Functional relations in Stokes multipliers \\ - Fun with $x^6 + \alpha x^2$ potential-
\parbox{0.9\textwidth}{ {\em \begin{center} Department of Physics
Faculty of Science
836 Ohya
Shizuoka 422
Japan \end{center} }}
We consider eigenvalue problems in quantum mechanics in one dimension. % Hamiltonians contain a class of double well potential terms, $x^6 + \alpha x^2$ , for example . % The space coordinate is continued to a complex plane and the connection problem of fundamental system of solutions is considered. % A hidden $U_q(\widehat{gl}(2|1))$ structure arises in "fusion relations" of Stokes multipliers. % With this observation, we derive coupled nonlinear integral equations which characterize the spectral properties of both $\pm \alpha$ potentials simultaneously.
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\begin{document}
\title{
Functional relations in Stokes multipliers 
\\
- Fun with $x^6 + \alpha x^2$ potential-
}
\author{ J. Suzuki \thanks{e-mail: sjsuzuk@ipc.shizuoka.ac.jp}\\
           \parbox{0.9\textwidth}{
        {\em
        \begin{center}
         Department of Physics, Faculty of Science, \\
	Shizuoka University, \\
	 836 Ohya, Shizuoka 422, \\
	 Japan
        \end{center}
        }}
       }
%
\maketitle
\begin{center}
-Dedicated to Professor R. J. Baxter on the occasion of his 60-th birthday-
\end{center}
\begin{abstract}
We consider eigenvalue problems in quantum mechanics
in one dimension.
%
Hamiltonians  contain  a class of double well potential terms,
$x^6 + \alpha x^2$ , for example .
%
The space coordinate is continued to a complex plane and 
the connection problem of fundamental system of solutions is considered.
%
A hidden $U_q(\widehat{gl}(2|1))$ structure arises
in "fusion relations" of Stokes multipliers.
%
With this observation, we derive coupled nonlinear integral
equations which characterize the spectral properties  of
both $\pm \alpha$ potentials simultaneously.
\end{abstract}

\noindent Key words: Spectral determinants, Stokes multipliers, fusion hierarchy,
nonlinear integral equations

%\noindent pacs number: 05.50.+q,11.55.Ds.
\clearpage
%
\section{Introduction}

The eigenvalue problem of a one-body 
1D Schr{\"o}dinger operator
is the most fundamental subject in quantum mechanics.
%
Still, it provides vivid materials of research.

Besides a few exceptions where eigenvalues and wavefunctions 
are obtainable explicitly, one may employ several tools for analysis,
e.g., the perturbation theory, the variational approach and so on.
%
Among them, the exact WKB method \cite{V1}-\cite{Takei}
is unique in the sense that it
provides non-perturbative 
information on the analytical structure of wavefunctions 
and spectral properties.
%
We analytically continue $x$, original coordinate variable, 
 to a complex number.
%
The whole complex plane is divided into several sectors.
%
In each sectors, there are two linearly independent solutions
, as Schr{\"o}dinger operator is the 2nd order differential
operator. 
%
They are referred to as  the fundamental system of solutions
(FSS) in the sector \cite{FBOOK}.
%
The relations among FSS in different sectors are central
issue in the connection problem.
%
The importance of the problem 
and consequently Stokes multipliers,
in the WKB problem has been deeply recognized 
and emphasized in early '80, especially in \cite{V1}.

Recently, a remarkable link has been established among
the spectral determinants of a 1D Schr{\"o}dinger operator
associated with the anharmonic oscillator, 
transfer matrices and ${\bf Q}$ operators in CFT  
possessing $U_q(\widehat{sl}(2))$
\cite{DT1}-\cite{JS}.
%
Here the spectral determinants imply 
$D(E)=\prod_{E_j \in {\rm eigenvalue}} ( 1- \frac{E}{E_j})$ 
and its generalizations.
%
A curious interplay between $D(E)$ and
 generalized Stokes multipliers is also found \cite{V1, DT2}.
%
In view of solvable models, a striking fact is that 
they share the same functional relations with transfer matrices 
in the fusion hierarchy  possessing $U_q(\widehat{sl}(2))$ 
\cite{DT2, Sbook, Baxbook,BLZ1}.
%
This allows for applications of the strong machineries in the study of
solvable models \cite{Baxbook}-\cite{SuzSpinS} 
to the studies of Stokes multipliers, spectral
determinants and so on.
%
Several results have been explicitly obtained 
for the anharmonic oscillator problem,
and  are extended to higher differential analogues \cite{DT3, JSAN}.

%------------------------------------------------
In this note,  we consider an anharmonic oscillator  
perturbed by a lower power potential term.
%
It belongs to a class of potentials discussed 
generally in \cite{V6,V7} with the 
exact resolution method.
%
To be precise, we consider the eigenvalue problem,
%
\begin{equation}
{\cal H}(x,\alpha) \Psi_k(x)=
\Bigl( -\frac{d^2}{dx^2} + x^{2M} + \alpha  x^{M-1} \Bigr ) \Psi_k(x)= E_k \Psi_k(x).
\label{hamil}
\end{equation}
%
Throughout this report we set $\hbar=1$ and $M >1$.


The spectral problem concerning this Hamiltonian turns out to be 
in a category to which one can apply the tools in
solvable models.

The sign of $\alpha$ seems to be crucial if one
considers the operator (\ref{hamil}) on the real axis.
%

\begin{equation*}
\boxed{\rm FIG. 1}
\end{equation*}

%
We will not expect much difference from the "pure" anharmonic
oscillator when $\alpha>0$, while we do expect change for $\alpha<0$
as the potential develops the double well.

It will be shown, however, that the negative $\alpha$ 
and the positive $\alpha$ problems 
are not separable when we discuss the global connection problem.
%
Roughly speaking, 
the negative $\alpha$ problem is coupled to 
the positive $\alpha$ problem by crossing a border line of
neighboring sectors and vice versa.
See \S 2 for precise arguments.
%
It may be then reasonable to consider a two-fold connection problem
(crossing two adjacent lines) , or more generally, 
relations between sectors
separated by even multiples of border lines.
%
Some of the Stokes multipliers, in the generalized connection problem, 
possess expressions corresponding to the eigenvalues of
the (fusion) transfer matrices of the 3 state Perk-Schulz (PS) model 
\cite{PerkSchulz81, Schulz81,Schulz83}
of which underlying symmetry is  $U_q(\widehat{gl}(2|1))$.
%
Others can not be directly equated with the (fusion) transfer matrices
but  have relations with the the 3 state PS model as well.
%
Thus  we conclude that the perturbation $\alpha  x^{M-1} $
breaks the $U_q(\widehat{sl}(2))$ symmetry of the "pure"
anharmonic oscillator but it brings the new symmetry 
$U_q(\widehat{gl}(2|1))$.
%
The deformation parameter $q$ is related to the exponent
of the perturbation by $q=\exp(i\frac{\pi}{M+1})$.
%
Through these findings, we can  
derive the nonlinear integral equations (NLIE) which characterize
the energy levels of both the negative $\alpha$ problem and 
the positive $\alpha$ problem simultaneously.


The paper is organized as follows.
In the next section, we will explore symmetries of solutions
to (\ref{hamil}). The precise definition of sectors is given.
%
The connection problem is addressed in section 3.
Certain components in fusion Stokes matrices are identified with
eigenvalues of fusion transfer matrices associated to $U_q(\widehat{gl}(2|1))$.
%
The spectral determinant is  explicitly parameterized by 
FSS in a sector.
%
The coupled NLIE are  then derived in section4,
which determine energy levels.
%
We will perform analytical and numerical checks on the consistency
of our result in section 5.
%
Section 6 is devoted to summary and discussions on open problems.


%==========================================================

\section{Asymptotic Expansion and Symmetry of solutions}

Let $\phi(x,\alpha,E) $ be an entire function of $(x,\alpha, E)$
and a  solution to
${\cal H}(x,\alpha) \phi(x,\alpha,E) = E \phi(x,\alpha,E) $.

The solution, 
which decays exponentially at $x \rightarrow \infty$, is of primary interest.
%
By employing the argument in \cite{HS}, we immediately find
its asymptotic behavior,

\begin{eqnarray}
\phi(x, \alpha, E) & \sim&  x^{-M/2 -\alpha/2} 
 \exp(-\frac{x^{M+1}}{M+1}), \label{asymptotics} \\
%
\partial_x \phi(x, \alpha, E) & \sim&  x^{M/2 -\alpha/2} 
 \exp(-\frac{x^{M+1}}{M+1}). \label{asymptotics2}
%
\end{eqnarray}
The validity of the above expansion is not restricted to
the real axis, but extends to
the wedge, $|{\rm arg} x |< \frac{3 \pi}{2M+2}$ \cite{ Sbook, HS}.

The second order linear differential equation admits another
independent solution.
%
To specify it, or to deal with the global problem, it is
convenient to extend  $x$ to the complex plane
as mentioned in introduction.
%
Then, as in the case of $\alpha=0$, the solution exhibits 
a symmetry  by rotating the complex $x$ plane by
a specific angle.

The direct calculation proves the following.
%
\begin{thm}
Let $\phi(x,\alpha,E)$ be the above solution and $q=\exp(i\frac{\pi}{M+1})$.
%
Then $\phi(q^{-1} x, q^{M+1} \alpha, q^2 E)$ is also the solution
to the differential equation, ${\cal H}(x,\alpha) \phi= E \phi$.
\end{thm}

This is the desired second solution  which
grows exponentially on the positive real axis:
$
x^{-M/2 +\alpha/2} 
 \exp(\frac{x^{M+1}}{M+1})$ for $ x \rightarrow \infty$
%
We note that $q^{M+1} \alpha=-\alpha$.
%
This deserves an attention.
%
As mentioned in introduction, the potential assumes 
the completely different
structure for $\alpha$ positive and $\alpha$ negative on the real axis. 
%
The rotation in the complex $x$ plane by angle 
$\frac{\pi}{M+1}$, however, couples these two problems.
%
Thus we shall treat the Hamiltonians with $\pm \alpha$
{\it simultaneously}.
%
Similar pairing of differential equations
is found for the positive and the negative
angular momentum terms in a class of 
3rd order differential equations\cite{DT3}.

%
This observation is crucial in our approach and
can be generalized further.
%
To state it, we prepare some notations.

Hereafter  $\alpha$ always takes a non-negative real value.
%
By ${\cal H}^{(\epsilon)}(x,\alpha)$, we mean
the Schr{\"o}dinger operator,
$$
 -\frac{d^2}{dx^2} + x^{2M} +\epsilon \alpha x^{M-1}
$$
where $\epsilon=\pm 1$.

Let ${\cal S}_k$ be a sector in the plane satisfying
$$
|{\rm arg} x -\frac{k \pi}{M+1}  | \le \frac{\pi}{2M+2}.  
$$

\begin{equation*}
\boxed{\rm FIG. 2}
\end{equation*}

The FSS depends on the sector.
%
We  conveniently define
$$
y_j^{(\epsilon)}:= 
\frac{q^{j/2-\epsilon \alpha/2}}{\sqrt{2 i}}
 \phi(xq^{-j}, \epsilon \alpha, q^{2j} E).
$$

\begin{thm} 
For the ${\cal H}^{(\epsilon)}(x,\alpha)$, the FSS
in the sector ${\cal S}_{j}$ is given by
$(y_j^{(\epsilon_j)}, y_{j+1}^{(\epsilon_{j+1})})$ where
$\epsilon_j=\epsilon (-1)^{j}$.
\end{thm}

For $\alpha=0$ case, this has been argued in \cite{Sbook, DT2}.
%
It is easily checked that $y_j^{(\epsilon_j)}$ is the 
sub-dominant solution in ${\cal S}_j$; it tends to zero
as $x$ tends to infinity along in any direction in the sector.


In the next section, we consider the global connection problem
of these FSS in the complex $x$ plane.
%
%------------------------------------------------------------------
%

\section{Fusion Stokes multipliers, $U_q(\widehat{gl}(2|1))$ structure
 and Spectral Determinants}

We introduce the Wronskian matrix 
\begin{equation}
\Phi_j^{(\epsilon)}(x):=
\begin{pmatrix}
y_j^{(\epsilon)},            &  y_{j+1}^{(-\epsilon)}   \\
\partial_x y_j^{(\epsilon)},   & \partial_x y_{j+1}^{(-\epsilon)} \\
\end{pmatrix}.
\end{equation}
%
and the Wronskian $W_k^{(\epsilon)}:
={\hbox {\rm det}} \Phi_k^{(\epsilon)}(x)$.

The linear dependence of the solution can be easily verified
by evaluating the Wronskian at ${\cal S}_{j+1/2}$ 
using the asymptotic expansions (\ref{asymptotics}) and
(\ref{asymptotics2}) .
%
The present normalization yields $W_k^{(\epsilon)}=1$.

 Let  ${\cal M}_{j,1}^{(\epsilon)}$ 
be the Stokes matrix connecting the Wronskian matrices 
$\Phi_j^{(\epsilon)}(x)$ and $\Phi_{j+1}^{(-\epsilon)}(x)$,
\begin{equation}
\Phi_{j}^{(\epsilon)}(x) =
 \Phi_{j+1}^{(-\epsilon)}(x){\cal M}_{j,1}^{(\epsilon)}.
 \label{single}
\end{equation}
%
It permits an explicit parameterization

\begin{equation}
{\cal M}_{j,1}^{(\epsilon)}:=
\begin{pmatrix}
\tau^{(\epsilon)}_j,            &  1  \\
-1,   &                            0 \\
\end{pmatrix}, 
\end{equation}
where $ \tau^{(\epsilon)}_j $ is referred to as the Stokes multiplier.
%
We have two remarks.
%
First, the (1,1) element is the function of $\alpha$ and $q^{2 j}E$.
We omit the dependency on $\alpha$. The dependency on $E$ is indicated 
by the index $j$.
%
Second, the (2,1) element,$-1$, 
is a consequence of the present normalization
of the Wronskian. 

To be more specific, we consider the operator ${\cal H}^{(+)}(x,\alpha)$
and start from the positive real axis (or more generally ${\cal S}_0$). 
%
The initial FSS is $(y_0^{(+)}, y_1^{(-)})$.

\begin{equation*}
\boxed{\rm FIG. 3}
\end{equation*}

A linear relation follows from (\ref{single}) 
 between this FSS and $(y_1^{(-)}, y_2^{(+)})$,
the FSS at the neighboring sector ${\cal S}_1$, 
\begin{equation}
y_0^{(+)}= \tau^{(+)}_0 y_1^{(-)} -y_2^{(+)}. 
\label{taupy}
\end{equation}
Similarly, the FSS in ${\cal S}_2$ is linked to the FSS in ${\cal S}_1$
by
\begin{equation}
y_1^{(-)}=\tau^{(-)}_0 y_2^{(+)} -y_3^{(-)}. 
\label{taumy}
\end{equation}


Judging from the upper indices  which indicate the corresponding signs of $\alpha$,
it may be natural to introduce a generalized Stokes matrix 
${\cal M}_{0,2}^{(+)}$ connecting
 FSS $(y_0^{(+)}, y_1^{(-)})$ and $(y_2^{(+)}, y_3^{(-)})$.
%
It is simply obtained by the matrix multiplication, 

\begin{equation}
{\cal M}_{0,2}^{(+)}= {\cal M}_{1,1}^{(-)} {\cal M}_{0,1}^{(+)}=
\begin{pmatrix}
\tau^{(-)}_1 \tau^{(+)}_0-1,            &  \tau^{(-)}_1  \\
-\tau^{(+)}_0,   &                            -1 \\
\end{pmatrix}.
\label{fusion2}
\end{equation}

Equations (\ref{taupy}) and (\ref{taumy}) yield $\tau$'s in terms of $y$'s.
%
The (1,1) component in (\ref{fusion2}), hereafter denoted by
$T_{1,1}(E)$,  is then represented 
 in terms of $y$ as, 
\begin{equation}
T_{1,1}(E)=
\tau^{(-)}_1 \tau^{(+)}_0-1=
\frac{y_0^{(+)}}{y_2^{(+)}}+
\frac{y_0^{(+)}y_3^{(-)}}{y_2^{(+)}y_1^{(-)}} +
\frac{y_3^{(-)}}{y_1^{(-)}}.
\label{pstransfer}
\end{equation}
The dependence of $E$ in the rhs is implicitly
indicated by indices of $y$.
%
We will comment on this representation in terms of a solvable model later.

There is another expression using both $y$'s and $\partial y$'s.
This form is of practical use in the following generalization.
%
By applying the Cramer formula to (\ref{single}),
we immediately obtain

\begin{equation}
\tau^{(\epsilon)}_j = 
\begin{vmatrix}
y_j^{(\epsilon)},            &  y_{j+2}^{(\epsilon)}  \\
\partial_x y_j^{(\epsilon)},   &  \partial_x y_{j+2}^{(\epsilon)} \\
\end{vmatrix}.
\label{qwronskian}
\end{equation}
%
Note that we use the fact that the Wronskian is normalized to be unity.
The (1,1) entries in (\ref{fusion2}) is then given by
$$
\begin{vmatrix}
y_0^{(+)},            &  y_{3}^{(-)}  \\
\partial_x y_0^{(+)},   &  \partial_x  y_{3}^{(-)}  \\
\end{vmatrix}.
$$


One can further generalize the above result.
%
Naturally, the "fusion" Stokes matrix ${\cal M}_{j,2 k}^{(+)}$ 
is defined  which 
relates FSS of ${\cal S}_j$ to ${\cal S}_{j+2k}$.
%
Explicitly, it is given by 


\begin{equation}
{\cal M}_{0,2k}^{(+)}=
\begin{pmatrix}
%--(1,1)----
\begin{vmatrix}
y_0^{(+)},            &  y_{2k+1}^{(-)}  \\
\partial_x y_0^{(+)},   &  \partial_x  y_{2k+1}^{(-)}  \\
\end{vmatrix}, &
%---(1.,2)-----
\begin{vmatrix}
y_1^{(-)},            &  y_{2k+1}^{(-)}  \\
\partial_x y_1^{(-)},   &  \partial_x  y_{2k+1}^{(-)}  \\
\end{vmatrix}  \\
%--(2,1)---
-\begin{vmatrix}
y_0^{(+)},            &  y_{2k}^{(+)}  \\
\partial_x y_0^{(+)},   &  \partial_x  y_{2k}^{(+)}  \\
\end{vmatrix},   &  
%--(2,2)---
-\begin{vmatrix}
y_1^{(-)},            &  y_{2k}^{(+)}  \\
\partial_x y_1^{(-)},   &  \partial_x  y_{2k}^{(+)}  \\
\end{vmatrix} \\
\end{pmatrix},
\label{fusion2d}
\end{equation}
for $j=0$.
%
We can prove the above using the induction on $k$ most easily.
%
Similar formula holds for ${\cal M}_{0,2 k}^{(-)}$ by replacing all
upper indices $+ \leftrightarrow -$.

We are now ready to relate an entry in a fusion Stokes matrix to
the spectral determinant.
%
Hereafter we assume $M=2m-1$.
%
It follows from the above argument that
\begin{equation}
%\Phi_{0}^{(+)} &=& \Phi_{2m}^{(+)} {\cal M}_{0,2 m}^{(+)} \\
%{\hbox or } \\ 
\Phi_{2m }^{(+)}= \Phi_{0}^{(+)} ({\cal M}_{0,2 m}^{(+)})^{-1}.
%
\label{posneg}
\end{equation}

\begin{equation*}
\boxed{\rm FIG. 4}
\end{equation*}

 $y_{0}^{(+)}$( $y_{2m}^{(+)}$) stands for 
the subdominant solution on the positive (negative)
real axis. 
%
They tend to zero asymptotically in their proper region, being appropriate
basis for the eigenfunction.  
%
The eq.(\ref{posneg}) tells, however, that $y_{2 m }^{(+)}$ is combined to
 {\it both} $y_{0}^{(+)}$ {\it and} $y_{1}^{(-)}$ by rotating the complex
 plane by $-\pi$, 
 
\begin{eqnarray*}
y_{2m}^{(+)} &=& 
(c_1 y_0^{(+)} +c_2 y_1^{(-)} )/{\rm det} {\cal M}_{0,2 m}^{(+)}\\
c_1&=& ({\cal M}_{0,2 m}^{(+)})_{1,1}, \quad 
c_2= -({\cal M}_{0,2 m}^{(+)})_{2,1}.
\end{eqnarray*}

%
This is an obstacle in constructing an eigenfunction defined
on the whole real axis.
%
The prescription is to demand that the coefficient of
 $y_{1}^{(-)}$ must vanish if $E$ is an eigenvalue.
% 
Consequently, it is proportional to the spectral determinant.
 
%
The coefficient is essentially equal to
the (2,1) component of ${\cal M}_{0,2 m}^{(+)}$, and 
it reads in terms of the original $\phi$ function as 

$$
\frac{q^{\alpha(m+1)}}{2}
\begin{vmatrix}
\phi(x,\alpha, E),            & \phi(-x,\alpha, E) \\
\partial_x \phi(x,\alpha, E),   &  \partial_x  \phi(-x,\alpha,E)  \\
\end{vmatrix}.
$$

The $x$ dependencies are spurious as the entities are
products of Stokes multipliers which are obviously $x$ independent.
%
We  adopt the simplest choice  $x=0$.
%
The coefficient is now proportional to 
$\phi(0,\alpha,E) \partial_x \phi(x,\alpha,E)|_{x=0}$.
%
Thus we conclude that for an eigenvalue $E^{(+)}_j$ of ${\cal H}^{(+)}(x,\alpha)$,
\begin{equation}
\phi(0, \alpha, E^{(+)}_j) =0 \quad\text{ or  } \quad
 \partial_x \phi(x, \alpha, E^{(+)}_j)|_{x=0}=0 
\end{equation}
must hold.

We can repeat the same argument starting from 
${\cal H}^{(-)}(x,\alpha)$ on the positive real axis.

%
The above observation may lead to the identification
\begin{eqnarray}
\phi(0,\epsilon \alpha,E) &\sim& 
 D^{(\epsilon)}_-(E):= \prod_j (1- E^{(\epsilon)}_{-,j}),\\
%
\partial_x \phi(x,\epsilon \alpha,E)|_{x=0} &\sim& 
 D^{(\epsilon)}_+(E) := \prod_j (1- E^{(\epsilon)}_{+,j}) . 
\label{ident}
\end{eqnarray}

The lower sign signifies the parity:
the positive parity means a contribution from
symmetric wave function.
%
The product must be taken
over  eigenvalues with the corresponding parity.
%
The total set eigenvalues $\{ E^{(\epsilon)}_j   \}$  of 
${\cal H}^{(\epsilon)}(x,\alpha)$ 
consists of two subsets,
%
$\{ E^{(\epsilon)}_j   \}$ =$\{ E^{(\epsilon)}_{+,j}   \} \cup 
\{ E^{(\epsilon)}_{-,j}  \}   $ and
$D^{(\epsilon)}(E) = D^{(\epsilon)}_{+}(E) D^{(\epsilon)}_-(E)$.

%
%The similar results for models
%defined only on the positive axis can be derived by more 
%elementary argument \cite{BLZnew, V6}.

We comment on the relation of the present result
to an existing solved model.
%
$T_{1,1}(E)$ in (\ref{pstransfer}) can be represented,
utilizing (\ref{ident}), as
\begin{equation}
T_{1,1}(E)=
%
q^{\alpha-1} \frac{D^{(+)}_-(E)}{D^{(+)}_-(q^4 E)}
+ 
q^{2 \alpha} \frac{D^{(+)}_-(E)D^{(-)}_-(q^6 E) }
      {D^{(+)}_-(q^4 E) D^{(-)}_-(q^2 E)}
+
q^{\alpha+1} \frac{D^{(-)}_-(q^6 E) }
      { D^{(-)}_-(q^2 E)},
\label{dvf11}
\end{equation}
where we safely choose  $x=0$ in the rhs. 

The above expression has similarity to
the dressed vacuum form (DVF) of the (unfused) transfer matrix for
the 3-state PS model with grading $(+,-,+)$.
%
The latter can be found in eq.(3.1) and (3.2) of \cite{Schulz83, JKSfusion}.
%
The spectral parameter $v$ corresponds to energy in the Schr{\"o}dinger operator,
precisely,  $E=\exp(\frac{2 \pi v}{M+1})$.

The spectral determinants have the following identification to 
the eigenvalues of Baxter's $Q$ operators,
\begin{eqnarray*}
%
D^{(+)}_-(E) &=& Q_2(v+\frac{j-2}{2}i)  \\
%
D^{(-)}_-(E) &=& Q_1(v+\frac{j-1}{2}i).  \\
\end{eqnarray*}
%
Those with positive parity may be identified with the
second solutions of  Baxter's $Q$ operators.

The scalar factors (vacuum expectation values)
$f_a(x), g_a(x)$ in  \cite{JKSfusion} 
depend on the choice of the quantum space.
%
We assume that the present quantum space space gives the simple
scalars as in (\ref{dvf11}).
%
In this sense, $T_{1,1}(E)$ exhibits 
the hidden $U_q(\widehat{gl}(2|1))$ symmetry
behind the present Schr{\"o}dinger operator,
just as in the $U_q(\widehat{sl}(2))$ symmetry for $\alpha=0$ problem.
%
This coincidence can be observed further.
%
 We have checked up to certain value of $k$
that the (1,1) element and the (2,2) element of ${\cal M}^{(+)}_{0,2 k}$
coincide with DVF of symmetric fusion transfer matrices
$\Lambda^{(1)}_{k}$ and $-\Lambda^{(1)}_{k-1}$ in
\cite{JKSfusion}, respectively.
%
The interpretation of the (1,2) and the (2,1) element, in terms of 
fusion transfer matrices, is still an open
problem.
%
%In spite of the lack of it, 
%they are also intimately related to the PS model as they
%are expressible by $Q-$ functions associated to the model.


One can adopt another description of $T_{1,1}$.
%
The (2,1) component of (\ref{single}) results
%
\begin{equation}
\tau^{(\epsilon)}_0 =\frac{\partial y_0^{(\epsilon)}}{  \partial y_1^{(-\epsilon)} }
+\frac{\partial y_2^{(\epsilon)} } {  \partial y_1^{(-\epsilon)} }  . 
\label{tqdash}
\end{equation}

Proceeding as above, we arrive at,

\begin{equation}
T_{1,1}(E)=
%
q^{\alpha+1} \frac{D^{(+)}_+(E)}{D^{(+)}_+(q^4 E)}
+ 
q^{2 \alpha} \frac{D^{(+)}_+(E)D^{(-)}_+(q^6 E) }
      {D^{(+)}_+(q^4 E) D^{(-)}_+(q^2 E)}
+
q^{\alpha-1} \frac{D^{(-)}_+(q^6 E) }
      { D^{(-)}_+(q^2 E)}.
\label{dvf11dash}
\end{equation}

In the next section, we determine the energy levels by utilizing the
above results.

%=============================================================


\section{Nonlinear Integral equations for eigenvalue problem}


%
The Bethe ansatz equations follow from the pole-free
property of $T_{1,1}(E) $ on the real $E$ axis,
%
\begin{eqnarray}
%
q^{\alpha-1} \frac{D^{(-)}_+ (q^{2}E^{(+)}_{+,j}  )}
                  {D^{(-)}_+( q^{-2}E^{(+)}_{+,j}   )  }
&=& 
q^{-\alpha-1} \frac{D^{(+)}_+ (q^{2}E^{(-)}_{+,j}  )}
                  {D^{(+)}_+( q^{-2}E^{(-)}_{+,j}   )  }
=-1,   \\
%
q^{\alpha+1} \frac{D^{(-)}_- (q^{2}E^{(+)}_{-,j}  )}
                  {D^{(-)}_-( q^{-2}E^{(+)}_{-,j}   )  }
&=& 
q^{-\alpha+1} \frac{D^{(+)}_- (q^{2}E^{(-)}_{-,j}  )}
                  {D^{(+)}_-( q^{-2}E^{(-)}_{-,j}   )  }
=-1 .
%
\end{eqnarray}
%
To their analysis, we apply 
the strong machinery in solvable models, the
method of nonlinear integral equations.
%
Hereafter we shall confine ourselves to the case
$0\le \alpha \le M$ where energies are nonnegative.
%
The simple pattern of energy spectrum permits the 
following simple-mind choice of auxiliary functions,
\begin{eqnarray}
a^{(\epsilon)}_{\epsilon'}(E) &:=& q^{\epsilon \alpha -\epsilon'}
\frac{D^{(-\epsilon)}_{\epsilon'} (q^{2} E )}
                  {D^{(-\epsilon)}_{\epsilon'} ( q^{-2} E )  } 
  \label{smalla} \\
%
A^{(\epsilon)}_{\epsilon'}(E) &:=& 1+a^{(\epsilon)}_{\epsilon'}(E).  
\label{capitala}
\end{eqnarray}
Thus 
\begin{equation}
A^{(\epsilon)}_{\epsilon'}(E^{(\epsilon)}_{\epsilon', j})=0.
\end{equation}
Remember  that  $\epsilon (=\pm 1)$ represents the signature
of the perturbation while $\epsilon' (=\pm 1)$ denotes the parity.


In addition, we need  some inputs about the asymptotic behaviors 
from the  WKB method.
%
Fortunately, they are already available \cite{DT1} as the existence of
lower power term does not alter them.

\begin{eqnarray}
\ln D^{(\epsilon)}_{\pm}(E) &\sim& \frac{a_0}{2} (-E)^{\mu},
\qquad |E| \rightarrow \infty, |{\rm arg } (-E) | < \pi
\label{wkb1} \\
%
b_0 E_j^{(\epsilon)}  &\sim& 2 \pi (j+\frac{1}{2}), \quad j \rightarrow \infty 
\label{wkbapp}  \\
%
\mu&=& \frac{M+1}{2M}, \quad a_0=\frac{b_0}{2 \sin \mu \pi}, \nonumber \\
%
b_0 &=& \frac{\pi^{1/2} \Gamma(\frac{1}{2M})}{M \Gamma(\frac{3}{2}+\frac{1}{2M})}. 
\nonumber
\end{eqnarray}

There might be several routes to reach  nonlinear integral equations
among $a^{(\epsilon)}_{(\epsilon')}$ and
$A^{(\epsilon)}_{(\epsilon')}$.
%
Here we choose the quickest way \cite{DT1, DT3, DeVe92,BLZ2} 
which fully exploits the fact
that zeros of  $D^{(\epsilon)}_{\pm}(E)$ are
on the positive real $\theta$ axis.
%
In addition, we assume that there are no zeros of $A^{(\epsilon)}_{\pm}(E)$
inside the narrow strip including the positive real axis 
other than those from zeros of  $D^{(\epsilon)}_{\pm}(E)$.

Apparently we have
\begin{eqnarray*}
\log a^{(\epsilon)}_{\epsilon'}(E)  &=& 
 \frac{(\epsilon \alpha - \epsilon') \pi}{M+1} i
+ \sum_{j} F(\frac{E}{E^{(-\epsilon)}_{\epsilon',j}})  \\
%
F(E) &=& \log \frac{1- q^2 E}{1-q^{-2} E}.
\end{eqnarray*}

The above assumption allows the representation of  
the summation part by an integral over contour ${\cal C}_E$
which surrounds the positive real axis counterclockwise,

\begin{equation}
\log a^{(\epsilon)}_{\epsilon'}(E)  = 
\frac{(\epsilon \alpha -\epsilon') \pi}{M+1} i
+ \frac{1}{2\pi i} \int_{ {\cal C}_E } dE' F(\frac{E}{E'}) \partial_{E'}
 \log A^{(-\epsilon)}_{\epsilon'}(E') .
\label{nlie1}
\end{equation}

For convenience, we introduce a variable 
$\theta$ \cite{DT1} by
\begin{equation*}
E =  \exp(\theta/\mu) /\nu^2  \qquad
%
 \nu=(2M+2)^{-\frac{1}{2\mu}} /\Gamma(\frac{1}{2\mu})
\end{equation*}
which originates from the matching condition of the WKB result (\ref{wkb1})
and the $Q-$ operator analysis \cite{BLZ2, BLZ4}.

Let  $\mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta),
 \mathfrak{A}^{(\epsilon)}_{\epsilon'}(\theta)$ be
auxiliary functions defined in (\ref{smalla}), (\ref{capitala}) 
regarded as functions of $\theta$.
%
Then eq.(\ref{nlie1}) reads,
%
\begin{eqnarray*}
\log \mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta)  &=& 
\frac{(\epsilon \alpha -\epsilon') \pi}{M+1} i
+ \frac{1}{2\pi i} \int_{ {\cal C}_\theta } d\theta'  
G(\theta-\theta') \partial_{\theta'}
 \log \mathfrak{A}^{(-\epsilon)}_{\epsilon'}(\theta') \\
%
G(\theta) &=& \log \bigl( q^2 
 \frac{\sinh ( \frac{M \theta}{M+1}+i \frac{\pi}{M+1})  }
       {\sinh(\frac{M \theta}{M+1}-i \frac{\pi}{M+1})}                        
	                 \bigr )   \nonumber 
%
\end{eqnarray*}
where ${\cal C}_\theta$  encircles the whole real axis counterclockwise.

For the reason which will be supplemented,  we shall keep 
$\mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta)$
in the lower half plane but use  
$1/\mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta)$ in the upper half plane. 

This requirement  modifies the above expression as
\begin{eqnarray}
\log \mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta)  &=& 
\frac{(\epsilon \alpha -\epsilon') \pi}{M+1} i
- \frac{1}{2\pi i} 
  \int_{-\infty}^{\infty} \partial_{\theta} G(\theta-\theta'+ i 0) 
  \log \mathfrak{a}^{(-\epsilon)}_{\epsilon'} (\theta'-i 0)   \nonumber\\
& & + \frac{1}{\pi} \Im \Bigl( \int_{-\infty}^{\infty} d\theta'
 \partial_{\theta} G(\theta-\theta'+i 0) 
 \log \mathfrak{A}^{(-\epsilon)}_{\epsilon'}(\theta'-i 0)  \Bigr) \label{nlie2},
\end{eqnarray}
%
where  $\theta$ is assumed to possess small negative imaginary part.
%
The property  $({a}^{(\epsilon)}_{\epsilon'} (\theta) )^*
= \frac{1}{ {a}^{(\epsilon)}_{\epsilon'} (\theta^*)}$ is employed in the above
transformation.

We solve (\ref{nlie2}) in terms of 
$\log \mathfrak{a}^{(\epsilon)}_{\epsilon'} (\theta)$ to reach the final
expression of NLIE, 

\begin{eqnarray}
\ln \mathfrak{a}^{(\epsilon)}_{\epsilon'}(\theta) &=&
%
-\frac{i}{2} b_0 \nu^{-2 \mu} {\rm e}^{\theta} + 
\frac{\pi}{2} i (-\epsilon' + \epsilon \frac{\alpha}{M})   \nonumber\\
%
&+& 2 i \Im \{ \int_{-\infty}^{\infty} K_{1}(\theta-\theta'+i0)
         \ln \mathfrak{A}^{(\epsilon)}_{\epsilon'}(\theta'-i0) d\theta' \nonumber\\
& &   + \int_{-\infty}^{\infty} K_{2}(\theta-\theta'+i0)
                    \ln \mathfrak{A}^{(-\epsilon)}_{\epsilon'}(\theta'-i0) d\theta'
   \}.
\label{nlie}
\end{eqnarray}
%
The kernel functions read
\begin{eqnarray*}
K_1(\theta) &=& 
-\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{iw \theta} 
\frac{ \sinh^2 \frac{\pi (M-1) w}{2 M}}{\sinh\pi w \sinh \frac{\pi w}{M}}  \\
%
K_2(\theta) &=& 
-\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{iw \theta} 
\frac{\sinh \frac{\pi (M+1) w}{2 M} \sinh \frac{\pi (M-1)w}{2 M}  }
 {\sinh\pi w \sinh \frac{\pi w}{M}}.  \\
\end{eqnarray*}


Few remarks are in order.
%
\begin{enumerate}
%
\item As a consequence of connection rules, the integral equations are coupled
for auxiliary functions related to the positive and the negative coefficient
 of $x^{M-1}$ .
%
\item On the other hand,  equations with different parities are decoupled.
%
\item The constants are determined from the consistency by putting $\theta
\rightarrow -\infty$.  The $\alpha$ dependence is only summarized 
in these constants.
%
\item  The first term in the rhs is determined so that we recover the
result from the WKB method (\ref{wkb1}) 
by dropping contributions of integrals.
%
Clearly, $ \mathfrak{a}^{(\epsilon)}_{\epsilon'}(\theta)$ is  bounded 
in the upper-half plane.  This explains our choice of appropriate half planes
for auxiliary functions.

%
\end{enumerate}


The eigenvalues $\{E^{(\epsilon)}_{j,\pm} \}$ are evaluated by
$$
\ln \mathfrak{a}^{(\epsilon)}_{\pm}(\theta^{(\epsilon)}_{j,\pm})=(2 j+1) \pi i,
\quad \text{  and } \quad 
E^{(\epsilon)}_{j,\pm}=  \exp(\theta^{(\epsilon)}_{j,\pm}/\mu)/ \nu^2.
$$

More explicitly

\begin{eqnarray}
& & \frac{1}{2} b_0 \nu^{-2 \mu} {\rm e}^{\theta^{(\epsilon)}_{j,\epsilon'}}
=( 2 j+1- \epsilon' \frac{1}{2} + \epsilon \frac{\alpha}{2 M} ) \pi
           \nonumber \\
& & +2  \Im \{ 
   \int_{-\infty}^{\infty} K_{1}(\theta^{(\epsilon)}_{j,\epsilon'}-\theta'+i0)
              \ln \mathfrak{A}^{(\epsilon)}_{\epsilon'}(\theta') d\theta'
 + \int_{-\infty}^{\infty} K_{2}(\theta^{(\epsilon)}_{j,\epsilon'}-\theta'+i0)
                \ln \mathfrak{A}^{(-\epsilon)}_{\epsilon'}(\theta') d\theta'
           \}
           \nonumber \\
& &  \qquad(j \ge 0).  \label{detenergy}
\end{eqnarray}

We present examples of  numerical solutions
to (\ref{nlie2})  in Fig. 5.
The real and the imaginary parts of $\ln \mathfrak{A}^{(\pm)}_+$ are depicted
for $M=3, \alpha=1$.

\begin{equation*}
\boxed{\rm FIG. 5}
\end{equation*}



\section{benchmarks}
We shall check the nonlinear integral equations analytically for limiting cases
and numerically.

\noindent (1) $\alpha=0$ case \par
\noindent By putting,  
$\mathfrak{a}^{(+)}_{\pm}(E)=\mathfrak{a}^{(-)}_{\pm}(E)$
the coupled NLIE reduce to an identical integral equation.
%
Immediately seen, the result coincides with
the nonlinear integral equation in \cite{DT1}.
\par
\noindent (2) $\alpha=M$ case \par
\noindent In this case, we have a duality in energy spectra;
$ \{E^{(+)}_j \}$ coincide with $ \{E^{(-)}_j \}$ ,
except for $E^{(-)}_0=0$ in the latter.  
%
This degeneracy can be easily explained by the following representation
of the Hamiltonians \cite{facto, cann},
\begin{eqnarray*}
{\cal H}^{(-)}(x,\alpha=M) &=& {\cal D}^{\dagger} {\cal D}  \\
%
{\cal H}^{(+)}(x,\alpha=M) &=& {\cal D} {\cal D}^{\dagger} \\
%
{\cal D} &=& \frac{1}{i} \frac{d}{d x} -i x^M.
\end{eqnarray*}
Once an eigenvector 
${\cal H}^{(-)}(x,M) \psi^{(-)}_j =E^{(-)}_j \psi^{(-)}_j $
is found,
we can construct the eigenvector for ${\cal H}^{(+)}(x,\alpha)$
with the same energy by  $\psi^{(+)}_{j-1}:= {\cal D}  \psi^{(-)}_{j}$.
%
Only the exception is the $j=0$ case where ${\cal D}  \psi^{(-)}_{j=0}=0$.
%
It is interesting that the asymptotic form (\ref{asymptotics})
from  the WKB type argument
is exact for all $x$ in this case.
 \footnote
{I thank V.V. Bazhanov, R.J. Baxter and B. Nienhuis for pointing
out the explicit eigenfunction for  $\psi^{(-)}_0 $   right after 
my talk.}

The above facts can be also  verified from (\ref{nlie}).
%
Note that the rhs can be treated as the mod $2\pi i$ quantity.
%
Then the  choice $\alpha=M$ leads to the same coupled equations under 
identifications
%
${\mathfrak a}^{(+)}_+(\theta ) \leftrightarrow 
      {\mathfrak a}^{(-)}_-(\theta )$,
$ {\mathfrak a}^{(+)}_-(\theta ) \leftrightarrow
     {\mathfrak a}^{(-)}_+(\theta )$.
%
This explains the degeneracy of the spectra as it consists both
 from the negative and the positive parity contributions.
 %
The zero energy case must be treated more separately.
By choosing $j=0, \epsilon=-\epsilon'=-1, \alpha=M$, 
we  find the first term in lhs of (\ref{detenergy})
is null.  So the first order approximation is 
$\theta^{(-)}_{+, 0}=-\infty$.  Actually this is exact as we determine
the constant terms so that NLIE are consistent in
$\theta \rightarrow -\infty$. See remark 3 after (\ref{nlie}).
This solution gives the missing energy 0.

Finally we present the preliminary numerical results for $M=3$.
%
% \begin{figure}[hbtp]
%\centering
%  \includegraphics[width=6cm]{g3.ps}
%\caption{First two energy levels from
%the IMSL routine, the NLIE and the WKB method. 
%Horizontal axis denotes $\alpha$.}
%\label{comp}
%\end{figure}

% Fig. \ref{comp} shows the results from the  IMSL package, the (naive) 

\begin{table}[hbtp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$\alpha$&      IMSL  0-th&  IMSL 1st&    WKB  0th&   WKB 1st&  NLIE 0th& NLIE 1st\\
\hline
-2.5000&            0.22909& 2.3741&  $\diamond$&	2.36641 &  0.22872&	2.37175\\
-2.0000&            0.44007&  2.7962&       0*&	2.73228&        0.43969&	2.79688\\
-1.5000&            0.63726&  3.2028&        0.17736 &	3.09594&    0.63673&    3.20230\\
-1.0000&           0.81664&  3.5949&      0.38490&	3.45603&        0.81478& 3.59506\\
-0.50000&          0.98599&  3.9732&     0.59582 &  3.81142&   0.98547& 3.97303\\
0.0000&            1.1448&  4.3385 &       0.8008&  4.16123& 	1.1440&    4.3382\\
0.50000&           1.2943&  4.6917&      0.99516&  4.50476&  1.2931&  4.6918\\
1.0000&            1.4356&  5.0333&      1.1768& 4.84147&  1.43596&  5.0336\\
1.5000&            1.5696&  5.3642&     1.3456 &  5.17101&   1.57034&  5.3640\\
2.0000&            1.6972&  5.6850&     1.5024&  5.49313&   1.69667&  5.6842\\
2.5000&            1.8189&  5.9962&     1.6487&  5.80773&   1.81861&  5.9960\\
\hline
\end{tabular}
%\caption{First two energy levels calculated by 
% the IMSL library, the (naive) WKB method and the NLIE method. 
% %
% We choose $M=3$ and adopt various $\alpha$.
% %
% See asterisk and $\diamond$ for the text.}
% \label{tab:compare}
\end{center}
\end{table}

Table 1 shows the results from the  
IMSL package (dsleig.f), the (naive)  WKB method
and those obtained by solving the nonlinear equations.
%
The agreement is  not yet precise enough (typically 3-4 digits).
%
Some implement is still in need for the numerical accuracy.
%
Nevertheless, the NLIE data already 
show much improvement from the (naive) WKB results.

By the (naive) WKB method, we mean a self-consistent determination of
$E^{(\epsilon)}_j $ by
\begin{equation}
\oint |p| dx =  
\int_{-x_0}^{x_0}
  \sqrt{E^{(\epsilon)}_j-x^6-\epsilon \alpha x^2} dx =
%
(j+\frac{1}{2}) \pi
\label{wkbnaive}
\end{equation}
where  $E^{(\epsilon)}_j-x_0^6-\epsilon \alpha x_0^2=0$.
%
Particularly, for the value with asterisk, this method has subtlety.
%
Immediately seen, 
$E^{(\epsilon)}_0=0, x_0=2^{1/4}$ is a formal solution to (\ref{wkbnaive})
for $j=0, \epsilon=-, \alpha=2$.
%
It however involves an isolated turning point of the 2nd order at the origin 
if $E^{(\epsilon)}_0=0$, 
which spoils the simple application of the condition  (\ref{wkbnaive}).
%
The value with $\diamond$ has similar difficultly.
%
We  however skip further discussion on the validity on the (naive)  WKB method
as it is out of the present subject.

Summarizing, we check the consistency of (\ref{nlie}) in some limiting cases
and by numerical methods.

\section{Summary and Discussion}

In this report, the eigenvalue problem has been addressed for 
 the 1D quantum systems of which Hamiltonians include
double well potentials.
%
We have successfully derived the coupled NLIE which determine
 energy levels of the systems with potential terms of 
$\pm \alpha x^{M-1} + x^{2 M}$
at the same time.

The essence of our strategy is to utilize the following
 correspondences
between 1D quantum mechanics and  1+1 D solvable models,
\vskip 0.9cm
\begin{tabular}{rrl}
energy &    $\Longleftrightarrow$ &     spectral parameter \\
Stokes multipliers&  $\Longleftrightarrow$ &     transfer matrices \\
eigenfunctions or derivatives at $x=0$ &  $\Longleftrightarrow$ &    
    vacuum expectation values of $Q$ operators \\
\end{tabular}
\vskip 0.9cm
%
We are then entitled to apply the strong machinery of the latter
developed since  Baxter's revolution.
%

There are several open questions.
\begin{enumerate}
\item 
In this report we confine ourselves to the simplest
case $\alpha \le M$. 
%
 For $\alpha > M$, the existence of negative eigenvalues
ruins the analyticity assumptions on auxiliary functions.
%
Still, formal expressions of NLIE are possible which are
similar to   excited states TBA  equations.
%
The integration contour is, however,  not so simple as 
described here.  
%
This is an apparent drawback in actual numerical investigations.
%
The clever choice of auxiliary functions may be desired.
%
\item The understanding is lacking on the intrinsic reason 
why affine symmetry like
$U_q(\widehat{sl}(2) )$ or  $U_q(\widehat{gl}(2|1) )$ 
comes into play in
this simple 1D quantum mechanical model.
%
\item This is somewhat related to the above, but is the 
 most intriguing question. 
%
 {\bf Where is the Yang-Baxter equation
in the 1D Schr{\"o}dinger operator problem?}.
%
Once this is known, the fusion hierarchy, useful in the present study,
 is a mere corollary of it.
\end{enumerate}
We hope to answer these in the future publication.




\section*{Acknowledgments}
The author thanks V. V. Bazhanov, 
A. Kuniba, J.M. Maillard, B. Nienhuis P. Pearce, and C. Richard 
for comments, discussions and encouragement.
%
He also would like to thank the organizers of "Baxter's revolution
in mathematical physics".


%\end{document}

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\end{thebibliography}
 
\newpage
\begin{center}
{FIGURE CAPTIONS} 
\end{center}

\noindent Fig 1. An anharmonic oscillator perturbed by
a positive or a negative perturbation term.

\noindent Fig 2. The complex plane is divided into sectors.
${\cal S}_0$ and ${\cal S}_1$ are indicated as examples.

\noindent Fig 3. The FSS in 
${\cal S}_0$ and ${\cal S}_1$ are related by the matrix  
${\cal M}_{0,1}^{(+)}$.

\noindent Fig 4. The connection of FSS on the negative and the positive real axis
is accomplished by  ${\cal M}_{0,2 m}^{(+)}$.

\noindent Fig 5. Left: the real part of $\ln \mathfrak{A}^{(\pm)}_+$,
Right: the imaginary part of  $\ln \mathfrak{A}^{(\pm)}_+$.

\begin{center}
{TABLE CAPTIONS} 
\end{center}


\noindent Table 1. 
First two energy levels calculated by  the IMSL library, the (naive) 
WKB method and the NLIE method.  We choose $M=3$ and adopt various $\alpha$.
See asterisk and $\diamond$ for the text.


\newpage

\begin{figure}[hbtp]
\centering
  \includegraphics[width=10cm]{potentia.eps}
%\caption{An anharmonic oscillator perturbed by
%a positive or a negative perturbation term.}
\caption{}
\label{pot}
\end{figure}

\newpage
 \begin{figure}[hbtp]
\centering
  \includegraphics[width=10cm]{sectors.eps}
%\caption{The complex plane is divided into sectors.
%${\cal S}_0$ and ${\cal S}_1$ are indicated as examples. }
\caption{}
\label{sec}
\end{figure}

\newpage
\begin{figure}[hbtp]
\centering
  \includegraphics[width=10cm]{StokesM.eps}
%\caption{The FSS in 
%${\cal S}_0$ and ${\cal S}_1$ are related by the matrix  ${\cal M}_{0,1}^{(+)}$.}
\caption{}
\label{stokesm}
\end{figure}

\newpage
\begin{figure}[hbtp]
\centering
  \includegraphics[width=10cm]{realaxis.eps}
%\caption{The connection of FSS on the negative and the positive real axis
%is accomplished by  ${\cal M}_{0,2 m}^{(+)}$.}
\caption{}
\label{connection}
\end{figure}

\newpage
 \begin{figure}[hbtp]
\centering
{  \includegraphics[width=5cm]{fig-pos-re.eps} \hspace{1cm}
  \includegraphics[width=5cm]{fig-pos-im.eps} }
%\caption{Left: the real part of $\ln \mathfrak{A}^{(\pm)}_+$,
%Right: the imaginary part of  $\ln \mathfrak{A}^{(\pm)}_+$.
%}
\caption{}
\label{afig}
\end{figure}


 
\end{document}



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