eigenvalues of the Dirichlet problem.

3) Sequence of +/- signs.

We showed in [BK24] that in order to recover the coefficients of the operator
${\mathcal{L}}$, three sequences are also required:

1) Sequence of branch points of the Riemann surface of the operator.

2) Sequence of eigenvalues of the 3-point Dirichlet problem.

3) Sequence of +/- signs.

We prove in the present paper that in the case of small coefficients McKean’s
transformation establishes a correspondence between these three sequences for
the operator ${\mathcal{L}}$ and the Schrödinger operator with an energy-
dependent potential.

Note that since our operator is non-self-adjoint, the branch points of the
Riemann surface and the spectrum of the 3-point problem can be non-real. The
spectra of the 2-periodic problem and the Dirichlet problem for the
Schrodinger operator with an energy-dependent potential can also be complex.
In our article [BK21] we proved that in the case of a sufficiently small
potential the spectrum of the Schrödinger operator is real. Using this fact,
in the present paper we prove that for small coefficients the branch points of
the Riemann surface and the spectrum of the 3-point problem of the operator
${\mathcal{L}}$ are real.

The inverse spectral problem for the Schrödinger operator with the periodic
potential in the class of potentials from $L^{2}({\mathbb{T}})$ is proved by
Marchenko and Ostrovskii [MO75] and Korotyaev [K99]. Some inverse results for
the 3-order operator ${\mathcal{L}}$ are obtained by McKean [McK81]. There,
for the case of small smooth coefficients, a uniqueness theorem for recovering
the potential from spectral data was proved. The problem in the certain
classes of non-smooth coefficients is more difficult. Some results in this
direction are presented in our article [BK24].

###  1.2. Second order operators

Recall the results for the Schrödinger operator $-f^{\prime\prime}+Vf$ with
the potential $V\in{\mathcal{H}}$, acting on $L^{2}({\mathbb{R}})$. The
spectrum is pure absolutely continuous and consists of the intervals
$[E_{n-1}^{+},E_{n}^{-}]$ separated by gaps, where
$E_{0}^{+}<E_{1}^{-}\leqslant E_{1}^{+}<...$ are the eigenvalues of the
2-periodic problem $f(0)=f(2),f^{\prime}(0)=f^{\prime}(2)$. The spectrum of
the Dirichlet problem $f(0)=f(1)=0$ consists of the simple eigenvalues
$\mathfrak{m}_{1}<\mathfrak{m}_{2}<...$. This eigenvalues are zeros of the
entire function $\varphi(1,\cdot)$, where $\varphi(x,\lambda)$ is the
fundamental solution of the equation

$-f^{\prime\prime}+Vf=\lambda f,$ (1.3)

satisfying the initial conditions
$\varphi|_{x=0}=0,\varphi^{\prime}|_{x=0}=1$. The eigenvalues satisfy
$\mathfrak{m}_{n}\in[E_{n}^{-},E_{n}^{+}]$ for all $n\in{\mathbb{N}}$.
Introduce the norming constants
$\mathfrak{h}_{sn}:{\mathcal{H}}\to{\mathbb{R}}$ by

$\mathfrak{h}_{sn}(V)=2\pi
n\log|\varphi^{\prime}(1,\mathfrak{m}_{n}(V))|,\quad\log 1=0,\quad
n\in{\mathbb{N}}.$ (1.4)

Introduce the space $S$ of all real, strictly increasing sequences
$(s_{1},s_{2},...)$ such that $(s_{n}-(\pi n)^{2})\in\ell^{2}$. The following
result holds true, see, e.g., [PT87].

The mapping $V\to(\mathfrak{m}_{n}(V),\mathfrak{h}_{sn}(V))$ is a real
analytic isomorphism between ${\mathcal{H}}$ and $S\oplus\ell^{2}$.

Remark. We solve the corresponding inverse problem for 3-rd order operator
under the 3-point Dirichlet conditions in [BK24x].

The inverse problem of recovering the potential $V$ from the periodic spectrum
was solved by Korotyaev [K99]. Introduce the mapping
$\mathfrak{g}:{\mathcal{H}}\to\ell^{2}\oplus\ell^{2}$ from [K99] by

$\mathfrak{g}(V)=(\mathfrak{g}_{n}(V))_{n\in{\mathbb{N}}},\quad
V\in{\mathcal{H}},$ (1.5)

where

$\mathfrak{g}_{n}=(\mathfrak{g}_{cn},\mathfrak{g}_{sn})\in{\mathbb{R}}^{2},\quad|\mathfrak{g}_{n}|={E_{n}^{+}-E_{n}^{-}\over
2},$

the vectors $(\mathfrak{g}_{cn},\mathfrak{g}_{sn})$ are given by

$\mathfrak{g}_{cn}={E_{n}^{+}+E_{n}^{-}\over 2}-\mathfrak{m}_{n},\quad$ (1.6)
$\mathfrak{g}_{sn}=\Big{|}{(E_{n}^{+}-E_{n}^{-})^{2}\over
4}-\mathfrak{g}_{cn}^{2}\Big{|}^{1\over
2}\mathop{\mathrm{sign}}\nolimits\mathfrak{h}_{sn}.$ (1.7)

Introduce the Fourier transformations
$\Phi_{cn},\Phi_{sn}:{\mathcal{H}}\to{\mathbb{R}},n\in{\mathbb{N}}$, by

$f_{cn}=\Phi_{cn}f=\int_{0}^{1}f(x)\cos 2\pi nxdx,\ \
f_{sn}=\Phi_{sn}f=\int_{0}^{1}f(x)\sin 2\pi nxdx.$

and $\Phi_{n}=(\Phi_{cn},\Phi_{sn})_{n\in{\mathbb{N}}}.$ The following results
were proved by Korotyaev [K99].

The mapping $\mathfrak{g}$, given by (1.5)–(1.7), is a real analytic
isomorphism between ${\mathcal{H}}$ and $\ell^{2}\oplus\ell^{2}$. Moreover, if
$n\to+\infty$, then the functions $\mathfrak{g}_{n}$ satisfy

$\mathfrak{g}_{n}(V)=(V_{cn},V_{sn})+O(n^{-1}),\qquad{\partial\mathfrak{g}_{n}\over\partial
V}=\Phi_{n}+O(n^{-1}).$

Remark. We solve the corresponding inverse problem for 3-rd order operator in
[BK24].

###  1.3. McKean’s transformation

Consider the differential equation

$(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy=\lambda
y,\qquad\lambda\in{\mathbb{C}},$ (1.8)

in the class

$C^{[2]}({\mathbb{R}})=\\{y:y,y^{\prime},y^{[2]}\in
AC_{loc}({\mathbb{R}})\\},$

where

$y^{[2]}=y^{\prime\prime}+py.$ (1.9)

McKean’s transformation is based on the following simple result.

######  Lemma 1.1.

Let $p,q\in L_{loc}^{1}({\mathbb{R}}),\lambda\in{\mathbb{C}}$. Assume that Eq.
(1.8) has a solution $\eta\in C^{[2]}({\mathbb{R}})$ such that $\eta(x)\neq 0$
for all $x\in{\mathbb{R}}$. Let, in addition, $y\in C^{[2]}({\mathbb{R}})$ be
a solution to Eq. (1.8). Let the function $f$ has the form

$f=\eta^{3\over
2}\Big{(}{y\over\eta}\Big{)}^{\prime}={y^{\prime}\eta-y\eta^{\prime}\over\sqrt{\eta}}.$
(1.10)

Then $f,f^{\prime},f^{\prime\prime}+{p\over 2}f\in AC_{loc}({\mathbb{R}})$,
and $f$ satisfies the equation

$-f^{\prime\prime}+Vf=Ef,$ (1.11)

where

$E={3z^{2}\over 4}={3\lambda^{2\over 3}\over 4},\quad\lambda=\Big{(}{4\over
3}E\Big{)}^{3\over 2},\quad z=\lambda^{1\over
3},\quad\mathop{\mathrm{arg}}\nolimits z\in(-{\pi\over 3},{\pi\over
3}],\quad\mathop{\mathrm{arg}}\nolimits\lambda\in(-\pi,\pi],$ (1.12)

the energy-dependent potential $V$ has the form

$V=E-{p\over 2}-{3\over 2}{\eta^{[2]}\over\eta}+{3\over
4}\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}.$ (1.13)

Moreover, $V+{p\over 2}\in AC_{loc}({\mathbb{R}})$.

Remark. 1) In the unperturbed case $\psi=0$ we have $V=0$.

2) If the coefficients $p,q$ are 1-periodic and
$\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$, then the potential $V$ is also
1-periodic with respect to the variable $x$, and $V+{p\over 2}\in
AC({\mathbb{T}})$.

3) Eq. (1.11) was the subject of our article [BK21].

###  1.4. Ramifications

Now we consider the periodic coefficients. Let
$\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$. Introduce the fundamental solutions
$\varphi_{1},\varphi_{2},\varphi_{3}$ of Eq. (1.8) satisfying the conditions
$\varphi_{j}^{[k-1]}|_{x=0}=\delta_{jk}$, $j,k=1,2,3$, where $y^{[0]}=y$,
$y^{[1]}=y^{\prime}$, $y^{[2]}=y^{\prime\prime}+py$. Each of the functions
$\varphi_{j}(x,\cdot),j=1,2,3,x\in[0,1]$ is entire and real on ${\mathbb{R}}$.
We define the monodromy matrix by

$M(\lambda)=\big{(}\varphi_{j}^{[k-1]}(1,\lambda)\big{)}_{j,k=1}^{3}.$ (1.14)

The matrix-valued function $M$ is entire. The characteristic polynomial $D$ of
the monodromy matrix is given by

$D(\tau,\lambda)=\mathop{\mathrm{det}}\nolimits(M(\lambda)-\tau
1\\!\\!1_{3}),\quad(\tau,\lambda)\in{\mathbb{C}}^{2},$ (1.15)

here and below $1\\!\\!1_{N}$ is the $N\times N$ identity matrix. An
eigenvalue of $M$ is called a multiplier, it is a zero of the polynomial
$D(\cdot,\lambda)$. Each $M$ has exactly $3$ (counting with multiplicities)
multipliers $\tau_{j},j=1,2,3$, which satisfy $\tau_{1}\tau_{2}\tau_{3}=1.$ In
particular, each $\tau_{j}\neq 0$ for all $\lambda\in{\mathbb{C}}$.

Introduce the Floquet solutions $\mathfrak{f}_{k},k=1,2,3$, of Eq. (1.8)
satisfying the conditions

$\mathfrak{f}_{k}(x+1,\lambda)=\tau_{k}(\lambda)\mathfrak{f}_{k}(x,\lambda),\qquad\mathfrak{f}_{k}(0,\lambda)=1.$
(1.16)

Three roots $\tau_{1},\tau_{2}$ and $\tau_{3}$ of the equation
$D(\cdot,\lambda)=0$ constitute three distinct branches of some analytic
function $\tau$ of the variable $\lambda$ that has only algebraic
singularities in ${\mathbb{C}}$. Thus the function $\tau$ is analytic on a
3-sheeted Riemann surface ${\mathcal{R}}$, Fig. 3 c), see [BK24xxx]. Moreover,
for each $x\in{\mathbb{R}}$ the functions
$\mathfrak{f}_{1}(x,\cdot),\mathfrak{f}_{2}(x,\cdot),\mathfrak{f}_{3}(x,\cdot)$
constitute three branches of a function meromorphic on the surface
${\mathcal{R}}$.

There are only a finite number of the algebraic singularities (ramifications)
in any bounded domain. In order to describe these points we introduce the
discriminant $\rho$ of the polynomial $D(\cdot,\lambda)$ by

$\rho=(\tau_{1}-\tau_{2})^{2}(\tau_{1}-\tau_{3})^{2}(\tau_{2}-\tau_{3})^{2}.$
(1.17)

The function $\rho(\lambda)$ is entire. A zero of $\rho$ is a ramification of
the Riemann surface ${\mathcal{R}}$.

If $\psi=0$, then the function $\rho$ and its zeros $r_{n}^{0,\pm}$ have the
form

$\rho_{0}=64\sin^{2}{\sqrt{3}z\over 2}\sin^{2}{\sqrt{3}\omega z\over
2}\sin^{2}{\sqrt{3}\omega^{2}z\over 2},\quad
r_{n}^{0,+}=r_{n}^{0,-}=\Big{(}{2\pi n\over\sqrt{3}}\Big{)}^{3},\ \
n\in{\mathbb{Z}},\quad\omega=e^{i{2\pi\over 3}}.$ (1.18)

If the coefficients $p,q$ are 1-periodic and
$\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$, then the potential $V$, given by
(1.13), is also 1-periodic with respect to the variable $x$, and $V+{p\over
2}\in AC({\mathbb{T}})$. Consider the 2-periodic problem

$f(2)=f(0),\qquad f^{\prime}(2)=f^{\prime}(0),$ (1.19)

for Eq. (1.11). Let each of the potentials $V(x,\cdot),x\in{\mathbb{R}}$, be
analytic on some neighborhood of $E_{o}\in{\mathbb{C}}$ and let the problem
(1.11), (1.19) with $E=E_{o}$ has a non-trivial solution. Then $E_{o}$ is an
eigenvalue of this problem, see [BK21].

We prove the following result about the ramification.

######  Theorem 1.2.

Let $\psi\in{\mathcal{H}}_{\mathbb{C}}^{2}$. Assume that there exists a simple
multiplier $\tau$ for Eq. (1.8) with some $\lambda_{o}\in{\mathbb{C}}$ and the
corresponding Floquet solution $\eta\in C^{[2]}({\mathbb{R}})$ does not vanish
for all $x\in{\mathbb{R}}$. Let McKean’s transformation be given by
(1.10)–(1.13). Then each of the potentials $V(x,\cdot),x\in{\mathbb{R}}$, is
analytic on some neighborhood of $E_{o}={3\over 4}\lambda_{o}^{2\over 3}$.
Moreover, $\lambda_{o}$ is a ramification of Eq. (1.8) iff $E_{o}$ is an
eigenvalue of the 2-periodic problem (1.11), (1.19).

Remark. The idea of proving this result was proposed by McKean [McK81].
Following [McK81], in the proof we use two substitutions in (1.10): we take
$g=\mathfrak{f}_{1}$ in the first substitution, and we take
$g=\mathfrak{f}_{2}$ in the second one, here
$\mathfrak{f}_{1},\mathfrak{f}_{2}$ are the Floquet solutions corresponding
the multipliers $\tau_{1},\tau_{2}$, respectively.

###  1.5. Three-point Dirichlet problem

We also consider the operator ${\mathcal{L}}_{dir}$ acting on $L^{2}(0,2)$ and
given by

${\mathcal{L}}_{dir}y=(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy,\qquad
y(0)=y(1)=y(2)=0.$ (1.20)

The spectrum $\sigma({\mathcal{L}}_{dir})$ of ${\mathcal{L}}_{dir}$ is pure
discrete and satisfies

$\sigma({\mathcal{L}}_{dir})=\\{\lambda\in{\mathbb{C}}:{\mathscr{B}}(\lambda)=0\\},$
(1.21)

where ${\mathscr{B}}$ is the entire function given by

${\mathscr{B}}(\lambda)=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}\varphi_{2}(1,\lambda)&\varphi_{3}(1,\lambda)\\\
\varphi_{2}(2,\lambda)&\varphi_{3}(2,\lambda)\end{pmatrix}.$ (1.22)

In the unperturbed case $\psi=0$ the function ${\mathscr{B}}$ has the form

${\mathscr{B}}_{0}(\lambda)=-{8\over 3\sqrt{3}\lambda}\sin{\sqrt{3}z\over
2}\sin{\sqrt{3}\omega z\over 2}\sin{\sqrt{3}\omega^{2}z\over 2}.$ (1.23)

All eigenvalues are simple, real, and have the form

$\mu_{n}^{o}=\Big{(}{2\pi n\over\sqrt{3}}\Big{)}^{3},\quad
n\in{\mathbb{Z}}\setminus\\{0\\}.$ (1.24)

Consider the Dirichlet problem

$f(0)=f(1)=0,$ (1.25)

for Eq. (1.11). Let each of the potentials $V(x,\cdot),x\in{\mathbb{R}}$, be
analytic on some neighborhood of $E_{o}\in{\mathbb{C}}$ and let the problem
(1.11), (1.25) with $E=E_{o}$ has a non-trivial solution. Then $E_{o}$ is an
eigenvalue of this problem, see [BK21].

Consider the transpose operator

$\widetilde{\mathcal{L}}y=-(y^{\prime\prime}+py)^{\prime}-py^{\prime}+qy.$
(1.26)

Note that the operators
${\mathcal{L}}(\psi)=-\widetilde{\mathcal{L}}(\psi_{*})$ and
$-{\mathcal{L}}(\psi_{*}^{-})$ are unitarily equivalent, where

$\psi_{*}=(p,-q),\quad\psi^{-}(x)=\psi(1-x),\quad\psi_{*}^{-}(x)=\psi_{*}(1-x),\quad
x\in{\mathbb{R}}.$ (1.27)

In particular, if
$y(\cdot,\lambda,\psi),(\lambda,\psi)\in{\mathbb{C}}\times{\mathcal{H}}\oplus{\mathcal{H}}$,
is a solution of Eq. (1.8) for some $\lambda\in{\mathbb{C}}$, then
$\tilde{y}(\cdot,\lambda,\psi)=y(\cdot,-\lambda,\psi_{*})$ is a solution of
the equation

$-(\tilde{y}^{\prime\prime}+p\tilde{y})^{\prime}-p\tilde{y}^{\prime}+q\tilde{y}=\lambda\tilde{y}.$
(1.28)

The function $\tilde{y}=y^{\prime}\mathfrak{f}_{3}-y\mathfrak{f}_{3}^{\prime}$
in the numerator of the substitution (1.10) satisfies Eq. (1.28), see Lemma
2.2.

We also consider the operator $\widetilde{\mathcal{L}}_{dir}$ which is a
restriction of the operator $\widetilde{\mathcal{L}}$ onto the functions
satisfying the 3-point conditions $y(0)=y(1)=y(2)=0$.

We prove the following result about the 3-point eigenvalue.

######  Theorem 1.3.

Let $\psi\in{\mathcal{H}}_{\mathbb{C}}^{2}$. Assume that there exists a simple
multiplier $\tau$ for Eq. (1.8) with some $\lambda\in{\mathbb{C}}$ and the
corresponding Floquet solution $\eta\in C^{[2]}({\mathbb{R}})$ does not vanish
for all $x\in{\mathbb{R}}$. Let McKean’s transformation be given by the
identities (1.10)–(1.13).

Let $\lambda$ be an eigenvalue of the 3-point Dirichlet problem for Eq. (1.28)
and let $\tilde{y}$ be the corresponding eigenfunction. If, in addition,
$\eta^{[2]}(0)\neq 0$, then $E={3\over 4}\lambda^{2\over 3}$ is an eigenvalue
of the Dirichlet problem (1.11), (1.25) and $f={\tilde{y}\over\sqrt{\eta}}$ is
the corresponding eigenfunction.

Conversely, let $E$ be an eigenvalue of the Dirichlet problem (1.11), (1.25)
and let $f$ be the corresponding eigenfunction. Then $\lambda=({4\over
3}E)^{3\over 2}$ is an eigenvalue of the 3-point Dirichlet problem for Eq.
(1.28) and $\tilde{y}=f\sqrt{\eta}$ is the corresponding eigenfunction.

Remark. In the proof we use the substitution (1.10), where
$y^{\prime}\eta-y\eta^{\prime}$ is the eigenfunction $\widetilde{y}$ of the
3-point problem for Eq. (1.28).

###  1.6. Localization of the ramifications

Theorems 1.2 and 1.3 prove that McKean’s transformation introduced in Lemma
1.1 transforms the ramification to the eigenvalue of the 2-periodic problem,
and the 3-point eigenvalue to the eigenvalue of the Dirichlet problem for the
Schrödinger operator. Next we consider the case of small coefficients. In this
case, we prove a global one-to-one correspondence between the set of positive
(excluding $r_{0}^{\pm}$) ramifications and the set of 2-periodic eigenvalues
of the Schrödinger operator, and a similar correspondence between the set of
positive 3-point eigenvalues and the set of the Dirichlet eigenvalues for the
Schrödinger operator. The same correspondences are established for negative
ramifications and 3-point eigenvalues. Moreover, in this case it is possible
to prove the reality of ramifications (except $r_{0}^{\pm}$) and 3-point
eigenvalues.

Introduce the domains ${\mathcal{D}}_{n},n\in{\mathbb{Z}}$, by

${\mathcal{D}}_{0}=\Big{\\{}\lambda\in{\mathbb{C}}:|\lambda|<1\Big{\\}},\ \
{\mathcal{D}}_{n}=\Big{\\{}\lambda\in{\mathbb{C}}:\big{|}z-{2\pi
n\over\sqrt{3}}\big{|}<1\Big{\\}},\ \
{\mathcal{D}}_{-n}=\big{\\{}\lambda\in{\mathbb{C}}:-\lambda\in{\mathcal{D}}_{n}\big{\\}},\
\ n\in{\mathbb{N}}.$ (1.29)

Always below $\varepsilon_{0}>0$ is a fixed small enough number. Introduce the
ball

${\mathcal{B}}(\varepsilon_{0})=\\{\psi\in{\mathcal{H}}\oplus{\mathcal{H}}:\|\psi\|<\varepsilon\\},$

and the corresponding ball ${\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$ in
${\mathcal{H}}_{\mathbb{C}}\oplus{\mathcal{H}}_{\mathbb{C}}$. In [BK24xxx] and
[BK24xx] we prove the following results.

Let $\psi\in{\mathcal{B}}(\varepsilon_{0})$. Then

i) There are exactly two (counting with multiplicity) ramifications
$r_{n}^{\pm}$ in each domain ${\mathcal{D}}_{n},n\in{\mathbb{Z}}$ and there
are no ramifications in
${\mathbb{C}}\setminus\cup_{n\in{\mathbb{Z}}}{\mathcal{D}}_{n}$. Moreover,

$r_{n}^{\pm}(\psi)=-r_{-n}^{\mp}(\psi_{*})=-r_{-n}^{\mp}(\psi_{*}^{-})=r_{n}^{\pm}(\psi^{-})\quad\quad\forall\
\ n\in{\mathbb{Z}}.$ (1.30)

ii) There exists the multiplier $\tau_{3}$, satisfying the estimate

$|\tau_{3}(\lambda)-e^{z}|\leqslant{C\|\psi\|\over|z|}|e^{z}|,$ (1.31)

for all $\lambda\in{\mathscr{D}}_{3}$ and for some $C>0$, where

${\mathscr{D}}_{3}={\mathbb{C}}_{+}\setminus\cup_{n\in\\{0\\}\cup{\mathbb{N}}}\overline{{\mathcal{D}}_{-n}}.$
(1.32)

Moreover, if $\lambda\in{\mathscr{D}}_{3}$, then $\tau_{3}(\lambda)$ is the
simple multiplier and the corresponding Floquet solution
$\mathfrak{f}_{3}(x,\lambda)$ does not vanish for all $x\in{\mathbb{R}}$. Each
function $\tau_{3}$ and $\mathfrak{f}_{3}(x,\cdot),x\in{\mathbb{R}}$, is
analytic on ${\mathscr{D}}_{3}$, $\tau_{3}$ is real for real $\lambda$, and
satisfies

$\tau_{3}(\lambda)>0,\quad\forall\ \ \lambda>1.$ (1.33)

Introduce images ${\mathcal{S}}$ and ${\mathcal{S}}_{n}$ of the domains
${\mathscr{D}}_{3}$ and ${\mathcal{D}}_{n}$ in the $E$-plane by

${\mathcal{S}}=\Big{\\{}E\in{\mathbb{C}}:\lambda=\Big{(}{4\over
3}E\Big{)}^{3\over 2}\in{\mathscr{D}}_{3}\Big{\\}},$ (1.34)

and

${\mathcal{S}}_{n}=\Big{\\{}E\in{\mathbb{C}}:\lambda=\Big{(}{4\over
3}E\Big{)}^{3\over
2}\in{\mathcal{D}}_{n}\Big{\\}}=\Big{\\{}E\in{\mathbb{C}}:|\sqrt{E}-\pi
n|<{\sqrt{3}\over 2}\Big{\\}},\quad n\in{\mathbb{N}}.$ (1.35)

Due to the statement iii), if $\psi\in{\mathcal{B}}(\varepsilon_{0})$, then
for any $\lambda\in{\mathscr{D}}_{3}$ we may take $\eta=\mathfrak{f}_{3}$ in
McKean’s transformation (1.10)–(1.13). Then the potential
$V(x,E),x\in{\mathbb{R}}$, is analytic with respect to $E$ on the domain
${\mathcal{S}}$.

In order to consider the Schrödinger operator with the energy-dependent
potential we assume $p^{\prime},q\in L^{2}({\mathbb{T}})$. Introduce the
Sobolev space ${\mathcal{H}}_{1}=H^{1}({\mathbb{T}})$. Introduce the norm
$\|\psi\|_{1}$ in the space ${\mathcal{H}}_{1}\oplus{\mathcal{H}}$ by

$\|\psi\|_{1}^{2}=\|p\|^{2}+\|p^{\prime}\|^{2}+\|q\|^{2}.$

Let ${\mathcal{H}}_{1,{\mathbb{C}}}$ be the complexification of the real space
${\mathcal{H}}_{1}$. Introduce the ball

${\mathcal{B}}_{1}(\varepsilon_{0})=\\{\psi\in{\mathcal{H}}_{1}\oplus{\mathcal{H}}:\|\psi\|<\varepsilon\\},$

and the corresponding ball ${\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$
in ${\mathcal{H}}_{1,{\mathbb{C}}}\oplus{\mathcal{H}}_{\mathbb{C}}$. We prove
the following results about the ramifications.

######  Theorem 1.4.

Let $\psi\in{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$. Then

i) There are exactly two (counted with multiplicity) eigenvalues $E_{n}^{\pm}$
of the 2-periodic problem (1.11),(1.19) in each domain
${\mathcal{S}}_{n},n\in{\mathbb{N}}$, and there are no eigenvalues in the
domain ${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. If,
in addition, $\psi$ is real, then the eigenvalues
$E_{n}^{\pm},n\in{\mathbb{N}}$, are real and satisfy

$2<E_{1}^{-}\leqslant E_{1}^{+}<E_{2}^{-}\leqslant E_{2}^{+}<...$ (1.36)

ii) The ramifications $r_{n}^{\pm}$ satisfy

$E_{n}^{\pm}(\psi)=E_{n}^{\pm}(\psi^{-})={3\over 4}(r_{n}^{\pm}(\psi))^{2\over
3}\quad\forall\ \ n\in{\mathbb{N}}.$ (1.37)
$E_{n}^{\pm}(\psi_{*})=E_{n}^{\pm}(\psi_{*}^{-})={3\over
4}(-r_{-n}^{\mp}(\psi))^{2\over 3}\quad\forall\ \ n\in{\mathbb{N}}.$ (1.38)

If $\psi$ is real, then all $r_{n}^{\pm},n\in{\mathbb{Z}}\setminus\\{0\\}$,
are real and satisfy

$...<r_{-1}^{-}\leqslant r_{-1}^{+}<r_{1}^{-}\leqslant
r_{1}^{+}<r_{2}^{-}\leqslant r_{2}^{+}<...$

Remark. Proof of Theorem 1.4 is based on our results from [BK21].

###  1.7. Localization of 3-point eigenvalues and norming constants

In [BK24xx] we prove the following results.

Let $\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$. Then there is
exactly one simple eigenvalue $\mu_{n}$ of the operator ${\mathcal{L}}_{dir}$
and exactly one simple eigenvalue $\widetilde{\mu}_{n}$ of the operator
$\widetilde{\mathcal{L}}_{dir}$ in each domain
${\mathcal{D}}_{n},n\in{\mathbb{Z}}\setminus\\{0\\}$. Moreover,

$\mu_{n}(\psi)=-\mu_{-n}(\psi_{*}^{-})=-\widetilde{\mu}_{-n}(\psi_{*})=\widetilde{\mu}_{n}(\psi^{-}),\quad\forall\
\ n\in{\mathbb{Z}}\setminus\\{0\\}.$ (1.39)

In order to define norming constants we need to consider the transpose
operator $\widetilde{\mathcal{L}}_{dir}$. If
$\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$, then in each domain
${\mathcal{D}}_{n},n\in{\mathbb{Z}}_{0}$, there is exactly one real simple
eigenvalue $\widetilde{\mu}_{n}$ of this operator. Let $\widetilde{y}_{n}(x)$
be the corresponding eigenfunction such that
$\widetilde{y}_{n}^{\prime}(0)=1$. We introduce the norming constants
$h_{sn},n\in{\mathbb{N}}$, by

$h_{sn}=8(\pi n)^{2}\log|\widetilde{y}_{n}^{\prime}(1)\tau_{3}^{-{1\over
2}}(\widetilde{\mu}_{n})|,\quad\tau_{3}^{1\over 2}(\widetilde{\mu}_{n})>0.$
(1.40)

We formulate our theorem about the 3-point eigenvalues and norming constants.

######  Theorem 1.5.

Let $\psi\in{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$. Then

i) There is exactly one simple eigenvalue $\mathfrak{m}_{n}$ of the Dirichlet
problem (1.11), (1.25) in each domain ${\mathcal{S}}_{n},n\in{\mathbb{N}}$,
and there are no eigenvalues in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. If, in
addition, $\psi$ is real, then all eigenvalues
$\mathfrak{m}_{n},n\in{\mathbb{N}}$, are real and satisfy

$\mathfrak{m}_{1}<\mathfrak{m}_{2}<...,\quad\mathfrak{m}_{n}\in[E_{n}^{-},E_{n}^{+}],\quad\forall\
\ n\in{\mathbb{N}}.$ (1.41)

ii) The eigenvalues $\mu_{n}$ of the 3-point problem satisfy

$\mathfrak{m}_{n}(\psi^{-})={3\over 4}(\mu_{n}(\psi))^{2\over 3},\quad\forall\
\ n\in{\mathbb{N}},$ (1.42) $\mathfrak{m}_{n}(\psi)={3\over
4}\big{(}\widetilde{\mu}_{n}(\psi)\big{)}^{2\over 3}={3\over
4}\big{(}-\mu_{-n}(\psi_{*})\big{)}^{2\over 3},\quad\forall\ \
n\in{\mathbb{N}}.$ (1.43)

If $\psi$ is real, then all $\mu_{n},n\in{\mathbb{Z}}\setminus\\{0\\}$, are
real and satisfy

$\mu_{n}\in[r_{n}^{-},r_{n}^{+}].$ (1.44)

iii) The norming constants satisfy

$\mathfrak{h}_{sn}={h_{sn}\over 4\pi n},\quad\forall\ \ n\in{\mathbb{N}}.$
(1.45)

Remark. The relation (1.44) was obtained in [McK81] for the case of small
$p,q\in C^{\infty}({\mathbb{T}})$ in a different way, without using McKean’s
transformation. We use McKean’s transformation in this paper, which gives a
simpler proof for a wider class of coefficients.

##  2\. McKean’s transformation

###  2.1. Transformation of Eq. (1.8)

We prove our first results about McKean’s transformation.

Proof of Lemma 1.1. Let $p,q\in
L_{loc}^{1}({\mathbb{R}}),\lambda\in{\mathbb{C}}$. Assume that Eq. (1.8) has a
solution $\eta\in C^{[2]}({\mathbb{R}})$ and $\eta(x)\neq 0$ for all
$x\in{\mathbb{R}}$. Let $y\in C^{[2]}({\mathbb{R}})$ be a solution to Eq.
(1.8) and let the function $f$ has the form (1.10). Then

$f={y^{\prime}\eta-y\eta^{\prime}\over\sqrt{\eta}},\quad
f^{\prime}={y^{[2]}\eta-y\eta^{[2]}\over\sqrt{\eta}}-{f\eta^{\prime}\over
2\eta}.$

This yields

$\displaystyle
f^{\prime\prime}={(y^{[2]})^{\prime}\eta-y(\eta^{[2]})^{\prime}\over\sqrt{\eta}}+{y^{[2]}\eta^{\prime}-y^{\prime}\eta^{[2]}\over\sqrt{\eta}}-{y^{[2]}\eta-y\eta^{[2]}\over\sqrt{\eta}}{\eta^{\prime}\over
2\eta}-{f^{\prime}\eta^{\prime}\over 2\eta}-{f\over
2}\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{\prime}$ (2.1)
$\displaystyle={(y^{[2]})^{\prime}\eta-y(\eta^{[2]})^{\prime}\over\sqrt{\eta}}+{y^{[2]}\eta^{\prime}-y^{\prime}\eta^{[2]}\over\sqrt{\eta}}-{y^{[2]}\eta-y\eta^{[2]}\over\sqrt{\eta}}{\eta^{\prime}\over\eta}+{f\over
4}\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}-{f\over
2}\biggl{(}{\eta^{\prime\prime}\over\eta}-\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}\biggr{)}$
$\displaystyle={(y^{[2]})^{\prime}\eta-y(\eta^{[2]})^{\prime}\over\sqrt{\eta}}-f{\eta^{[2]}\over\eta}+{f\over
4}\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}-{f\over
2}\biggl{(}{\eta^{[2]}-p\eta\over\eta}-\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}\biggr{)}.$

The functions $y,\eta$ satisfy equation (1.8), which yields

$(y^{[2]})^{\prime}\eta-y(\eta^{[2]})^{\prime}=-p(y^{\prime}\eta-y\eta^{\prime})=-p\sqrt{\eta}f.$

Substituting this identity into (2.1) we obtain (1.11)–(2.3). Moreover,
${\eta^{\prime}\over\eta},{\eta^{[2]}\over\eta}-({\eta^{\prime}\over\eta})^{2}\in
AC_{loc}({\mathbb{R}})$. The identities (1.13) and (2.3) give $V+{p\over 2}\in
AC_{loc}({\mathbb{R}})$ and $f,f^{\prime},f^{\prime\prime}+{p\over 2}f\in
AC_{loc}({\mathbb{R}})$.

###  2.2. Analysis of the potential

Assume that $p,q\in L_{loc}^{1}({\mathbb{R}})$. In what follows, we will
mainly deal with the solution $\eta(x,\lambda)$ to Eq. (1.8) that satisfies
the following conditions:

a) $\eta(\cdot,\lambda)\in C^{[2]}({\mathbb{R}})$ for all
$\lambda\in{\mathscr{G}}$, where ${\mathscr{G}}\subset{\mathbb{C}}$ is a
domain such that $S_{\delta}\subset{\mathscr{G}}$ for some $\delta\in(0,\pi)$,
$S_{\delta}=\\{\lambda\in{\mathbb{C}}:|\mathop{\mathrm{arg}}\nolimits\lambda|<\delta\\}$,

b) $\eta(x,\cdot)$ is analytic in ${\mathscr{G}}$,

c) $\eta(x,\lambda)\neq 0$ for all
$(x,\lambda)\in{\mathbb{R}}\times{\mathscr{G}}$,

d) $\eta$ satisfies

$\sup_{x\in{\mathbb{R}}}|e^{-zx}\eta(x,\lambda)-1|=o(1),\ \
\sup_{x\in{\mathbb{R}}}|z^{-1}e^{-zx}\eta^{\prime}(x,\lambda)-1|=o(1),\ \
\sup_{x\in{\mathbb{R}}}|z^{-2}e^{-zx}\eta^{[2]}(x,\lambda)-1|=o(1),$ (2.2)

as $|\lambda|\to\infty,\lambda\in S_{\delta}$. For this case we introduce the
function

$\xi={\eta^{\prime}\over\eta}-z.$ (2.3)

Then the definition (2.3) implies

$\xi^{\prime}={\eta^{[2]}\over\eta}-\Big{(}{\eta^{\prime}\over\eta}\Big{)}^{2}-p,$
(2.4)

the definition (2.3) and the identity (2.4) give $\xi,\xi^{\prime}+p\in
AC_{loc}({\mathbb{R}})$. Moreover, the definition (2.3) and the asymptotics
(2.2) yield

$\sup_{x\in{\mathbb{R}}}|z^{-1}\xi(x,\lambda)|=o(1),\quad\sup_{x\in{\mathbb{R}}}|z^{-2}(\xi^{\prime}(x,\lambda)+p(x))|=o(1),$
(2.5)

as $|\lambda|\to\infty,\lambda\in S_{\delta}$. The definition (1.13) gives

$V=E-2p-{3\over 2}\xi^{\prime}-{3\over 4}(\xi+z)^{2}.$ (2.6)

######  Lemma 2.1.

Let $p,q\in L_{loc}^{1}({\mathbb{R}})$, let $\lambda\in{\mathscr{G}}$, and let
the function $\xi$ be defined by (2.3). Then

i) The function $\xi$ satisfies the differential equation

$(\xi^{\prime}+p)^{\prime}+3z(\xi^{\prime}+p)+3z^{2}\xi+3\xi(\xi^{\prime}+p)+3z\xi^{2}+\xi^{3}-p\xi-
zp+q=0.$ (2.7)

ii) The vector-valued function $X$ given by

$X=\begin{pmatrix}X_{1}\\\
X_{2}\end{pmatrix}=\begin{pmatrix}-\omega^{2}\xi+{i\over\sqrt{3}z}(\xi^{\prime}+p)\\\
-\omega\xi-{i\over\sqrt{3}z}(\xi^{\prime}+p)\end{pmatrix},$ (2.8)

satisfies: $X(\cdot,\lambda)\in AC_{loc}({\mathbb{R}})$ and

$X^{\prime}=i\sqrt{3}z\Omega X+W+{\mathcal{K}}[X]{\mathcal{I}}_{-},$ (2.9)

where

$\Omega=\begin{pmatrix}\omega&0\\\ 0&-\omega^{2}\end{pmatrix},\quad
W={i\over\sqrt{3}}\Big{(}p{\mathcal{I}}_{+}-{q\over
z}{\mathcal{I}}_{-}\Big{)},\quad{\mathcal{I}}_{+}=\begin{pmatrix}\omega\\\
-\omega^{2}\end{pmatrix},\quad{\mathcal{I}}_{-}=\begin{pmatrix}1\\\
-1\end{pmatrix},$ (2.10)
${\mathcal{K}}[X]={ip\over\sqrt{3}z}(X_{1}+X_{2})+3(X_{1}+X_{2})(\omega
X_{1}-\omega^{2}X_{2})-i\sqrt{3}(X_{1}+X_{2})^{2}-{i\over\sqrt{3}z}(X_{1}+X_{2})^{3}.$
(2.11)

iii) The potential $V(x,E)$ satisfies

$V=-{p\over 2}-{3z\over 2}(\omega^{2}X_{1}+\omega X_{2})-{3\over
4}(X_{1}+X_{2})^{2},\quad z=\Big{(}{4\over 3}E\Big{)}^{1\over 2}.$ (2.12)

Proof. i) The definition (2.3) implies
$\eta(x)=\eta(0)e^{zx+\int_{0}^{x}\xi(s)ds}$. Substituting this identity into
Eq. (1.8) we obtain (2.7).

ii) We rewrite Eq. (2.7) in the vector form

${\mathscr{X}}^{\prime}=A_{0}{\mathscr{X}}+B+A_{1}[{\mathscr{X}}],$ (2.13)

on ${\mathbb{R}}$, where

${\mathscr{X}}=\begin{pmatrix}{\mathscr{X}}_{1}\\\
{\mathscr{X}}_{2}\end{pmatrix}=\begin{pmatrix}\xi\\\
\xi^{\prime}+p\end{pmatrix}\in AC_{loc}({\mathbb{R}}),\quad
A_{0}=\begin{pmatrix}0&1\\\ -3z^{2}&-3z\end{pmatrix},$ $B=\begin{pmatrix}-p\\\
zp-q\end{pmatrix},\quad A_{1}[{\mathscr{X}}]=\begin{pmatrix}0\\\
p{\mathscr{X}}_{1}-3{\mathscr{X}}_{1}{\mathscr{X}}_{2}-3z{\mathscr{X}}_{1}^{2}-{\mathscr{X}}_{1}^{3}\end{pmatrix}.$

Introduce the matrix

$U=\begin{pmatrix}1&1\\\ i\sqrt{3}\omega
z&-i\sqrt{3}\omega^{2}z\end{pmatrix},\quad
U^{-1}=\begin{pmatrix}-\omega^{2}&{i\over\sqrt{3}z}\\\
-\omega&-{i\over\sqrt{3}z}\end{pmatrix}.$

Multiplying Eq. (2.13) by $U^{-1}$ and using

$U^{-1}A_{0}U=i\sqrt{3}z\Omega,\qquad
X=\begin{pmatrix}-\omega^{2}&{i\over\sqrt{3}z}\\\
-\omega&-{i\over\sqrt{3}z}\end{pmatrix}\begin{pmatrix}\xi\\\
\xi^{\prime}+p\end{pmatrix}=U^{-1}{\mathscr{X}},$
$W={i\over\sqrt{3}}\begin{pmatrix}\omega p-{1\over z}q\\\ -\omega^{2}p+{1\over
z}q\end{pmatrix}=\begin{pmatrix}(\omega^{2}+{i\over\sqrt{3}})p-{i\over\sqrt{3}z}q\\\
(\omega-{i\over\sqrt{3}})p+{i\over\sqrt{3}z}q\end{pmatrix}=U^{-1}B,\quad{\mathcal{K}}[X]{\mathcal{I}}_{-}=U^{-1}A_{1}[UX],$

we obtain (2.9).

iii) The definition (1.13) implies

$V=-{p\over 2}-{3\over 2}{\mathscr{X}}_{2}-{3z\over
2}{\mathscr{X}}_{1}-{3\over 4}{\mathscr{X}}_{1}^{2}.$ (2.14)

The definition (2.8) yields

${\mathscr{X}}_{1}=X_{1}+X_{2},\quad{\mathscr{X}}_{2}=i\sqrt{3}z(\omega
X_{1}-\omega^{2}X_{2}).$

Substituting these identities into (2.14) and using the identities
$1+i\sqrt{3}\omega=\omega^{2},1-i\sqrt{3}\omega^{2}=\omega$ we obtain (2.12).

Remark. If $\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$ and the solution $\eta$
is the Floquet solution, then $\xi$, and therefore $V$, is an 1-periodic
function, and $\xi,\xi^{\prime}+p,V+{p\over 2}\in AC({\mathbb{T}})$.

###  2.3. Solutions of the transpose equation.

In the following lemma we prove that the numerator $y^{\prime}g-yg^{\prime}$
in the definition (1.10) is a solution of the transpose equation (1.28) and
study relations between solutions to Eqs. (1.8) and (1.28).

######  Lemma 2.2.

Let $p,q\in L_{loc}^{1}({\mathbb{R}}),\lambda\in{\mathbb{C}}$. Then

i) If $f,g\in C^{[2]}({\mathbb{R}})$ are two linearly independent solutions of
Eq. (1.8), then $\tilde{y}=fg^{\prime}-f^{\prime}g$ is a solution of Eq.
(1.28).

ii) Let $\tilde{y}\in C^{[2]}({\mathbb{R}})$ be a non-trivial solution of Eq.
(1.28), let $g\in C^{[2]}({\mathbb{R}})$ be a solution of Eq. (1.8) such that
$g(x)\neq 0,x\in{\mathbb{R}}$, and let $f\in C^{[2]}({\mathbb{R}})$ satisfy

$\tilde{y}=fg^{\prime}-f^{\prime}g.$ (2.15)

Then

$(f^{\prime\prime}+pf)^{\prime}+pf^{\prime}+qf-\lambda f={(G,JY)\over g^{2}},$
(2.16)

where
$\lambda=z^{3},G=(g,g^{\prime},g^{[2]})^{\top},\tilde{Y}=(\tilde{y},\tilde{y}^{\prime},\tilde{y}^{[2]})^{\top}$,

$(G,J\tilde{Y})=g\tilde{y}^{[2]}-g^{\prime}\tilde{y}^{\prime}+g^{[2]}\tilde{y}=\mathop{\mathrm{const}}\nolimits.$
(2.17)

Proof. i) We have

$\tilde{y}=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f&g\\\
f^{\prime}&g^{\prime}\end{pmatrix},\quad\tilde{y}^{\prime}=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f&g\\\
f^{[2]}&g^{[2]}\end{pmatrix},$
$\tilde{y}^{[2]}=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f&g\\\
(f^{[2]})^{\prime}&(g^{[2]})^{\prime}\end{pmatrix}+\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f^{\prime}&g^{\prime}\\\
f^{[2]}&g^{[2]}\end{pmatrix}-p\tilde{y}=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f^{\prime}&g^{\prime}\\\
f^{[2]}&g^{[2]}\end{pmatrix},$
$(\tilde{y}^{[2]})^{\prime}=-p\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f&g\\\
f^{[2]}&g^{[2]}\end{pmatrix}+\mathop{\mathrm{det}}\nolimits\begin{pmatrix}f^{\prime}&g^{\prime}\\\
(f^{[2]})^{\prime}&(g^{[2]})^{\prime}\end{pmatrix}=-p\tilde{y}^{\prime}-(\lambda-q)\tilde{y}.$

Therefore, $\tilde{y}$ satisfies Eq. (1.28).

ii) The identity (2.15) gives

$f=g\sigma,\quad\sigma=\int_{0}^{x}{\tilde{y}(s)\over g^{2}(s)}ds.$

Then

$f^{\prime}=g^{\prime}\sigma+g\sigma^{\prime},\quad
f^{[2]}=g^{[2]}\sigma+2g^{\prime}\sigma^{\prime}+g\sigma^{\prime\prime}.$

Using

$\sigma^{\prime}={\tilde{y}\over
g^{2}},\quad\sigma^{\prime\prime}={\tilde{y}^{\prime}\over
g^{2}}-{2\tilde{y}g^{\prime}\over g^{3}},$

we obtain

$f^{[2]}=g^{[2]}\sigma+{\tilde{y}^{\prime}\over g}.$

Then

$(f^{[2]})^{\prime}=(g^{[2]})^{\prime}\sigma+g^{[2]}\sigma^{\prime}+{\tilde{y}^{[2]}-p\tilde{y}\over
g}-{\tilde{y}^{\prime}g^{\prime}\over g^{2}},$

which yields

$(f^{[2]})^{\prime}+pf^{\prime}+(q-\lambda)f=(g^{[2]}+pg)\sigma^{\prime}+{\tilde{y}^{[2]}-p\tilde{y}\over
g}-{\tilde{y}^{\prime}g^{\prime}\over g^{2}}={g^{[2]}\tilde{y}\over
g^{2}}+{\tilde{y}^{[2]}\over g}-{\tilde{y}^{\prime}g^{\prime}\over g^{2}}.$

This gives (2.16). The identities

$\displaystyle-(\tilde{y}^{[2]})^{\prime}-p\tilde{y}^{\prime}+(q-\lambda)\tilde{y}=0,$
$\displaystyle(g^{[2]})^{\prime}+pg^{\prime}+(q-\lambda)g=0$

give

$(\tilde{y}^{[2]})^{\prime}g+\tilde{y}(g^{[2]})^{\prime}+p\tilde{y}^{\prime}g+p\tilde{y}g^{\prime}=0.$

This implies

$\big{(}\tilde{y}^{[2]}g+\tilde{y}g^{[2]}\big{)}^{\prime}-\tilde{y}^{\prime}\big{(}g^{[2]}-pg\big{)}-\big{(}\tilde{y}^{[2]}-p\tilde{y}\big{)}g^{\prime}=0.$

Then we have

$\big{(}\tilde{y}^{[2]}g+\tilde{y}g^{[2]}-\tilde{y}^{\prime}g^{\prime}\big{)}^{\prime}=0,$

which yields (2.17).

##  3\. Local results

###  3.1. Transformation of the ramifications

We consider the ramifications.

Proof of Theorem 1.2. Assume, for definiteness, that $\tau=\tau_{3}$, the
proofs for $\tau=\tau_{1}$ and $\tau=\tau_{2}$ are similar. Assume that
$\tau(\lambda_{o})$ for some $\lambda_{o}\in{\mathbb{C}}$ is a simple
multiplier and the corresponding Floquet solution $\eta=\mathfrak{f}_{3}$
satisfies: $\eta(x,\lambda_{o})\neq 0$ for all $x\in{\mathbb{R}}$. Then the
function $\tau$ is analytic on some neighborhood ${\mathcal{U}}$ of
$\lambda_{o}$ and $\eta(x,\cdot),x\in{\mathbb{R}}$, is also analytic on
${\mathcal{U}}$ and $\eta(x,\lambda)\neq 0$ for all
$(x,\lambda)\in{\mathbb{R}}\times{\mathcal{U}}$, see [BK24xxx]. Then each of
the potentials $V(x,\cdot),x\in{\mathbb{R}}$, is analytic on ${\mathcal{U}}$.

In this case $\lambda_{o}$ is a ramification for Eq. (1.8) iff
$\tau_{1}(\lambda_{o})=\tau_{2}(\lambda_{o})$. Let
$\mathfrak{f}_{1},\mathfrak{f}_{2}$ be the corresponding Floquet solutions
defined by (1.16). Then $f_{1}=\eta^{3\over
2}({\mathfrak{f}_{1}\over\eta})^{\prime}$ is the Floquet solution of Eq.
(1.11) satisfying $f_{1}(x+1,E_{o})=t_{1}(E_{o})f_{1}(x,E_{o})$, where
$t_{1}=\tau_{3}^{1\over 2}\tau_{1}$ is a multiplier for Eq. (1.11),
$E_{o}={3\over 4}\lambda_{o}^{2\over 3}$. Similarly, $f_{2}=\eta^{3\over
2}({\mathfrak{f}_{2}\over\eta})^{\prime}$ is the Floquet solution of Eq.
(1.11) satisfying $f_{2}(x+1,E_{o})=t_{2}(E_{o})f_{2}(x,E_{o})$, where
$t_{2}=\tau_{3}^{1\over 2}\tau_{2}$ is the other multiplier for Eq. (1.11). We
have $\tau_{1}(\lambda_{o})=\tau_{2}(\lambda_{o})$ iff
$t_{1}(E_{o})=t_{2}(E_{o})$. Therefore, $\lambda_{o}$ is a ramification for
Eq. (1.8) iff $E_{o}={3\over 4}\lambda_{o}^{2\over 3}$ is an eigenvalue of the
2-periodic problem for Eq. (1.11).

###  3.2. Monodromy matrices

Introduce the monodromy matrix $M(\lambda)$ for Eq. (1.8) by (1.14) and the
corresponding monodromy matrix $\tilde{M}(\lambda)$ for Eq. (1.28). They are
entire $3\times 3$ matrix-valued functions. They satisfy

$\tilde{M}^{\top}JM=J,\quad J=\begin{pmatrix}0&0&1\\\ 0&-1&0\\\
1&0&0\end{pmatrix},$ (3.1)

see [BK24xx]. Recall that the matrix $M$ has three eigenvalues (counting with
multiplicity) $\tau_{1},\tau_{2},\tau_{3}$. The following lemma shows that the
matrix $\tilde{M}^{\top}$ (and then the matrix $\tilde{M}$) has three
eigenvalues $\tau_{1}^{-1},\tau_{2}^{-1},\tau_{3}^{-1}$.

######  Lemma 3.1.

Let $\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$. Then

i) $\tau\in{\mathbb{C}}\setminus\\{0\\}$ is an eigenvalue of the matrix $M$
(and then $M^{\top}$) iff $\tau^{-1}$ is an eigenvalue of the matrix
$\tilde{M}$.

ii) Let, in addition, ${\mathcal{F}}$ be the eigenvector of the matrix
$M^{\top}$ corresponding to the eigenvalue $\tau$. Then $J{\mathcal{F}}$ is
the eigenvector of the matrix $\tilde{M}$ corresponding to the eigenvalue
$\tau^{-1}$, here $J$ is given by (3.1).

Proof. i) The identity (3.1) gives $M=J(\tilde{M}^{\top})^{-1}J.$ Let
$\tau\in{\mathbb{C}}\setminus\\{0\\}$ be an eigenvalue of the matrix $M$. The
identities

$\displaystyle\mathop{\mathrm{det}}\nolimits(M-\tau
1\\!\\!1_{3})=\mathop{\mathrm{det}}\nolimits\big{(}J(\tilde{M}^{\top})^{-1}J-\tau
1\\!\\!1_{3}\big{)}=\mathop{\mathrm{det}}\nolimits\big{(}(\tilde{M}^{\top})^{-1}-\tau
1\\!\\!1_{3}\big{)}$
$\displaystyle=-\tau(\mathop{\mathrm{det}}\nolimits\tilde{M}^{\top})^{-1}\mathop{\mathrm{det}}\nolimits(\tilde{M}^{\top}-\tau^{-1}1\\!\\!1_{3})=-\tau(\mathop{\mathrm{det}}\nolimits\tilde{M})^{-1}\mathop{\mathrm{det}}\nolimits(\tilde{M}-\tau^{-1}1\\!\\!1_{3})$

give the statement.

ii) Let $\tau\in{\mathbb{C}}\setminus\\{0\\}$ be an eigenvalue of the matrix
$M^{\top}$ and let ${\mathcal{F}}$ be the corresponding eigenvector. The
identity (3.1) implies $M^{\top}=J\tilde{M}^{-1}J$. Then the identity
$M^{\top}{\mathcal{F}}=\tau{\mathcal{F}}$ gives
$J\tilde{M}^{-1}J{\mathcal{F}}=\tau{\mathcal{F}}$, then we have
$\tilde{M}^{-1}J{\mathcal{F}}=\tau J{\mathcal{F}}$. This yields
$\tilde{M}J{\mathcal{F}}=\tau^{-1}J{\mathcal{F}}$, which proves the statement.

###  3.3. Monodromy matrix and three-point eigenvalues

We prove the following results about the three-point problem.

######  Lemma 3.2.

Let $\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$.

i) If the monodromy matrix $M(\lambda),\lambda\in{\mathbb{C}}$, has the
eigenvector of the form $(0,a,b)^{\top},a,b\in{\mathbb{C}}$, then $\lambda$ is
the eigenvalue of the three-point problem for Eq. (1.8) and

$y=a\varphi_{2}+b\varphi_{3}$

is the corresponding eigenfunction.

ii) If $\lambda\in{\mathbb{C}}$ is the eigenvalue of the three-point problem
for Eq. (1.8), then the corresponding eigenfunction has the form

$y(x)=\mathop{\mathrm{det}}\nolimits\begin{pmatrix}\varphi_{2}(x,\lambda)&\varphi_{3}(x,\lambda)\\\
\varphi_{2}(2,\lambda)&\varphi_{3}(2,\lambda)\end{pmatrix},\quad
x\in{\mathbb{R}}.$ (3.2)

Moreover,

$y(x+1)=Ay(x)$ (3.3)

for all $x\in{\mathbb{R}}$ and for some $A\in{\mathbb{C}}\setminus\\{0\\}$,
that is, $y$ is the Floquet solution with the multiplier $A$. Moreover, the
eigenvector of the monodromy matrix $M$, corresponding the eigenvalue $A$, has
the form $(0,y^{\prime}(0),y^{[2]}(0))^{\top}$.

Proof. i) Let the matrix $M(\lambda)$, has the eigenvector $(0,a,b)^{\top}$.
Then we have

$\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(1)=M\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(0)=M\begin{pmatrix}0\\\ a\\\
b\end{pmatrix}=A\begin{pmatrix}0\\\ a\\\ b\end{pmatrix}=\begin{pmatrix}0\\\
aA\\\ bA\end{pmatrix},$

for some $A\in{\mathbb{C}}$, and

$\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(2)=M^{2}\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(0)=M^{2}\begin{pmatrix}0\\\ a\\\
b\end{pmatrix}=A^{2}\begin{pmatrix}0\\\ a\\\
b\end{pmatrix}=\begin{pmatrix}0\\\ aA^{2}\\\ bA^{2}\end{pmatrix}.$

This yields $y(0)=y(1)=y(2)=0$, that is, $y=a\varphi_{2}+b\varphi_{3}$ is a
solution of the three-point problem.

ii) The function $y$, given by (3.2), satisfies Eq. (1.8) and the three-point
Dirichlet conditions. Moreover, due to the periodicity of $p,q$, we have

$y(x+1)=A_{1}\varphi_{1}(x)+A_{2}\varphi_{2}(x)+A_{3}\varphi_{3}(x)$ (3.4)

for some constant $A_{1},A_{2},A_{3}$. The condition $y(1)=0$ gives $A_{1}=0$.
The condition $y(2)=0$ implies

$A_{2}\varphi_{2}(2)+A_{3}\varphi_{3}(2)=0,$

which yields $A_{3}=-A_{2}{\varphi_{2}(2)\over\varphi_{3}(2)}$. Substituting
these identities into (3.4) we obtain

$y(x+1)=A_{2}\Big{(}\varphi_{2}(x)-{\varphi_{2}(2)\over\varphi_{3}(2)}\varphi_{3}(x)\Big{)}.$

This gives (3.3).

The Floquet solution $y$ satisfies

$y=y(0)\varphi_{1}+y^{\prime}(0)\varphi_{2}+y^{[2]}(0)\varphi_{3}.$

The identity (3.3) yields

$M\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(0)=\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(1)=A\begin{pmatrix}y\\\ y^{\prime}\\\
y^{[2]}\end{pmatrix}(0),$

therefore, $(y(0),y^{\prime}(0),y^{[2]}(0))^{\top}$ is the eigenvector of the
monodromy matrix $M$, corresponding the eigenvalue $A$. The condition $y(0)=0$
gives that this eigenvector has the form
$(0,y^{\prime}(0),y^{[2]}(0))^{\top}$.

###  3.4. Additional results about the transpose equation

If $y,g$ are solutions of Eq. (1.8), then $\tilde{y}=y^{\prime}g-yg^{\prime}$
is a solution of Eq. (1.28). We prove the following converse result.

######  Lemma 3.3.

Let $\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$. Let $\tilde{y}\in
C^{[2]}({\mathbb{R}})$ be a solution of the three-point problem for Eq. (1.28)
for some $\lambda\in{\mathbb{C}}$ and let there exist the Floquet solution
$g\in C^{[2]}({\mathbb{R}})$ of Eq. (1.8) with the simple multiplier such that
$g$ does not vanish for all $x\in{\mathbb{R}}$ and $g^{[2]}(0)\neq 0$. Then
there exists the solution $y\in C^{[2]}({\mathbb{R}})$ of Eq. (1.8) such that
$\tilde{y}=y^{\prime}g-yg^{\prime}$.

Proof. Let $\tilde{y}$ be a solution of the three-point problem for Eq.
(1.28). Due to Lemma 2.2 ii), there exists the function $y$ such that

$\tilde{y}=yg^{\prime}-y^{\prime}g.$ (3.5)

Moreover, $y$ satisfies

$(y^{\prime\prime}+py)^{\prime}+py^{\prime}+qy-\lambda y={C\over g^{2}},$
(3.6)

where

$C=g\tilde{y}^{[2]}-g^{\prime}\tilde{y}^{\prime}+g^{[2]}\tilde{y}=\mathop{\mathrm{const}}\nolimits.$
(3.7)

Furthermore, $\widetilde{y}$ is the Floquet solution, that is,

$\tilde{y}(x+1)=A\tilde{y}(x)$ (3.8)

for all $x\in{\mathbb{R}}$ and for some constant $A\neq 0$, see Lm 3.2. Then

$\tilde{y}(n)=0,\quad\tilde{y}^{\prime}(n)=A^{n}\tilde{y}^{\prime}(0),\quad\tilde{y}^{[2]}(n)=A^{n}\tilde{y}^{[2]}(0),\quad\forall\
n\in{\mathbb{Z}}.$

The identity $g(x+1)=\tau g(x)$, $\tau$ is the multiplier for $g$, implies

$g(n)=\tau^{n}g(0),\quad g^{\prime}(n)=\tau^{n}g^{\prime}(0),\quad
g^{[2]}(n)=\tau^{n}g^{[2]}(0),\quad\forall\ n\in{\mathbb{Z}}.$

The identity (3.7) yields

$C=\tau^{n}A^{n}\big{(}g(0)\tilde{y}^{[2]}(0)-g^{\prime}(0)\tilde{y}^{\prime}(0)\big{)}=\mathop{\mathrm{const}}\nolimits\quad\forall\
n\in{\mathbb{Z}}.$ (3.9)

This yields $C=0$ or $A=\tau^{-1}$. If $A=\tau^{-1}$, then the identity (3.8)
gives that $\tilde{y}$ is the Floquet solution of Eq. (1.28) with the
multiplier $\tau^{-1}$ and $\tilde{y}=\alpha\varphi_{2}+\beta\varphi_{3}$ for
some $\alpha,\beta\in{\mathbb{R}}$. Then the Floquet solution $g$ of Eq. (1.8)
satisfying the multiplier $\tau$ has the form
$g=\beta\varphi_{1}-\alpha\varphi_{2}$, see Lm 3.1. This yields
$g^{[2]}(0)=0$, which contradicts the conditions of the lemma. Thus, $C=0$ and
the identities (3.6) and (3.5) show that $y$ is the solution of Eq. (1.8) such
that $\tilde{y}=y^{\prime}g-yg^{\prime}$.

###  3.5. Transformation of the three-point eigenvalues

We are ready to prove our first results about the three-point eigenvalues.

Proof of Theorem 1.3. We have proved in Theorem 1.2 that if $\tau(\lambda)$ is
a simple multiplier for some $\lambda\in{\mathbb{C}}$ and the Floquet solution
$\eta\in C^{[2]}({\mathbb{R}})$ of Eq. (1.8) does not vanish for all
$x\in{\mathbb{R}}$, then the potential $V(x,\cdot),x\in{\mathbb{R}}$, is
analytic on some neighborhood of $E$.

Let, in addition, $\lambda$ be an eigenvalue of the three-point Dirichlet
problem for Eq. (1.28), let $\tilde{y}$ be the corresponding eigenfunction,
and let $\eta^{[2]}(0)\neq 0$. Due to Lemma 3.3, there exists the solution $y$
of Eq. (1.8) such that $\tilde{y}=y^{\prime}\eta-y\eta^{\prime}$. Then
$E={3\over 4}\lambda^{2\over 3}$ is an eigenvalue of the Dirichlet problem for
Eq. (1.11) and $f={\tilde{y}\over\sqrt{\eta}}$ is the corresponding
eigenfunction.

Conversely, let $E$ be an eigenvalue of the Dirichlet problem for Eq. (1.11)
and let $f$ be the corresponding eigenfunction. Then $f(0)=f(1)=0$ and
$f(x)=C\varphi(x)$ for some constant $C$, where $\vartheta,\varphi$ are the
standard solution
$-y^{\prime\prime}+Vy=Ey,\varphi(0)=\vartheta^{\prime}(0)=0,\vartheta(0)=\varphi^{\prime}(0)=1$.
Using the periodicity $V$ we obtain $\varphi(x+1)=A\vartheta(x)+B\varphi(x)$
for some constant $A,B$. The condition $\varphi(1)=A=0$ gives
$\varphi(x+1)=B\varphi(x)$, which yields $\varphi(2)=B\varphi(1)=0$,
therefore, $f(2)=0$. Then the function $\tilde{y}=f\sqrt{\eta}$ is a solution
of the three-point Dirichlet problem for Eq. (1.28) and $\lambda=({4\over
3}E)^{3\over 2}$ is the corresponding eigenvalue.

##  4\. Estimates of the potential

###  4.1. Preliminaries

We prove that the spectrum of the Schrödinger operator with an energy-
dependent potential is real using the results of our paper [BK21]. For this we
need estimates of the potential.

Introduce the diagonal matrix

${\mathcal{E}}=\big{(}1\\!\\!1_{2}-e^{i\sqrt{3}z\Omega}\big{)}^{-1}e^{i\sqrt{3}z\Omega
t}=\begin{pmatrix}{\mathcal{E}}_{1}&0\\\
0&{\mathcal{E}}_{2}\end{pmatrix},\quad{\mathcal{E}}_{1}={e^{i\sqrt{3}z\omega
t}\over
1-e^{i\sqrt{3}z\omega}},\quad{\mathcal{E}}_{2}={e^{-i\sqrt{3}z\omega^{2}t}\over
1-e^{-i\sqrt{3}z\omega^{2}}},$ (4.1)

we used the definition (2.10) of $\Omega$. Moreover,

$\big{(}1\\!\\!1_{2}-e^{i\sqrt{3}z\Omega}\big{)}^{-1}={1\\!\\!1_{2}-e^{-i\sqrt{3}z\Omega^{*}}\over
D},$ (4.2)

where

$\Omega^{*}=\begin{pmatrix}\omega^{2}&0\\\ 0&-\omega\end{pmatrix},\quad
D(\lambda)=(1-e^{i\sqrt{3}z\omega})(1-e^{-i\sqrt{3}z\omega^{2}})=4\sin{\sqrt{3}\omega
z\over 2}\sin{\sqrt{3}\omega^{2}z\over 2}.$ (4.3)

The zeros of the function $D$ in ${\mathbb{C}}\setminus\\{0\\}$ are simple and
have the form $-({2\pi n\over\sqrt{3}})^{3},n\in{\mathbb{N}}$. We need the
following simple estimates.

######  Lemma 4.1.

i) Let $(x,\lambda)\in{\mathbb{R}}\times{\mathbb{C}}_{+},f\in
L^{2}({\mathbb{T}})$. Then

$\Big{|}\int_{0}^{1}e^{i\sqrt{3}z\Omega t}f(x-t)ds\Big{|}\leqslant 2\|f\|.$
(4.4)

ii) If $\lambda\in{\mathscr{D}}_{3}$, the domain ${\mathscr{D}}_{3}$ is given
by (1.32), then the following estimates hold true:

$|1-e^{i\sqrt{3}\omega z}|\geqslant{1\over
2},\quad|1-e^{-i\sqrt{3}\omega^{2}z}|\geqslant{1\over 2},$ (4.5)
$|D(\lambda)|\geqslant{1\over 4}.$ (4.6)

Moreover,

$\Big{|}\big{(}1\\!\\!1_{2}-e^{i\sqrt{3}z\Omega}\big{)}^{-1}\Big{|}\leqslant
16.$ (4.7)

Proof. i) The definition (2.10) of $\Omega$ implies

$|e^{i\sqrt{3}z\Omega x}|\leqslant|e^{i\sqrt{3}z\omega
x}|+|e^{-i\sqrt{3}z\omega^{2}x}|=2e^{-{3\over 2}x\mathop{\mathrm{Re}}\nolimits
z}\cosh\Big{(}{\sqrt{3}\over 2}x\mathop{\mathrm{Im}}\nolimits
z\Big{)},\qquad(x,\lambda)\in{\mathbb{R}}_{+}\times{\mathbb{C}}.$

We have $-\sqrt{3}\mathop{\mathrm{Re}}\nolimits
z+\mathop{\mathrm{Im}}\nolimits z\leqslant 0$ for all
$\mathop{\mathrm{arg}}\nolimits z\in[0,{\pi\over 3}]$. Then $e^{-{3\over
2}x\mathop{\mathrm{Re}}\nolimits z}\cosh({\sqrt{3}\over
2}x\mathop{\mathrm{Im}}\nolimits z)<1$, which yields (4.4).

ii) Note that $\mathop{\mathrm{Im}}\nolimits(\omega z)\geqslant
0,\mathop{\mathrm{Im}}\nolimits(\omega^{2}z)\leqslant 0$, for all
$\lambda\in{\mathbb{C}}$. Let $\lambda\in{\mathscr{D}}_{3}$. Then

$|1-e^{i\sqrt{3}\omega z}|=2|e^{i{\sqrt{3}\omega z\over
2}}|\Big{|}\sin{\sqrt{3}\omega z\over 2}\Big{|}\geqslant{1\over
2}e^{-{\sqrt{3}\over 2}\mathop{\mathrm{Im}}\nolimits(\omega
z)}e^{{\sqrt{3}\over 2}|\mathop{\mathrm{Im}}\nolimits(\omega z)|}={1\over 2},$
$|1-e^{-i\sqrt{3}\omega^{2}z}|=2|e^{-i{\sqrt{3}\omega^{2}z\over
2}}|\Big{|}\sin{\sqrt{3}\omega^{2}z\over 2}\Big{|}\geqslant{1\over
2}e^{{\sqrt{3}\over
2}\mathop{\mathrm{Im}}\nolimits(\omega^{2}z)}e^{{\sqrt{3}\over
2}|\mathop{\mathrm{Im}}\nolimits(\omega^{2}z)|}={1\over 2},$

where we used the estimates

$\Big{|}\sin{\sqrt{3}\omega z\over 2}\Big{|}\geqslant{1\over
4}e^{{\sqrt{3}\over 2}|\mathop{\mathrm{Im}}\nolimits\omega
z|},\quad\Big{|}\sin{\sqrt{3}\omega^{2}z\over 2}\Big{|}\geqslant{1\over
4}e^{{\sqrt{3}\over
2}|\mathop{\mathrm{Im}}\nolimits\omega^{2}z|},\quad\lambda\in{\mathscr{D}}_{3},$
(4.8)

see [PT87, Lm 2.1]. This yields (4.5). The definition (4.3) yields (4.6). The
estimate (4.6) and the identity (4.2) give

$\Big{|}\big{(}1\\!\\!1_{2}-e^{i\sqrt{3}z\Omega}\big{)}^{-1}\Big{|}\leqslant
4\big{|}1\\!\\!1_{2}-e^{-i\sqrt{3}z\Omega^{*}}\big{|}\leqslant
4\big{(}|1-e^{-i\sqrt{3}z\omega^{2}}|+|1-e^{i\sqrt{3}z\omega}|\big{)}\leqslant
16,$

which yields (4.7).

###  4.2. Transformation to an integral equation in the periodic case

In the case of periodic coefficients $p,q$ we rewrite Eq. (2.9) into the form
of the integral equation (4.9).

######  Lemma 4.2.

Let $(\lambda,\psi)\in{\mathbb{C}}\setminus\\{({2\pi
n\over\sqrt{3}})^{3},n\in{\mathbb{Z}}\\}\times{\mathcal{H}}\oplus{\mathcal{H}}$.
Let $X(\cdot,\lambda)\in AC({\mathbb{T}})$ be an 1-periodic solution of Eq.
(2.13). Then $X$ satisfies the non-linear integral equation

$X=G[X],$ (4.9)

where

$G[X]=X^{0}+F[X],$ (4.10)
$X^{0}(x,\lambda)={i\over\sqrt{3}}\int_{0}^{1}{\mathcal{E}}(t,\lambda)\Big{(}p(x-t){\mathcal{I}}_{+}-{q(x-t)\over
z}{\mathcal{I}}_{-}\Big{)}dt$ (4.11)

(${\mathcal{E}},{\mathcal{I}}_{\pm}$ are given by (4.1) and (2.10)) is the
1-periodic solution of the equation

$(X^{0})^{\prime}=i\sqrt{3}z\Omega X^{0}+W,$ (4.12)

and the integral operator $F$ is given by

$F[X](x,\lambda)=\int_{0}^{1}{\mathcal{E}}(t,\lambda){\mathcal{K}}[X](x-t,\lambda){\mathcal{I}}_{-}dt,$
(4.13)

Proof. Let $\lambda\in{\mathscr{D}}_{3}$, $f\in L^{1}({\mathbb{T}})$. Then the
1-periodic solution $X\in AC({\mathbb{T}})$ of the equation

$X^{\prime}=i\sqrt{3}z\Omega X+f$ (4.14)

has the form

$X(x,\lambda)=\int_{x-1}^{x}{\mathcal{E}}(x-t,\lambda)f(t)dt=\int_{0}^{1}{\mathcal{E}}(t,\lambda)f(x-t)dt.$
(4.15)

Put $f=W+{\mathcal{K}}[X]{\mathcal{I}}_{-}$ in (4.14), then we conclude that
the 1-periodic solution of Eq. (2.13) satisfies the integral equation (4.16),

$X=X^{0}+F[X],$ (4.16)

and then (4.9), where

$X^{0}(x,\lambda)=\int_{0}^{1}{\mathcal{E}}(t,\lambda)W(x-t,\lambda)dt.$

satisfies Eq. (4.12), and the integral operator $F$ is given by (4.13). The
definition (2.10) gives (4.11).

###  4.3. Solvability of Eq. (4.9)

Introduce the spaces $C({\mathbb{T}})$ of continuous scalar and 2-vector
functions on the circle equipped with the norm

$\|f\|_{\infty}=\max_{x\in{\mathbb{T}}}|f(x)|.$

Introduce the balls

${\mathscr{B}}(\delta)=\\{{\mathcal{A}}\in
C({\mathbb{T}}):\|{\mathcal{A}}\|_{\infty}<\delta\\}\subset
C({\mathbb{T}}),\quad\delta>0,$

of 2-vector functions on the circle.

We consider the mapping $G:C({\mathbb{T}})\to C({\mathbb{T}})$ given by
(4.10). We will show that there exists $\delta_{0}>0$ such that the mapping
$G$ is a contraction on the ball ${\mathscr{B}}(\delta_{0})$. Then Eq. (4.9)
has a unique solution in this ball. In order to obtain this solution we
construct the sequence
$(X^{n})_{n=0}^{\infty}\subset{\mathscr{B}}(\delta_{0})$ of iterations. We
take $X^{0}$ as a first approximation and then the next approximations
$X^{n},n\in{\mathbb{N}}$, are given by $X^{n}=G[X^{n-1}]$ for all
$n\in{\mathbb{N}}$. Then the solution $X$ of equation (4.9) has the form

$X=\lim_{n\to\infty}X^{n}.$

Moreover,

$X^{n}=X^{0}+\sum_{k=1}^{n}(X^{k}-X^{k-1}),$

which yields

$X=X^{0}+\sum_{k=1}^{\infty}(X^{k}-X^{k-1}).$ (4.17)

######  Lemma 4.3.

i) Let
$(\lambda,\psi)\in{\mathscr{D}}_{3}\times{\mathcal{H}}\oplus{\mathcal{H}}$ and
let 2-vector-valued functions $X,Y\in C({\mathbb{T}})$. Then

$\big{\|}G[X]-G[Y]\big{\|}_{\infty}\leqslant{64\over\sqrt{3}}\|X-Y\|_{\infty}\Big{(}\|p\|+9\big{(}\|X\|_{\infty}+\|Y\|_{\infty}\big{)}+\big{(}\|X\|_{\infty}+\|Y\|_{\infty}\big{)}^{2}\Big{)}.$
(4.18)

ii) Let
$(\lambda,\psi)\in{\mathscr{D}}_{3}\times{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$.
If, in addition, $\max\\{\|X\|_{\infty},\|Y\|_{\infty}\\}\leqslant\delta_{0}$
for some $\delta_{0}>0$ small enough, then

$\big{\|}G[X]-G[Y]\big{\|}_{\infty}\leqslant{1\over 2}\|X-Y\|_{\infty}.$
(4.19)

iii) Let
$(\lambda,\psi)\in{\mathscr{D}}_{3}\times{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$
and let $X^{0}(\cdot,\lambda)\in{\mathscr{B}}(\delta_{0})$. Then Eq. (4.9) has
the unique solution $X$ in the ball ${\mathscr{B}}(\delta_{0})$. Moreover,
this solution satisfies

$\|X(\cdot,\lambda)\|_{\infty}\leqslant C\|X^{0}(\cdot,\lambda)\|_{\infty},$
(4.20)

for some $C>0$. Each of the functions
$X(\cdot,\cdot,\psi),\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$, is
analytic on the domain ${\mathscr{D}}_{3}$. Each of the functions
$X(\cdot,\lambda,\cdot),\lambda\in{\mathscr{D}}_{3}$, is analytic on the ball
${\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$.

Proof. i) Let $\lambda\in{\mathscr{D}}_{3}$. The definition (4.13) and the
estimate (4.7) imply

$\big{|}F[X](x)\big{|}\leqslant 32\|{\mathcal{K}}[X]{\mathcal{I}}_{-}\|$
(4.21)

for all $x\in{\mathbb{T}}$ and for a 2-vector-valued function $X\in
C({\mathbb{T}})$, here and below in this proof
$F[X](x)=F[X](x,\lambda),X^{0}(x)=X^{0}(x,\lambda),...$ Then the definition
(4.10) gives

$\big{|}X^{1}(x)-X^{0}(x)\big{|}=\big{|}G[X^{0}](x)-X^{0}(x)\big{|}=\big{|}F[X^{0}](x)\big{|}\leqslant
32\|{\mathcal{K}}[X^{0}]{\mathcal{I}}_{-}\|,$ (4.22)

for all $x\in{\mathbb{T}}$. Moreover,

$\big{|}G[X](x)-G[Y](x)\big{|}=\big{|}F[X](x)-F[Y](x)\big{|}\leqslant
32\|{\mathcal{K}}[X]{\mathcal{I}}_{-}-{\mathcal{K}}[Y]{\mathcal{I}}_{-}\|$
(4.23)

for two 2-vector-valued functions $X,Y\in C({\mathbb{T}})$. The definitions
(2.10) and (2.11) give

$\|{\mathcal{K}}[X]{\mathcal{I}}_{-}\|=2\|{\mathcal{K}}[X]\|,\quad\big{\|}{\mathcal{K}}[X]{\mathcal{I}}_{-}-{\mathcal{K}}[Y]{\mathcal{I}}_{-}\big{\|}=2\big{\|}{\mathcal{K}}[X]-{\mathcal{K}}[Y]\big{\|}.$
(4.24)

The definition (2.11) and the estimate $|z|>1$ imply

$\|{\mathcal{K}}[X]\|\leqslant{\|X\|_{\infty}\over\sqrt{3}}\big{(}\|p\|+9\|X^{0}\|_{\infty}+\|X\|_{\infty}^{2}\big{)}.$
(4.25)

Moreover,

$\displaystyle{\mathcal{K}}[X]-{\mathcal{K}}[Y]={ip\over\sqrt{3}z}(X_{1}-Y_{1}+X_{2}-Y_{2})+3(X_{1}+X_{2})\big{(}\omega(X_{1}-Y_{1})+\omega^{2}(Y_{2}-X_{2})\big{)}$
$\displaystyle+3(X_{1}-Y_{1}+X_{2}-Y_{2})(\omega
Y_{1}-\omega^{2}Y_{2})+i\sqrt{3}\big{(}(Y_{1}+Y_{2})^{2}-(X_{1}+X_{2})^{2}\big{)}$
$\displaystyle+{i\over\sqrt{3}z}\big{(}(Y_{1}+Y_{2})^{3}-(X_{1}+X_{2})^{3}\big{)},$

which yields

$\Big{\|}{\mathcal{K}}[X]-{\mathcal{K}}[Y]\Big{\|}\leqslant{\|X-Y\|_{\infty}\over\sqrt{3}}\Big{(}\|p\|+9\big{(}\|X\|_{\infty}+\|Y\|_{\infty}\big{)}+\big{(}\|X\|_{\infty}+\|Y\|_{\infty}\big{)}^{2}\Big{)}.$

Then the estimate (4.23) and the identities (4.24) give (4.18).

ii) Let
$(\lambda,\psi)\in{\mathscr{D}}_{3}\times{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$.
Then $|z|>1$ and $\|p\|\leqslant\varepsilon_{0}$. If
$\max\\{\|X\|_{\infty},\|Y\|_{\infty}\\}\leqslant\delta_{0}$, then the
estimate (4.18) yields (4.19).

iii) Due to (4.19), the mapping $G$ is a contraction on the ball
${\mathscr{B}}(\delta_{0})\subset C({\mathbb{T}})$. Then equation (4.9) has
the unique solution $X$ in this ball. The estimate (4.19) implies

$\|X^{k}-X^{k-1}\|_{\infty}=\|G[X^{k-1}]-G[X^{k-2}]\|_{\infty}\leqslant{1\over
2}\|X^{k-1}-X^{k-2}\|_{\infty}\leqslant{1\over
2^{k-1}}\|X^{1}-X^{0}\|_{\infty}.$

This shows that the series (4.17) converges absolutely and uniformly on any
bounded subset of
${\mathscr{D}}_{3}\times{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$. Each
term is an analytic function on ${\mathscr{D}}_{3}$ for fixed
$\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$, and on
${\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$ for fixed
$\lambda\in{\mathscr{D}}_{3}$. Then the sum is also an analytic function.
Summing the majorants we obtain

$\|X-X^{0}\|_{\infty}\leqslant\|X^{1}-X^{0}\|_{\infty}\Big{(}1+\sum_{k=2}^{\infty}{1\over
2^{k-1}}\Big{)}=2\|X^{1}-X^{0}\|_{\infty}\leqslant
64\|{\mathcal{K}}[X^{0}]{\mathcal{I}}_{-}\|\leqslant
128\|{\mathcal{K}}[X^{0}]\|,$

where we used (4.22) and (4.24). The estimate (4.25) gives

$\|X(\cdot,\lambda)-X^{0}(\cdot,\lambda)\|_{\infty}\leqslant
C\|X^{0}(\cdot,\lambda)\|_{\infty},$

for some $C>0$. Using the estimate

$\|X\|_{\infty}\leqslant\|X^{0}\|_{\infty}+\|X-X^{0}\|_{\infty},$

we obtain (4.20).

###  4.4. Estimates in the class $p^{\prime},q\in L^{2}({\mathbb{T}})$

We have the following estimate of $X_{0}$ and $X$ in the case $p^{\prime},q\in
L^{2}({\mathbb{T}})$.

######  Lemma 4.4.

Let $\psi\in{\mathcal{H}}_{1,{\mathbb{C}}}\oplus{\mathcal{H}}_{\mathbb{C}}$
and let $\lambda\in{\mathscr{D}}_{3}$. Then

i) The function $X^{0}$, given by (4.11), satisfies

$\big{\|}X^{0}(\cdot,\lambda)\big{\|}_{\infty}\leqslant{19\|\psi\|_{1}\over|z|}.$
(4.26)

ii) The function $F[X^{0}]$, given by (4.13), satisfies

$\|F[X^{0}](x,\lambda)\|_{\infty}\leqslant{C\|\psi\|_{1}^{2}\over|z|^{3}},$
(4.27)

for some $C>0$.

iii) Each of the functions $X(x,\cdot,\cdot),x\in{\mathbb{T}}$, is analytic
with respect to $\lambda$ on ${\mathscr{D}}_{3}$ for fixed
$\psi\in{\mathcal{B}}_{1,C}(\varepsilon_{0})$, and it is analytic with respect
to $\psi$ on ${\mathcal{B}}_{1,C}(\varepsilon_{0})$ for fixed
$\lambda\in{\mathscr{D}}_{3}$. Moreover, if
$(\lambda,\psi)\in{\mathscr{D}}_{3}\times{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$,
then

$\|X(\cdot,\lambda)\|_{\infty}\leqslant{C\|\psi\|_{1}\over|z|},$ (4.28)
$\|X(\cdot,\lambda)-X^{0}(\cdot,\lambda)\|_{\infty}\leqslant{C\|\psi\|_{1}^{2}\over|z|^{3}},$
(4.29)

for some $C>0$.

Proof. i) We have

$\displaystyle\int_{0}^{1}e^{i\sqrt{3}z\omega t}p(x-t)dt={1\over
i\sqrt{3}z\omega}\Big{(}(e^{i\sqrt{3}z\omega}-1)p(x)+\int_{0}^{1}e^{i\sqrt{3}z\omega
t}p^{\prime}(x-t)dt\Big{)},$
$\displaystyle\int_{0}^{1}e^{-i\sqrt{3}z\omega^{2}t}p(x-t)dt=-{1\over
i\sqrt{3}z\omega^{2}}\Big{(}(e^{-i\sqrt{3}z\omega^{2}}-1)p(x)+\int_{0}^{1}e^{-i\sqrt{3}z\omega^{2}t}p^{\prime}(x-t)dt\Big{)}.$

This yields

$\int_{0}^{1}e^{i\sqrt{3}z\Omega t}p(x-t)dt={\Omega^{-1}\over
i\sqrt{3}z}\Big{(}(e^{i\sqrt{3}z\Omega}-1\\!\\!1_{2})p(x)+\int_{0}^{1}e^{i\sqrt{3}z\Omega
t}p^{\prime}(x-t)dt\Big{)}.$

The definition (4.11) and the identity
$\Omega^{-1}{\mathcal{I}}_{+}=(1,1)^{\top}$, see (2.10), imply

$X^{0}(x,\lambda)={1\over 3z}\Big{(}-p(x)\begin{pmatrix}1\\\
1\end{pmatrix}+\int_{0}^{1}{\mathcal{E}}(t,\lambda)\big{(}p^{\prime}(x-t)\begin{pmatrix}1\\\
1\end{pmatrix}-i\sqrt{3}q(x-t){\mathcal{I}}_{-}\big{)}dt\Big{)},$ (4.30)

for all $x\in{\mathbb{R}}$, where ${\mathcal{E}}$ is given by (4.1). The
identity (4.30) and the estimates (4.4) and (4.7) give

$\big{\|}X^{0}(\cdot,\lambda)\big{\|}_{\infty}\leqslant{1\over
3|z|}\Big{(}\|p\|_{\infty}+32(\|p^{\prime}\|+\sqrt{3}\|q\|)\Big{)}\leqslant{1\over
3|z|}\Big{(}\|p\|+33\|p^{\prime}\|+32\sqrt{3}\|q\|\Big{)}.$

here we used the estimate $\|p\|_{\infty}\leqslant\|p\|+\|p^{\prime}\|$. This
yields (4.26).

ii) The definitions (4.13) and (2.10) imply

$F[X^{0}](x,\lambda)=\int_{0}^{1}{\mathcal{E}}(t,\lambda){\mathcal{I}}_{-}{\mathcal{K}}[X^{0}](x-t,\lambda)dt=\int_{x-1}^{x}{\mathcal{E}}(x-y,\lambda){\mathcal{I}}_{-}{\mathcal{K}}[X^{0}](y,\lambda)dy,$
(4.31)

where

${\mathcal{E}}{\mathcal{I}}_{-}=({\mathcal{E}}_{1},-{\mathcal{E}}_{2})^{\top}=\biggl{(}{e^{i\sqrt{3}z\omega
t}\over 1-e^{i\sqrt{3}z\omega}},-{e^{-i\sqrt{3}z\omega^{2}t}\over
1-e^{-i\sqrt{3}z\omega^{2}}}\biggr{)}^{\top},$

${\mathcal{E}},{\mathcal{E}}_{1}$, and ${\mathcal{E}}_{2}$ are given by (4.1)
and we used ${\mathcal{I}}_{-}=(1,-1)^{\top}$, see (2.10). The definition
(2.11) implies

${\mathcal{K}}[X^{0}]={ip\over\sqrt{3}z}(X_{1}^{0}+X_{2}^{0})+3(X_{1}^{0}+X_{2}^{0})(\omega
X_{1}^{0}-\omega^{2}X_{2}^{0})-i\sqrt{3}(X_{1}^{0}+X_{2}^{0})^{2}-{i\over\sqrt{3}z}(X_{1}^{0}+X_{2}^{0})^{3}.$
(4.32)

The identity (4.30) gives

$\alpha X_{1}^{0}+\beta X_{2}^{0}={(\beta-\alpha)p+A_{\alpha\beta}\over 3z},$

for all $\alpha,\beta\in{\mathbb{C}}$, where

$\displaystyle
A_{\alpha\beta}(x)=\int_{0}^{1}\Big{(}\big{(}\alpha{\mathcal{E}}_{1}(t,\lambda)+\beta{\mathcal{E}}_{2}(t,\lambda)\big{)}p^{\prime}(x-t)-i\sqrt{3}\big{(}\alpha{\mathcal{E}}_{1}(t,\lambda)-\beta{\mathcal{E}}_{2}(t,\lambda)\big{)}q(x-t)\Big{)}dt$
$\displaystyle=\int_{x-1}^{x}\Big{(}\big{(}\alpha{\mathcal{E}}_{1}(x-y,\lambda)+\beta{\mathcal{E}}_{2}(x-y,\lambda)\big{)}p^{\prime}(y)-i\sqrt{3}\big{(}\alpha{\mathcal{E}}_{1}(x-y,\lambda)-\beta{\mathcal{E}}_{2}(x-y,\lambda)\big{)}q(y)\Big{)}dy,$

therefore, $A_{\alpha\beta}^{\prime}\in L^{1}(0,1)$. The identity (4.32)
yields

${\mathcal{K}}[X^{0}]={iA_{11}\over
3\sqrt{3}z^{2}}\Big{(}4p-i\sqrt{3}A_{\omega,-\omega^{2}}-A_{11}-{A_{11}^{2}\over
9z^{2}}\Big{)}.$

Integrating by parts in (4.31) we obtain (4.27).

iii) The series (4.17) converges absolutely and uniformly on any bounded
subset of ${\mathscr{D}}_{3}\times{\mathcal{B}}_{1,C}(\varepsilon_{0})$. Each
term is an analytic function on ${\mathscr{D}}_{3}$ for fixed
$\psi\in{\mathcal{B}}_{1,C}(\varepsilon_{0})$ and on
${\mathcal{B}}_{1,C}(\varepsilon_{0})$ for fixed
$\lambda\in{\mathscr{D}}_{3}$. Then the sum is also analytic. The estimates
(4.20) and (4.26) give (4.28).

Eq. (4.9) gives

$\|X-X^{0}\|_{\infty}=\|F[X]\|_{\infty}\leqslant\|F[X]-F[X^{0}]\|_{\infty}+\|F[X^{0}]\|_{\infty}.$
(4.33)

The definitions (4.13) and (2.10) imply

$F[X](x)-F[X^{0}](x)=\int_{x-1}^{x}{\mathcal{E}}(x-y){\mathcal{I}}_{-}\Big{(}{\mathcal{K}}[X](y)-{\mathcal{K}}[X^{0}](y)\Big{)}dy,$
(4.34)

we omit $\lambda$ in the arguments. The definition (2.11) yields

$\displaystyle{\mathcal{K}}[X]-{\mathcal{K}}[X^{0}]\\!=\\!{ip(F_{1}[X]+F_{2}[X])\over\sqrt{3}z}\\!+\\!3(X_{1}+X_{2})(\omega
X_{1}-\omega^{2}X_{2})\\!-\\!3(X_{1}^{0}+X_{2}^{0})(\omega
X_{1}^{0}-\omega^{2}X_{2}^{0})$
$\displaystyle-i\sqrt{3}(X_{1}+X_{2})^{2}+i\sqrt{3}(X_{1}^{0}+X_{2}^{0})^{2}-{i(X_{1}+X_{2})^{3}\over\sqrt{3}z}+{i(X_{1}^{0}+X_{2}^{0})^{3}\over\sqrt{3}z},$

therefore, $({\mathcal{K}}[X]-{\mathcal{K}}[X^{0}])^{\prime}\in L^{1}(0,1)$.
Integration by parts in (4.34) and the estimates (4.26) and (4.28) yield

$\big{\|}F[X]-F[X^{0}]\big{\|}_{\infty}\leqslant{C\|\psi\|_{1}^{2}\over|z|^{3}},$

for some $C>0$. Substituting this estimate and (4.27) into (4.33) we obtain
(4.29).

Due to Lemma 4.4 ii), each function $X(x,\cdot,\cdot),x\in{\mathbb{T}}$, is
analytic on ${\mathscr{D}}_{3}$ for fixed
$\psi\in{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$, and on
${\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$ for fixed
$\lambda\in{\mathscr{D}}_{3}$. Then the identity (2.12) shows that each of the
potentials $V(x,\cdot,\cdot),x\in{\mathbb{T}}$, is analytic with respect to
$E$ on ${\mathcal{S}}$ for fixed
$\psi\in{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$, and it is analytic
with respect to $\psi$ on ${\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$
for fixed $E\in{\mathcal{S}}$.

##  5\. Second order equation. Proof of Theorem 1.4

###  5.1. The spectra are real

Each of the potential $V(x,\cdot),x\in{\mathbb{R}}$, is analytic on the domain
${\mathcal{S}}$, then there are well defined spectra of the 2-periodic problem
(1.19), the Dirichlet problem (1.25), and the quasi-periodic problems

$f(1)=e^{ik}f(0),\qquad f^{\prime}(1)=e^{ik}f^{\prime}(0),\qquad
k\in[0,2\pi),$ (5.1)

for Eq. (1.11) in this domain, see [BK21]. Moreover, these spectra are pure
discrete.

Introduce the half-planes

$\Pi_{r}=\\{E\in{\mathbb{C}}:\mathop{\mathrm{Re}}\nolimits E>r\\},\quad
r\in{\mathbb{R}}.$ (5.2)

The definition (1.34) gives

$\Pi_{3\over 4}\subset{\mathcal{S}}.$ (5.3)

where the half-planes $\Pi_{r},r\in{\mathbb{R}}$, are defined by (5.2).

######  Lemma 5.1.

i) The potential $V$ satisfies

$\Big{\|}V(x,E)+{p\over 2}\Big{\|}_{\infty}\leqslant{C\|\psi\|_{1}},$ (5.4)

for all
$(E,\psi)\in{\mathcal{S}}\times{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$
and for some $C>0$.

ii) Let $\psi\in{\mathcal{B}}_{1}(\varepsilon_{0})$. Then the spectra of the
problems (1.19), (1.25), and (5.1) for Eq. (1.11) in the half-plane $\Pi_{2}$
are all real.

Proof. i)The identity (2.12) and the estimate (4.28) imply (5.4).

ii) Recall the following result, see [BK21, Th 1.3]:

Let for some $a\in{\mathbb{R}}$ the potential $V$ satisfies:

1) For each $E\in\Pi_{a}$ the function $V$ is 1-periodic and $V(\cdot,E)\in
L^{1}({\mathbb{T}})$,

2) For almost every $x\in{\mathbb{T}}$ the function $V(x,\cdot)$ is real
analytic on the domain $\Pi_{a}$.

3) $\xi=\sup_{E\in\Pi_{a}}\|\mathop{\mathrm{Im}}\nolimits
V(x,E)\|_{\infty}<\infty.$

Then the spectra in the half-plane $\Pi_{a+r}$ are real:

$\mathfrak{S}\cap\Pi_{a+r}\subset(a+r,+\infty),\quad\text{where}\quad
r={\xi\over 2-\sqrt{3}}.$

The estimate (5.4) gives

$\|\mathop{\mathrm{Im}}\nolimits
V(x,E)\|_{\infty}\leqslant{C\|\psi\|_{1}},\quad\forall\quad(x,E)\in{\mathbb{T}}\times{\mathcal{S}}.$

Due to (5.3), we may take $a={3\over 4}$ and $\xi=C\|\psi\|_{1}$ in the above
result. Then $a+r<2$ and the spectra in the half-plane $\Pi_{2}$ are real.

###  5.2. Spectra

Let $\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$. Introduce the
fundamental solutions $\vartheta(x,E)$,
$\varphi(x,E),(x,E)\in{\mathbb{R}}\times{\mathcal{S}}$ of Eq. (1.11)
satisfying the initial conditions
$\vartheta(0,E)=\varphi^{\prime}(0,E)=1,\vartheta^{\prime}(0,E)=\varphi(0,E)=0$.
Introduce the Lyapunov function by

$\Delta(E)={1\over 2}\big{(}\vartheta(1,E)+\varphi^{\prime}(1,E)\big{)},$
(5.5)

Each of the functions $\vartheta(x,\cdot)$, $\varphi(x,\cdot)$,
$\vartheta^{\prime}(x,\cdot)$, $\varphi^{\prime}(x,\cdot)$,
$x\in{\mathbb{R}}$, and then $\Delta$, is analytic on ${\mathcal{S}}$, see
[BK21].

The spectrum of the 2-periodic problem (1.11), (1.19) is the set of zeros of
the functions $\Delta(E)\pm 1$ in the domain ${\mathcal{S}}$, the spectrum of
the quasi-periodic problem (1.11), (5.1) for $k\in[0,2\pi)$ is the set of
zeros of the functions $\Delta(E)-\cos k$ in ${\mathcal{S}}$, the spectrum of
the Dirichlet problem (1.11), (1.25) is the set of zeros of the function
$\varphi(1,E)$ in ${\mathcal{S}}$.

In the unperturbed case $\psi=0$ the spectrum of the 2-periodic problem
consists of the simple eigenvalue $E_{0}^{o,+}=0$ and the eigenvalues
$E_{n}^{o,\pm}=(\pi n)^{2},n\in{\mathbb{N}}$, of multiplicity 2, the spectrum
of the Dirichlet problem (1.11), (1.25) consists of the simple eigenvalues
$\mathfrak{m}_{n}^{o}=(\pi n)^{2},n\in{\mathbb{N}}$.

We prove the following results.

######  Lemma 5.2.

Let $\psi\in{\mathcal{B}}_{1,{\mathbb{C}}}(\varepsilon_{0})$. Then

i) There are exactly two (counting with multiplicity) eigenvalues
$E_{n}^{\pm}$ of the 2-periodic problem (1.11), (1.19) in each domain
${\mathcal{S}}_{n},n\in{\mathbb{N}}$, and there are no eigenvalues in the
domain ${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. If
$\psi$ is real, then all zeros $E_{n}^{\pm},n\in{\mathbb{N}}$, are real, may
be labeled by (1.36) and satisfy

$\displaystyle\Delta(E_{n}^{\pm})=(-1)^{n},\quad(-1)^{n}\Delta(E)>1\ \
\forall\ E\in(E_{n}^{-},E_{n}^{+}),\quad n\geqslant 1,$ (5.6)
$\displaystyle-1<\Delta(E)<1,\quad\forall\ \
E\in(2,E_{1}^{-})\cup\big{(}\cup_{n\geqslant
2}(E_{n-1}^{+},E_{n}^{-})\big{)},$ $\displaystyle\Delta^{\prime}(E)<1\ \
\forall\ E\in(2,E_{1}^{-})\cup\big{(}\cup_{n\geqslant
2}(E_{2n-1}^{+},E_{2n}^{-})\big{)},\quad\Delta^{\prime}(E)>1\ \ \forall\
E\in\cup_{n\geqslant 1}(E_{2n}^{+},E_{2n+1}^{-}).$

ii) There are exactly one simple eigenvalue $\mathfrak{m}_{n}$ of the
Dirichlet problem (1.11), (1.25) in each domain
${\mathcal{S}}_{n},n\in{\mathbb{N}}$, and there are no eigenvalues in the
domain ${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. If
$\psi$ is real, then these eigenvalues are real and satisfy (1.41).

Proof. i) The function $\Delta$ is analytic on ${\mathcal{S}}$. Moreover, we
have the estimate

$\big{|}\Delta(E)-\cos\sqrt{E}\big{|}\\!=\\!\big{|}\Delta(E)-1+2\sin^{2}{\sqrt{E}\over
2}\big{|}\\!=\\!\big{|}\Delta(E)+1-2\cos^{2}{\sqrt{E}\over
2}\big{|}\\!\leqslant\\!\|V(\cdot,E)\|e^{\|V(\cdot,E)\|+|\mathop{\mathrm{Im}}\nolimits\sqrt{E}|},$

for all $E\in{\mathcal{S}}$, see [BK21, Lm 3.2 ii)].

Let $E\in{\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{2n}$. The
definition (1.35) gives $|{\sqrt{E}\over 2}-\pi n|\geqslant{\sqrt{3}\over 4}$
for all $n\in{\mathbb{N}}$. Then

$|e^{|\mathop{\mathrm{Im}}\nolimits{\sqrt{E}\over 2}|}|<C|\sin{\sqrt{E}\over
2}|,$

for some $C>0$, see [PT87]. The estimate (5.4) gives

$\big{|}\Delta(E)-1+2\sin^{2}{\sqrt{E}\over 2}\big{|}\leqslant
C\|\psi\|_{1}e^{C\|\psi\|_{1}}e^{|\mathop{\mathrm{Im}}\nolimits\sqrt{E}|}<2|\sin^{2}{\sqrt{E}\over
2}|.$

Then, by Rouche’s theorem, the functions $\Delta(E)-1$ and
$\sin^{2}{\sqrt{E}\over 2}$ has the same number of zeros in each domain
${\mathcal{S}}_{2n},n\in{\mathbb{N}}$. Since the function
$\sin^{2}{\sqrt{E}\over 2}$ has exactly one zero of multiplicity two at each
point $2\pi n,n\in{\mathbb{Z}}$, and has no other zeros, the function
$\Delta(E)-1$ has exactly two zeros in each domain
${\mathcal{S}}_{2n},n\in{\mathbb{N}}$. Moreover, we have the estimates

$\big{|}\Delta(E)-1\big{|}\geqslant\big{|}2\sin^{2}{\sqrt{E}\over
2}\big{|}-\big{|}\Delta(E)-1+2\sin^{2}{\sqrt{E}\over 2}\big{|}>0,$

which shows that the function $\Delta(E)-1$ has no zeros in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{2n}$.

Similarly, if
$E\in{\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{2n-1}$, then

$\big{|}\Delta(E)+1-2\cos^{2}{\sqrt{E}\over 2}\big{|}<2|\cos^{2}{\sqrt{E}\over
2}|.$

Therefore, the function $\Delta(E)+1$ has exactly two zeros in each domain
${\mathcal{S}}_{2n-1},n\in{\mathbb{N}}$, and has no zeros in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{2n-1}$.

Thus, the function $\Delta^{2}(E)-1$ has exactly two zeros in each domain
${\mathcal{S}}_{n},n\in{\mathbb{N}}$, and has no zeros in the domain
$E\in{\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$.
Therefore, there are exactly two eigenvalues $E_{n}^{\pm}$ of the 2-periodic
problem (1.11), (1.19) in each domain ${\mathcal{S}}_{n},n\in{\mathbb{N}}$,
and there are no eigenvalues in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$.

Due to (5.3), $\Delta$ is analytic on the half-plane $\Pi_{3\over 4}$ and, if
$\psi$ is real, then $\Delta$ is real on the half-line $[{3\over 4},+\infty)$.
The whole spectrum of the 2-periodic problem in $\Pi_{2}$ is real, then all
zeros of the function $\Delta^{2}-1$ in $\Pi_{2}$ are real.

Moreover, the spectra of the quasi-periodic problems (1.11), (5.1) in the
half-plane $\Pi_{2}$ are real, then the standard Krein’s arguments give: if
$\Delta(\lambda)\in(-1,1)$, then $\Delta^{\prime}(\lambda)\neq 0$, if
$\Delta(\lambda)=\pm 1$ and $\Delta^{\prime}(\lambda)=0$, then
$\Delta(\lambda)\Delta^{\prime\prime}(\lambda)<0$.

These results show that the Lyapunov function on the half-line $[2,+\infty)$
oscillates in the similar way as in the case of the Schrodinger operator with
a potential, which does not depend on energy. This yields the statement and
the relations (5.6).

ii) Similarly, the function $\varphi(1,\cdot)$ is analytic in ${\mathcal{S}}$
and real on the half-line ${\mathcal{S}}\cup{\mathbb{R}}$. If
$E\in{\mathcal{S}}$, then we have the estimate

$\big{|}\varphi(1,E)-{\sin\sqrt{E}\over\sqrt{E}}\big{|}\leqslant{\|V(\cdot,E)\|\over\sqrt{E}}e^{\|V(\cdot,E)\|},$

see [BK21, Lm 3.2 ii)]. Let
$E\in{\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. The
definition (1.35) gives $|\sqrt{E}-\pi n|\geqslant{\sqrt{3}\over 2}$ for all
$n\in{\mathbb{N}}$. Then
$|e^{|\mathop{\mathrm{Im}}\nolimits\sqrt{E}|}|<C|\sin\sqrt{E}|,$ for some
$C>0$, see [PT87]. The estimate (5.4) gives

$\Big{|}\varphi(1,E)-{\sin\sqrt{E}\over\sqrt{E}}\Big{|}\leqslant
C\|\psi\|_{1}e^{C\|\psi\|_{1}}e^{|\mathop{\mathrm{Im}}\nolimits\sqrt{E}|}<\Big{|}{\sin\sqrt{E}\over\sqrt{E}}\Big{|}.$

Then, by Rouche’s theorem, the functions $\varphi(1,E)$ and $\sin\sqrt{E}$ has
the same number of zeros in each domain ${\mathcal{S}}_{n},n\in{\mathbb{N}}$,
and in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. Since the
function $\sin\sqrt{E}$ has exactly one simple zero at each point $\pi
n,n\in{\mathbb{Z}}$, and has no other zeros, the function $\varphi(1,E)$ has
exactly one simple zero in each domain ${\mathcal{S}}_{n},n\in{\mathbb{N}}$,
and has no the zeros in the domain
${\mathcal{S}}\setminus\cup_{n\in{\mathbb{N}}}{\mathcal{S}}_{n}$. If $\psi$ is
real, then these zeros are real and satisfy the estimates (1.41).

We have the identities

$\displaystyle\Delta^{2}(E)=\Big{(}{\vartheta(1,E)+\varphi^{\prime}(1,E)\over
2}\Big{)}^{2}=\Big{(}{\vartheta(1,E)-\varphi^{\prime}(1,E)\over
2}\Big{)}^{2}+\vartheta(1,E)\varphi^{\prime}(1,E)$ (5.7)
$\displaystyle=\Big{(}{\vartheta(1,E)-\varphi^{\prime}(1,E)\over
2}\Big{)}^{2}+\vartheta^{\prime}(1,E)\varphi(1,E)+1$

for all $E\in{\mathcal{S}}$. Let $E=\mathfrak{m}_{n},n\in{\mathbb{N}}$. Then
$\varphi(1,E)=0$ and we obtain $\Delta^{2}(E)\geqslant 1$. The relations (5.6)
give the second relations in (1.41).

###  5.3. Results for transpose operator

Let $\lambda\in{\mathscr{D}}_{1}$, where

${\mathscr{D}}_{1}={\mathbb{C}}_{+}\setminus\cup_{n\in\\{0\\}\cup{\mathbb{N}}}\overline{{\mathcal{D}}_{n}}=-{\mathscr{D}}_{3}.$

Then Eq. (1.28) has a simple multiplier
$\widetilde{\tau}_{1}(\lambda,\psi)=\tau_{3}^{-1}(-\lambda,\psi_{*})$ and the
corresponding Floquet solution

$\tilde{\mathfrak{f}}_{1}(x,\lambda,\psi)=\mathfrak{f}_{3}(x,-\lambda,\psi_{*}).$
(5.8)

This solution satisfies $\tilde{\mathfrak{f}}_{1}(x,\lambda)\neq 0$ for all
$(x,\lambda)\in{\mathbb{R}}\times{\mathscr{D}}_{1}$, see [BK24xxx]. Thus, we
may use this solution, as $\eta$, in the substitution (1.10) for Eq. (1.28).

In fact, let $\tilde{y}$ be a solution of Eq. (1.28) and let

$\tilde{f}=\tilde{\mathfrak{f}}_{1}^{3\over
2}\Big{(}{\tilde{y}\over\tilde{\mathfrak{f}}_{1}}\Big{)}^{\prime}.$

Then
$(\tilde{y}^{[2]})^{\prime}\tilde{\mathfrak{f}}_{1}-\tilde{y}(\tilde{\mathfrak{f}}_{1}^{[2]})^{\prime}=-p\tilde{\mathfrak{f}}_{1}^{1\over
2}f$ and we obtain

$-\tilde{f}^{\prime\prime}+\tilde{V}\tilde{f}=E\tilde{f},\quad E={3\over
4}\lambda^{2\over 3}\in{\mathcal{S}},$ (5.9)

where

$\tilde{V}=E-p-{\tilde{\mathfrak{f}}_{1}^{[2]}\over\tilde{\mathfrak{f}}_{1}}+{1\over
4}\Big{(}{\tilde{\mathfrak{f}}_{1}^{\prime}\over\tilde{\mathfrak{f}}_{1}}\Big{)}^{2}-{1\over
2}\biggl{(}{\tilde{\mathfrak{f}}_{1}^{[2]}-p\tilde{\mathfrak{f}}_{1}\over\tilde{\mathfrak{f}}_{1}}-\Big{(}{\tilde{\mathfrak{f}}_{1}^{\prime}\over\tilde{\mathfrak{f}}_{1}}\Big{)}^{2}\biggr{)}=E-{p\over
2}-{3\over
2}{\tilde{\mathfrak{f}}_{1}^{[2]}\over\tilde{\mathfrak{f}}_{1}}+{3\over
4}\Big{(}{\tilde{\mathfrak{f}}_{1}^{\prime}\over\tilde{\mathfrak{f}}_{1}}\Big{)}^{2}.$
(5.10)

The identity (5.8) shows that

$\tilde{V}(E,\psi_{*})=V(E,\psi),\quad E\in{\mathcal{S}}.$ (5.11)

Theorem 1.3 gives the following results.

######  Corollary 5.3.

Let $\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$.

i) Let $\lambda\in{\mathscr{D}}_{1}$ be an eigenvalue of the 3-point Dirichlet
problem for Eq. (1.8) and let $y$ be the corresponding eigenfunction. Then
$E={3\over 4}(-\lambda)^{2\over 3}\in{\mathcal{S}}$ is an eigenvalue of the
Dirichlet problem for Eq. (5.9) and
$\tilde{f}={y\over\sqrt{\tilde{\mathfrak{f}}_{1}}}$ is the corresponding
eigenfunction.

ii) Let $E\in{\mathcal{S}}$ be an eigenvalue of the Dirichlet problem for Eq.
(5.9) and let $\tilde{f}$ be the corresponding eigenfunction. Then
$\lambda=-({4\over 3}E)^{3\over 2}\in{\mathscr{D}}_{1}$ is the eigenvalue of
the three-point Dirichlet problem for Eq. (1.8) and
$y=\tilde{f}\sqrt{\tilde{\mathfrak{f}}_{1}}$ is the corresponding
eigenfunction.

###  5.4. Transformation of ramifications and 3-point eigenvalues

The following lemma establishes some relations between the set of the
ramifications and three-point eigenvalues for the third order operators and
the set of the 2-periodic and the three-point eigenvalues for the Schrödinger
operators.

######  Lemma 5.4.

Let $\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$. Then

i) Each number ${3\over 4}(r_{n}^{\pm})^{2\over 3},n\geqslant 1$, is an
eigenvalue of the 2-periodic problem for Eq. (1.11). Each number $({4\over
3}E_{n}^{\pm})^{3\over 2},n\geqslant 1$, is a ramification for Eq. (1.8).

ii) Each number ${3\over 4}(-\mu_{-n})^{2\over 3}\in{\mathcal{S}},n\geqslant
1$, is an eigenvalue of the Dirichlet problem for Eq. (5.9). The corresponding
eigenfunction has the form $y_{-n}\tilde{\mathfrak{f}}_{1}^{-{1\over 2}}$,
where $y_{-n}$ is the eigenfunction of the 3-point Dirichlet problem for Eq.
(1.8) corresponding to the eigenvalue $\mu_{-n}$, $\tilde{\mathfrak{f}}_{1}$
is the Floquet solution of Eq. (1.28) corresponding to the multiplier
$\widetilde{\tau}_{1}$.

Conversely, each number $-({4\over 3}\mathfrak{m}_{n})^{3\over 2},n\geqslant
1$, is an eigenvalue of the 3-point Dirichlet problem for Eq. (1.11) and
$\tilde{y}_{-n}=\tilde{f}_{n}\sqrt{\tilde{\mathfrak{f}}_{1}}$ is the
corresponding eigenfunction, where $\tilde{f}_{n}$ is the eigenfunction of the
Dirichlet problem for Eq. (5.9).

Proof. i) If $\psi\in{\mathcal{B}}_{\mathbb{C}}(\varepsilon_{0})$ and
$\lambda\in{\mathscr{D}}_{3}$, then the multiplier $\tau_{3}$ is simple, the
corresponding Floquet solution $\mathfrak{f}_{3}$ does not vanish for all
$x\in{\mathbb{R}}$, and $\mathfrak{f}_{3}^{[2]}(0)\neq 0$, see [BK24xxx]. Then
Theorem 1.2 i) gives the statement.

ii) Corollary 5.3 gives the statement.

Proof of Theorem 1.4. i) The results are proved in Lemma 5.2 i).

ii) Lemma 5.4 i) and the statement i) give the results.

Proof of Theorem 1.5 i)-ii). i) The results are proved in Lemma 5.2 ii).

ii) Let $n\in{\mathbb{N}}$. Lemma 5.4 gives (1.43). The identities (1.39)
gives $-\mu_{-n}(\psi)=\mu_{n}(\psi_{*}^{-})$. The identity (1.27) implies

${3\over 4}(\mu_{n}(\psi_{*}^{-}))^{2\over 3}=\mathfrak{m}_{n}(\psi_{*}),$

which yields (1.42).

Lemmas 5.1 ii) and 5.4 i) yield that all $\mu_{n}$ and
$r_{n}^{\pm},n\in{\mathbb{N}}$, are positive. Let $n\in{\mathbb{N}}$. The
relations (1.41) imply

$\mathfrak{m}_{n}(\psi^{-})\in[E_{n}^{-}(\psi^{-}),E_{n}^{+}(\psi^{-})].$

The identities (1.37) and (1.30) yield (1.44).

###  5.5. Norming constants for three-point problem

Let $n\in{\mathbb{N}}$. Norming constants for the Dirichlet problem for the
second order operator are introduced by

$\mathfrak{h}_{sn}=2\pi
n\log\big{(}(-1)^{n}\varphi^{\prime}(1,\mathfrak{m}_{n})\big{)},$

see (1.4). In the following theorem we express $\mathfrak{h}_{sn}$ in terms of
the eigenfunctions of the 3-point problem.

######  Proposition 5.5.

The function $\varphi$ and the norming constants $\mathfrak{h}_{sn}$ satisfy

$\varphi^{\prime}(1,\mathfrak{m}_{n})={\widetilde{y}^{\prime}(1,\widetilde{\mu}_{n})\over\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})}\tau_{3}^{-{1\over
2}}(\widetilde{\mu}_{n}),$ (5.12) $\mathfrak{h}_{sn}=2\pi
n\log\Big{(}(-1)^{n}{\widetilde{y}^{\prime}(1,\widetilde{\mu}_{n})\over\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})}\tau_{3}^{-{1\over
2}}(\widetilde{\mu}_{n})\Big{)},\quad n\in{\mathbb{N}}.$ (5.13)

where $\widetilde{y}(t,\widetilde{\mu}_{n})$ is the eigenfunction of the
operator $\widetilde{\mathcal{L}}_{dir}$, corresponding to the eigenvalue
$\widetilde{\mu}_{n}$, $\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})\neq 0$.

Proof. McKean’s transformation gives

$\varphi(t,\mathfrak{m}_{n})={C\widetilde{y}(t,\widetilde{\mu}_{n})\over\sqrt{\mathfrak{f}_{3}(t,\widetilde{\mu}_{n})}},$

for some $C>0$. This yields

$\varphi^{\prime}(t,\mathfrak{m}_{n})=C\Big{(}{\widetilde{y}^{\prime}(t,\widetilde{\mu}_{n})\over\sqrt{\mathfrak{f}_{3}(t,\widetilde{\mu}_{n})}}-{\widetilde{y}(t,\widetilde{\mu}_{n})\mathfrak{f}_{3}^{\prime}(t,\widetilde{\mu}_{n})\over\mathfrak{f}_{3}^{3\over
2}(t,\widetilde{\mu}_{n})}\Big{)}.$

Then, using
$\widetilde{y}(0,\widetilde{\mu}_{n})=\widetilde{y}(1,\widetilde{\mu}_{n})=\widetilde{y}(2,\widetilde{\mu}_{n})=0$,
we obtain

$\varphi^{\prime}(0,\mathfrak{m}_{n})={C\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})\over\sqrt{\mathfrak{f}_{3}(0,\widetilde{\mu}_{n})}},\quad\varphi^{\prime}(1,\mathfrak{m}_{n})={C\widetilde{y}^{\prime}(1,\widetilde{\mu}_{n})\over\sqrt{\mathfrak{f}_{3}(1,\widetilde{\mu}_{n})}}.$

The identity $\varphi^{\prime}(0,\mathfrak{m}_{n})=1$ gives

$C={\sqrt{\mathfrak{f}_{3}(0,\widetilde{\mu}_{n})}\over\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})},$

which yields

$\varphi^{\prime}(1,\mathfrak{m}_{n})={\widetilde{y}^{\prime}(1,\widetilde{\mu}_{n})\sqrt{\mathfrak{f}_{3}(0,\widetilde{\mu}_{n})}\over\widetilde{y}^{\prime}(0,\widetilde{\mu}_{n})\sqrt{\mathfrak{f}_{3}(1,\widetilde{\mu}_{n})}}.$

This gives (5.12), which implies (5.13).

Proof of Theorem 1.5 iii). The definition (1.40) and the identity (5.13) imply
(1.45).

##  6\. Riemann surfaces

###  6.1. Riemann surfaces for the case
$\psi=\mathop{\mathrm{const}}\nolimits$

Assume that $q=0$ and $p=p_{0}=\mathop{\mathrm{const}}\nolimits$. In this case
the multipliers have the form $\tau_{j}=e^{k_{j}},j=1,2,3$, where $k_{j}$ are
three zeros of the polynomial $k^{3}+2p_{0}k-\lambda$. The multiplier Riemann
surface ${\mathcal{R}}$ coincides with the surface defined by the algebraic
equation $k^{3}+2p_{0}k-\lambda=0$. There are exactly two branch points
$r_{0}^{\pm}=\pm{4\over 3}\sqrt{2\over 3}(-p_{0})^{3\over 2}$. The other
ramifications

$r_{n}^{+}=r_{n}^{-}={4\over 3\sqrt{3}}\big{(}2(\pi
n)^{2}-p_{0}\big{)}\big{(}(\pi n)^{2}-2p_{0}\big{)}^{1\over 2},\quad
r_{-n}^{\pm}=-r_{n}^{\pm},\quad n\in{\mathbb{N}},$

have multiplicity two and the corresponding branch points are degenerated.

Assume $p_{0}<0$ for definiteness, then $r_{0}^{+}=-r_{0}^{-}>0$. In order to
construct the surface ${\mathcal{R}}$ we take three replicas
${\mathcal{R}}_{1},{\mathcal{R}}_{2}$ and ${\mathcal{R}}_{3}$ of the slitted
complex plane:

${\mathcal{R}}_{1}={\mathbb{C}}\setminus[r_{0}^{+},+\infty),\quad{\mathcal{R}}_{2}={\mathbb{C}}\setminus\big{(}(-\infty,r_{0}^{-}]\cup_{n\in{\mathbb{Z}}}[r_{0}^{+},+\infty)\big{)},\quad{\mathcal{R}}_{3}={\mathbb{C}}\setminus(-\infty,r_{0}^{-}].$

We attach the upper (lower) edge of the slit $[r_{0}^{+},+\infty)$ on the
sheet ${\mathcal{R}}_{1}$ to the lower (upper) edge of the same slit on the
sheet ${\mathcal{R}}_{2}$. Similarly, the sheets ${\mathcal{R}}_{2}$ and
${\mathcal{R}}_{3}$ of the surface ${\mathcal{R}}$ are attached along the
slits $(-\infty,r_{0}^{-}]$, see Fig. 1 c).

Consider the parametrization of the surface ${\mathcal{R}}$ by the analytical
mapping ${\mathcal{Z}}(\lambda)=\lambda^{1\over 3},\lambda\in{\mathcal{R}}$,
Fig. 1 a). We have ${\mathcal{Z}}({\mathcal{R}})={\mathcal{Z}}$, where

${\mathcal{Z}}={\mathbb{C}}\setminus\cup_{j\in\\{0,2,3,5\\}}[0,z_{0,j}],\quad
z_{0,j}=(r_{0}^{+})^{1\over 3}e^{i{\pi\over 3}j},\quad 1^{1\over 3}=1.$

On the plane ${\mathcal{Z}}$ we identify with each other the corresponding
edges of the slits $[0,z_{0,j}]$, $n\in{\mathbb{N}}$, $j=0$ with $j=2$ and
$j=3$ with $j=5$: we identify the upper edge of the first slit with the upper
edge of the second slit, we identify the lower edge of the first slit with the
lower edge of the second slit. Moreover, we have
${\mathcal{Z}}({\mathcal{R}}_{j}^{\pm})={\mathcal{Z}}_{j}^{\pm},j=1,2,3$,
where ${\mathcal{R}}_{j}^{\pm}$ is a part of the sheet ${\mathcal{R}}_{j}$
over ${\mathbb{C}}_{\pm}$, and ${\mathcal{Z}}_{j}^{\pm}$ is the sector shown
in Fig. 1 a).

The Floquet solutions have the form $\mathfrak{f}_{j}=e^{k_{j}x},j=1,2,3$. For
each fixed $x\in{\mathbb{R}}$ the functions $\mathfrak{f}_{j}(x,\cdot)$
constitute three branches of a function $\mathfrak{f}(x,\cdot)$ analytic on
the surface ${\mathcal{R}}$. The definition (1.13) gives $V=E-2p_{0}-{3\over
4}k^{2}$ and it is analytic function on the surface ${\mathcal{E}}$, which is
the Riemann surface of the function $k(\lambda(E)),\lambda(E)=({4\over
3}E)^{3\over 2}$.

In order to construct the surface ${\mathcal{E}}$, see Fig. 1 b), we take two
replicas ${\mathcal{E}}_{1}$ and ${\mathcal{E}}_{2}$ of the slitted complex
plane:

${\mathcal{E}}_{1}={\mathcal{E}}_{2}={\mathbb{C}}\setminus\Big{(}(-\infty,0]\bigcup(\cup_{j\in\\{0,2\\}}[0,E_{0}^{+}]e^{i{2\pi\over
3}j})\Big{)},\quad E_{0}^{+}={3\over 4}(r_{0}^{+})^{2\over
3}=-{p_{0}\over\sqrt[3]{2}}.$

We attach the upper (lower) edge of the slit $(-\infty,0]$, on the sheet
${\mathcal{E}}_{1}$ to the lower (upper) edge of the same slit on the sheet
${\mathcal{E}}_{2}$. We identify with each other the corresponding edges of
the slits $[0,E_{0}^{+}]$ on the sheet ${\mathcal{E}}_{1}$ with the slits
$[0,E_{0}^{+}]e^{i{4\pi\over 3}}$ on the sheet ${\mathcal{E}}_{2}$: we
identify the upper edge of the first slit with the upper edge of the second
slit, we identify the lower edge of the first slit with the lower edge of the
second slit. Similarly, we identify the corresponding edges of the slits
$[0,E_{0}^{+}]e^{i{4\pi\over 3}}$ on the sheet ${\mathcal{E}}_{1}$ with the
slits $[0,E_{0}^{+}]$ on the sheet ${\mathcal{E}}_{2}$. Moreover, we have
${\mathcal{E}}({\mathcal{R}}_{j}^{\pm})={\mathcal{E}}_{j}^{\pm},j=1,2,3$,
where ${\mathcal{E}}_{j}^{\pm}$ are the sectors shown in Fig. 1 b). The points
$E_{n}^{+}e^{i{2\pi\over 3}j},j=0,2,n\in{\mathbb{N}},E_{n}^{+}={3\over
4}(r_{n}^{+})^{2\over 3}$, are degenerated ramifications of the surface
${\mathcal{E}}$. The small perturbation divides these points into pares of
simple ramifications.

${\mathcal{Z}}_{1}^{+}$${\mathcal{Z}}_{2}^{+}$${\mathcal{Z}}_{3}^{-}$${\mathcal{Z}}_{2}^{-}$${\mathcal{R}}_{1}$${\mathcal{R}}_{2}$${\mathcal{R}}_{3}$$r_{0}^{+}$$r_{1}^{-}$$r_{0}^{-}$$r_{-1}^{+}$${\mathcal{E}}_{1}^{+}$${\mathcal{E}}_{1,r}^{-}$${\mathcal{E}}_{1,l}^{-}$${\mathcal{E}}_{2}^{+}$${\mathcal{E}}_{3}^{-}$${\mathcal{E}}_{3,l}^{+}$${\mathcal{E}}_{3,r}^{+}$${\mathcal{E}}_{2}^{-}$a)b)c)${\mathcal{Z}}_{1,r}^{-}$${\mathcal{Z}}_{1,l}^{-}$${\mathcal{Z}}_{3,l}^{+}$${\mathcal{Z}}_{3,r}^{+}$$z_{0,0}$$z_{0,2}$$-z_{0,0}$$-z_{0,2}$$z_{1,0}$$z_{1,2}$$-z_{1,0}$$-z_{1,2}$$E_{0}^{+}$$E_{1}^{+}$$E_{0}^{+}e^{i{4\pi\over
3}}$$E_{1}^{+}e^{i{4\pi\over 3}}$ Figure 1. a) The multipliers $z$-plane, b)
$E$-surface ${\mathcal{E}}$, and c) $\lambda$-surface ${\mathcal{R}}$ for the
case $p=\mathop{\mathrm{const}}\nolimits<0$. Each double ramification
$r_{n}^{-}=r_{n}^{+}$ on the surface ${\mathcal{R}}$ (and the corresponding
ramifications $z_{n,j}=(r_{n}^{+})^{1\over 3}e^{i{\pi\over 3}j}$ on the plane
${\mathcal{Z}}$ and $E_{n}e^{i{\pi\over 3}j},E_{n}={3\over
4}(r_{n}^{+})^{2\over 3}$, on the surface ${\mathcal{E}}$) divides into two
simple ramifications on the surface ${\mathcal{R}}$ at small perturbation, see
Fig. 3 c)

Let $p_{0}\to 0$. Then $r_{0}^{\pm}\to 0,r_{n}^{\pm}\to(\pi
n)^{3},n\in{\mathbb{N}}$, and in the limit case we obtain the surfaces for the
unperturbed case $p_{0}=0$. The multiplier surface ${\mathcal{R}}^{0}$ for the
unperturbed case is the surface of the function $\lambda^{1\over 3}$ and it is
shown in Fig. 2 c). The mapping $z(\lambda)=\lambda^{1\over 3}$ gives the
simple parametrization ${\mathcal{R}}^{0}\to{\mathcal{Z}}^{0}={\mathbb{C}}$.
Moreover, we have
${\mathcal{Z}}^{0}({\mathcal{R}}_{j}^{0,\pm})={\mathcal{Z}}_{j}^{\pm},j=1,2,3$,
where ${\mathcal{R}}_{j}^{0,\pm}$ is a part of the sheet
${\mathcal{R}}_{j}^{0}$ over ${\mathbb{C}}_{\pm}$, and
${\mathcal{Z}}_{j}^{\pm}$ are the sectors shown in Fig. 2 a). The
corresponding Riemann surface ${\mathcal{E}}^{0}$ of the function
$k(\lambda(E))$ is the surface of the function $E^{1\over 2}$, see Fig. 2 b).
Note that in this unperturbed case we have $V=0$.

The function $f_{j}=\mathfrak{f}_{3}^{3\over
2}({\mathfrak{f}_{j}\over\mathfrak{f}_{3}})^{\prime},j=1,2$, is the Floquet
solution of Eq. (1.11) satisfying $f_{j}(x+1,E)=t_{j}(E)f_{j}(x,E)$, where
$t_{j}=\tau_{3}^{1\over 2}\tau_{j}$ is a multiplier for Eq. (1.11). Then the
Lyapunov function satisfies

$\Delta={1\over 2}(t_{1}+t_{2})={1\over 2}(\tau_{1}+\tau_{2})\tau_{3}^{1\over
2}={1\over 2}(T-\tau_{3})\tau_{3}^{1\over 2},$

where $T(E)=\mathop{\mathrm{Tr}}\nolimits M(1,\lambda(E))$. Since
$\mathop{\mathrm{Tr}}\nolimits M(1,\cdot)$ is an entire function, the function
$\Delta$ is analytic on the surface ${\mathcal{E}}$.

${\mathcal{Z}}_{1}^{+}$${\mathcal{Z}}_{2}^{+}$${\mathcal{Z}}_{3}^{-}$${\mathcal{Z}}_{2}^{-}$${\mathcal{R}}_{1}^{0}$${\mathcal{R}}_{2}^{0}$${\mathcal{R}}_{3}^{0}$$0$$\pi^{3}$$0$$-\pi^{3}$${\mathcal{E}}_{1}^{0,+}$${\mathcal{E}}_{1,r}^{0,-}$${\mathcal{E}}_{1,l}^{0,-}$${\mathcal{E}}_{2}^{0,+}$${\mathcal{E}}_{3}^{0,-}$${\mathcal{E}}_{3,l}^{0,+}$${\mathcal{E}}_{3,r}^{0,+}$${\mathcal{E}}_{2}^{0,-}$a)b)c)${\mathcal{Z}}_{1,r}^{-}$${\mathcal{Z}}_{1,l}^{-}$${\mathcal{Z}}_{3,l}^{+}$${\mathcal{Z}}_{3,r}^{+}$$\pi$$\pi
e^{i{2\pi\over 3}}$$-\pi$$-\pi e^{i{2\pi\over 3}}$${3\over 4}\pi^{2}$${3\over
4}\pi^{2}e^{i{4\pi\over 3}}$ Figure 2. a) The multipliers $z$-plane, b)
$E$-surface ${\mathcal{E}}$, and c) $\lambda$-surface ${\mathcal{R}}$ for the
case $\psi=0$. Each double ramification $r_{n}^{0}=(\pi n)^{3}$ on the surface
${\mathcal{R}}^{0}$ divides into two simple ramifications on the surface
${\mathcal{R}}$ at small perturbation, see Fig. 3 c)

###  6.2. Riemann surfaces for the case of small coefficients

Let $\psi\in{\mathcal{H}}\oplus{\mathcal{H}}$. Recall that for each
$x\in{\mathbb{R}}$ the functions
$\mathfrak{f}_{1}(x,\cdot),\mathfrak{f}_{2}(x,\cdot),\mathfrak{f}_{3}(x,\cdot)$
constitute three branches of a function $\mathfrak{f}(x,\cdot)$ meromorphic on
the multiplier Riemann surface ${\mathcal{R}}$ and the poles are not depending
on $x$.

If, in addition, $\|\psi\|\leqslant\varepsilon_{0}$, then for any
$\lambda\in{\mathscr{D}}_{3}$ we may take $g=\mathfrak{f}_{3}$ in McKean’s
transformation (1.10)–(1.13). Then each of the potentials
$V(x,\cdot),x\in{\mathbb{R}}$, is analytic on the domain ${\mathcal{S}}$ given
by (1.34). Moreover, it has a meromorphic extension onto the Riemann surface
${\mathcal{E}}$, which is a perturbation of the surface ${\mathcal{E}}^{0}$.

Describe the surface ${\mathcal{E}}$. Represent the construction of
${\mathcal{R}}$ from our paper [BK24xxx] for the case of small coefficients,
see Fig. 3 c). Assume $r_{0}^{-}=r_{0}^{+}=0$ for simplicity. We take three
replicas ${\mathcal{R}}_{1},{\mathcal{R}}_{2}$ and ${\mathcal{R}}_{3}$ of the
slitted complex plane:

${\mathcal{R}}_{1}={\mathbb{C}}\setminus\cup_{n\in{\mathbb{N}}}[r_{n-1}^{+},r_{n}^{-}],\quad{\mathcal{R}}_{2}={\mathbb{C}}\setminus\cup_{n\in{\mathbb{Z}}}[r_{n-1}^{+},r_{n}^{-}],\quad{\mathcal{R}}_{3}={\mathbb{C}}\setminus\cup_{n\leqslant
0}[r_{n-1}^{+},r_{n}^{-}].$

We attach the upper (lower) edge of the slit
$[r_{n-1}^{+},r_{n}^{-}],n\in{\mathbb{N}}$, on the sheet ${\mathcal{R}}_{1}$
to the lower (upper) edge of the same slit on the sheet ${\mathcal{R}}_{2}$.
Similarly, the sheets ${\mathcal{R}}_{2}$ and ${\mathcal{R}}_{3}$ of the
surface ${\mathscr{R}}$ are attached along the slits
$[r_{n-1}^{+},r_{n}^{-}],n\leqslant 0$.

Consider the parametrization of the surface ${\mathcal{R}}$ by the analytical
mapping ${\mathcal{Z}}(\lambda)=\lambda^{1\over 3},\lambda\in{\mathcal{R}}$,
Fig. 3 a). We have ${\mathcal{Z}}({\mathcal{R}})={\mathcal{Z}}$, where

${\mathcal{Z}}={\mathbb{C}}\setminus\cup_{n\in{\mathbb{N}},j\in\\{0,2,3,5\\}}[(r_{n}^{-})^{1\over
3},(r_{n}^{+})^{1\over 3}]e^{i{\pi\over 3}j},\quad 1^{1\over 3}=1.$

On the plane ${\mathcal{Z}}$ we identify with each other the corresponding
edges of the slits $[(r_{n}^{-})^{1\over 3},(r_{n}^{+})^{1\over
3}]e^{i{\pi\over 3}j}$, $n\in{\mathbb{N}}$, $j=0$ with $j=2$ and $j=3$ with
$j=5$: we identify the upper edge of the first slit with the upper edge of the
second slit, we identify the lower edge of the first slit with the lower edge
of the second slit. Moreover, we have
${\mathcal{Z}}({\mathcal{R}}_{j}^{\pm})={\mathcal{Z}}_{j}^{\pm},j=1,2,3$,
where ${\mathcal{R}}_{j}^{\pm}$ is a part of the sheet ${\mathcal{R}}_{j}$
over ${\mathbb{C}}_{\pm}$, and ${\mathcal{Z}}_{j}^{\pm}$ is the sector shown
in Fig. 3 a).

The definition (1.13) yields that for each $x\in{\mathbb{R}}$ the function
$V(x,\cdot)$ has a meromorphic extension from the domain ${\mathcal{S}}$ onto
a Riemann surface ${\mathcal{E}}$. Using the identity (1.12) we obtain the
following form for the surface ${\mathcal{E}}$, see Fig. 3 b). In order to
construct the surface ${\mathcal{E}}$ we take two replicas ${\mathcal{E}}_{1}$
and ${\mathcal{E}}_{2}$ of the slitted complex plane:

$\displaystyle{\mathcal{E}}_{1}={\mathbb{C}}\setminus\Big{(}(-\infty,0]\bigcup\cup_{n\in{\mathbb{N}}}\big{(}[E_{n}^{-}(\psi^{-}),E_{n}^{+}(\psi^{-})]\cup[E_{n}^{-}(\psi_{*}),E_{n}^{+}(\psi_{*})]e^{i{4\pi\over
3}}\big{)}\Big{)},$
$\displaystyle{\mathcal{E}}_{2}={\mathbb{C}}\setminus\Big{(}(-\infty,0]\bigcup\cup_{n\in{\mathbb{N}}}\big{(}[E_{n}^{-}(\psi_{*}),E_{n}^{+}(\psi_{*})]\cup[E_{n}^{-}(\psi^{-}),E_{n}^{+}(\psi^{-})]e^{i{4\pi\over
3}}\big{)}\Big{)}.$

We attach the upper (lower) edge of the slit $(-\infty,0]$, on the sheet
${\mathcal{E}}_{1}$ to the lower (upper) edge of the same slit on the sheet
${\mathcal{E}}_{2}$. We identify with each other the corresponding edges of
the slits $[E_{n}^{-}(\psi^{-}),E_{n}^{+}(\psi^{-})]$, $n\in{\mathbb{N}}$, on
the sheet ${\mathcal{E}}_{1}$ with the slits
$[E_{n}^{-}(\psi^{-}),E_{n}^{+}(\psi^{-})]e^{i{4\pi\over 3}}$,
$n\in{\mathbb{N}}$, on the sheet ${\mathcal{E}}_{2}$: we identify the upper
edge of the first slit with the upper edge of the second slit, we identify the
lower edge of the first slit with the lower edge of the second slit.
Similarly, we identify the corresponding edges of the slits
$[E_{n}^{-}(\psi_{*}),E_{n}^{+}(\psi_{*})]e^{i{4\pi\over 3}}$,
$n\in{\mathbb{N}}$, on the sheet ${\mathcal{E}}_{1}$ with the slits
$[E_{n}^{-}(\psi_{*}),E_{n}^{+}(\psi_{*})]$, $n\in{\mathbb{N}}$, on the sheet
${\mathcal{E}}_{2}$. Moreover, we have
${\mathcal{E}}({\mathcal{R}}_{j}^{\pm})={\mathcal{E}}_{j}^{\pm},j=1,2,3$,
where ${\mathcal{E}}_{j}^{\pm}$ are the sectors shown in Fig. 3 b).

The eigenvalue $\mu_{n}(t),n\in{\mathbb{N}}$, of the operator
${\mathcal{L}}_{dir}(\psi(\cdot+t))$ moves around the interval
$(r_{n}^{-},r_{n}^{+})$ making exactly $n$ rounds, as $t$ runs the interval
$[0,1]$, on the corresponding Riemann surface ${\mathcal{R}}$, see [McK81].
Respectively, the eigenvalue $\mathfrak{m}_{n}(t)$ of the Dirichlet problem
for the Schrödinger operator with the potential $V(E,\psi(\cdot+t))$ moves
around the interval $(E_{n}^{-},E_{n}^{+})$ making exactly $n$ rounds, as $t$
runs the interval $[0,1]$, on the corresponding Riemann surface
${\mathcal{E}}$, see Fig. 3.

${\mathcal{Z}}_{1}^{+}$${\mathcal{Z}}_{2}^{+}$${\mathcal{Z}}_{3}^{-}$${\mathcal{Z}}_{2}^{-}$${\mathcal{R}}_{1}$${\mathcal{R}}_{2}$${\mathcal{R}}_{3}$$r_{1}^{-}$$r_{1}^{+}$$r_{-1}^{+}$$r_{-1}^{-}$$\mu_{1}$$\mu_{1}^{1\over
3}$$\mathfrak{m}_{1}(\psi^{-})$$\mu_{-1}$$\mu_{-1}^{1\over
3}$$\mathfrak{m}_{1}(\psi_{*})$${\mathcal{E}}_{1}^{+}$${\mathcal{E}}_{1,r}^{-}$${\mathcal{E}}_{1,l}^{-}$${\mathcal{E}}_{2}^{+}$${\mathcal{E}}_{3}^{-}$${\mathcal{E}}_{3,l}^{+}$${\mathcal{E}}_{3,r}^{+}$${\mathcal{E}}_{2}^{-}$$E_{1}^{-}(\psi^{-})$$E_{1}^{+}(\psi^{-})$$E_{1}^{-}(\psi_{*})$$E_{1}^{+}(\psi_{*})$a)b)c)${\mathcal{Z}}_{1,r}^{-}$${\mathcal{Z}}_{1,l}^{-}$${\mathcal{Z}}_{3,l}^{+}$${\mathcal{Z}}_{3,r}^{+}$
Figure 3. a) The multipliers $z$-plane, b) $E$-surface ${\mathcal{E}}$, and c)
$\lambda$-surface ${\mathcal{R}}$ for the case $\psi$ small. For simplicity we
assume here $r_{0}^{+}=r_{0}^{-}=0$. On the $z$ and $E$-plane (fig. a,b) we
identify with each other the corresponding edges of the slits connected in the
figure by curves with arrows: we identify the upper edge of the first slit
with the upper edge of the second slit, we identify the lower edge of the
first slit with the lower edge of the second slit. The eigenvalue $\mu_{n}(t)$
($\mu_{-n}(t)$) of the the operator ${\mathcal{L}}_{dir}(\psi(\cdot+t))$ moves
around the interval $(r_{n}^{-},r_{n}^{+})$ ($(r_{-n}^{-},r_{-n}^{+})$) making
exactly $n$ round, as $t$ runs the interval $[0,1]$.

Acknowledgments. The work was supported by the RSF grant number 23-21-00023.

## References

  * [BK21] A.Badanin, E.Korotyaev, Hill’s operators with the potentials analytically dependent on energy. Journal of Differential Equations, 271 (2021), 638–664.
  * [BK24] A.Badanin, E.Korotyaev, Inverse problem for the L-operator in the Lax Pair of the Boussinesq’s equation on the circle, will be published in Funct. Anal. and Appl.
  * [BK24x] A.Badanin, E.Korotyaev. Inverse problem for 3-rd order operators under the 3-point Dirichlet conditions, to be published.
  * [BK24xx] A.Badanin, E.Korotyaev, Asymptotics for the Lax operator of the Boussinesq equation under the 3-point Dirichlet conditions, to be published.
  * [BK24xxx] A.Badanin, E.Korotyaev, Asymptotics for 3-rd order operators on the line with periodic small coefficients, to be published.
  * [K99] E.Korotyaev, Inverse Problem and the trace formula for the Hill Operator, II, Mathematische Zeitschrift 231(2) (1999) 345–368.
  * [MO75] V. A. Marchenko, I. V. Ostrovskii, A characterization of the spectrum of Hill’s operator. Mat. USSR Sb. 26 (1975) 493–554.
  * [McK81] H.McKean, Boussinesq’s equation on the circle, Com. Pure and Appl. Math. 34(1981) 599–691.
  * [PT87] J.Pöschel, E.Trubowitz, Inverse spectral theory, Academic Press, Boston, 1987.