数理解析研究所講究録
第 1563 巻 2007 年 124-140

124

Rotation number for the one-dimensional
Schr\"odinger operator with periodic singularpotentials
首都大学東京大学院理工学研究科数理情報科学専攻 新國裕昭 (Hiroaki Niikuni)

Department of Mathematics and Information Sciences,
Tokyo Metropolitan University

1.

Introduction and main result

In this article, we survey the results in [13, 14, 15]. In those papers, we study the onedimensional Schrodinger operatosr with singular potentials. In order to explain the motivation of our study, we describe its background. Such operators plays an important rolein solid state physics (see [10]) and have been studied in numerous work [1, 2, 5, 6, 8,
11, 16, 17]. In 1931, Kronig and Penney introduced the Hamiltonians which is formally
expressed as$L_{1}=- \frac{d^{2}}{dx^{2}}+\beta\sum_{l=-\infty}^{\infty}\delta(x-2\pi l)$

where

of

$L_{1}$

in$L^{2}(\mathbb{R})$is the Dirac delta function at the origin andis given through the boundary conditions on the lattice,$\beta\in \mathbb{R}\backslash \{0\}$$\delta(x)$$(L_{1}y)(x)=- \frac{d^{2}}{dx^{2}}y(x)$$Dom(L_{1})=\{y\in H^{2}(\mathbb{R}\backslash 2\pi \mathbb{Z})|$,$2\pi \mathbb{Z}$$x\in \mathbb{R}\backslash 2\pi \mathbb{Z}$. The precise definition
as folows.

,$(y(x+0)y(x+0))=(\begin{array}{ll}1 0\beta 1\end{array})(forx\in 2\pi \mathbb{Z}y(x-0)y(x-0))\}$

R. This operator
where $H^{2}(D)$ denotes the Sobolev space of order 2 on an open set
is the Hamiltonian for an electron in a one-dimensional crystal and is called KronigPenney Hamiltonian. The Dirac delta function is the most typical point interaction. The-interaction was widely generalized. In $[5, 6]$ , Gesztesy, Holden, and Kirsch inspired aof the form
new class of point interactions. They syudied the operator in$D\subset$$\delta$$L^{2}(\mathbb{R})$$(L_{2}y)(x)=- \frac{\theta}{dx^{2}}y(x)$,$x\in \mathbb{R}\backslash 2\pi \mathbb{Z}$

,

125

$Dom(L_{2})=\{y\in H^{2}(\mathbb{R}\backslash 2\pi \mathbb{Z})|$$(\begin{array}{ll}y(x +0)y(x +0)\end{array})=(\begin{array}{ll}1 \beta 0 1\end{array})(\begin{array}{ll}y(x -0)y’(x -0)\end{array})forx\in 2\pi \mathbb{Z}\}$.

This operator has the formal expressionin$L_{2}=- \frac{d^{2}}{dx^{2}}+\beta\sum_{l=-\infty}^{\infty}\delta’(x-2\pi l)$$L^{2}(\mathbb{R})$.

In [16], \v{S}eba found that the domain of any self-adjoint extension ofinof coupled type is expressed as$(-d^{2}/dx^{2})|_{C^{\infty}(R\backslash \{0\})}$$0$$L^{2}(\mathbb{R})$$\{y\in H^{2}(\mathbb{R}\backslash \{0\})|$$(_{y(+0)}y(+0))=cA(y(-0)y(-0))\}$

with $A\in SL(2,\mathbb{R}),$, and $|c|=1$ , where
denotes the special linear group(see also [2] and [1, Section K.1.4]). In [8], Hughes gave the Floquet-Bloch decompositionof the Schr\"odlinger operator inwith generalized point interaction on a latticedefined as$c\in \mathbb{C}$$SL(2,\mathbb{R})$$L^{2}(\mathbb{R})$$2\pi \mathbb{Z}$$(L_{3}y)(x)=- \frac{d^{2}}{dx^{2}}y(x)$$Dom(L_{3})=\{y\in H^{2}(\mathbb{R}\backslash 2\pi \mathbb{Z})|$,,$x\in \mathbb{R}\backslash 2\pi \mathbb{Z}$$(\begin{array}{ll}y(x +0)y(x +0)\end{array})for=cA(x\in 2\pi \mathbb{Z}y(x-0)y(x-0))\}$.

These backgrounds motivate us to vtudy the spectra of the onedimensional Schr\"odinger
operators with periodic generalized point interactions.
To define the operators, we introduce notations. We fix $n\in N=\{1,2,3, \ldots\}$ . Let$0=\kappa_{0}<\kappa_{1}<\cdots<\kappa_{n}=2\pi$
be a partition of the interval $(0,2\pi)$ . We put
for $j=1,2,$, , and. For,
and
we define the one-dimensional Schr\"odinger operatorin
as follows.
$\Gamma_{j}=\{\kappa_{j}\}+2\pi \mathbb{Z}$$\ldots$$n$$\Gamma=\Gamma_{1}\cup\Gamma_{2}\cup\cdots\cup\Gamma_{n}$$\{\theta_{j}\}_{j=1}^{n}\subset \mathbb{R}$$\{A_{j}\}_{-1}^{n}\subset SL_{2}(\mathbb{R})$$H=H(\theta_{1}, \theta_{2}, \ldots, \theta_{n}, A_{1}^{J-},A_{2}, \ldots,A_{n})$$L^{2}(\mathbb{R})$$(Hy)(x)=-y”(x)$ ,$Dom(H)=\{y\in H^{2}(\mathbb{R}\backslash \Gamma)|$$x\in \mathbb{R}\backslash \Gamma$,$(_{y(x+0)}y(x+0))=e^{\theta_{f}}A_{j}(forx\in\Gamma_{j},j=1,$(1.1)$2,..,ny’(x.-0)y(x-0))\}$

.

(1.2)

This operator $H$ is self-adjoint (see [13, Proposition 2.1]). Since the spectrum of $H$ isindependent of(see [14, Proposition $1.1(e)]$ ), we may put$\{\theta_{j}\}_{j=1}^{n}\subset \mathbb{R}$$\theta_{1}=\theta_{2}=\cdots=\theta_{n}=0$,

126which does not cause any loss of generality. Since $H$ has -periodic point interactions,the spectrum of $H$ has the band structure. According to the Floquet-Bloch theory, we
aslabel each band of the spectrum of $H$ . For $j\in N$ , we designate the $jth$ band of$2\pi$$\sigma(H)$$B_{j}=[\lambda_{2j-2}, \lambda_{2j-1}]$

The sequence$\{\lambda_{n}\}_{n=0}^{\infty}\subset \mathbb{R}$

.

(1.3)

satisfies the inequalities.

$\lambda_{0}<\lambda_{1}\leq\lambda_{2}<\lambda_{3}\leq\lambda_{4}<\cdots\leq\lambda_{2j-2}<\lambda_{2j-1}\leq\lambda_{2j}<\cdotsarrow\infty$

So, the consequtive bands$B_{j}$

and$B_{j+1}$

are separated by an open interval$G_{j}:=(\lambda_{2j-1)}\lambda_{2j})$,.
which is called the $jth$ gap ofwith two problems. One of the problems is to give
mainlydealt
In [13, 14, 15], weby the rvtation number. The other is toa characterization of the band edges ofdetermine the indices of the absent spectral gaps in a class of $H$ .
We quote the main theorem in [14]. For this purpose, we introduce the rotation number.
First, we consider the Schr\"odinger equation$\sigma(H)$$\sigma(H)$$-y”(x, \lambda)=\lambda y(x, \lambda)$,$(y(x+0,\lambda)y(x+0,\lambda))=A_{j}(_{y(x-0,\lambda)}y(x-0,\lambda))$$x\in \mathbb{R}\backslash \Gamma$,$x\in\Gamma_{j}$

,

(1.4)

,

$j=1,2,$ $\ldots,n$ ,(1.5)

where is a real parameter. We define the Pr\"ufer transform of a nontrivial solution
be the polar coordinates of $(y,y’)$ :to (1.4) and (1.5) as follows. Let$\lambda$$y(x, \lambda)$$(r,\omega)$$y=r\sin\omega$

Then we call the function$1,2,$ $\cdots n$ , we write$\omega=\omega(x, \lambda)$$y’=r$ cos,$w(x, \lambda)$.

the Pr\"ufer transform of$A_{j}=(\begin{array}{ll}a_{j} b_{j}c_{j} d_{j}\end{array})$Then,$\omega$$y(x, \lambda)$. For each $j=$
(1.6)

.

satisfies the equation$\omega’(x, \lambda)=\cos^{2}\omega(x, \lambda)+\lambda$

sin2$w(x, \lambda)$

,

(1.7)

$x\in \mathbb{R}\backslash \Gamma$

as well as the boundary conditionssin( sin $\omega(x-O,$ $\lambda)+d_{j}\cos w(x-O,$ )$=\cos\omega(x+O, \lambda)$ (sin $\omega(x-O,$ $\lambda)+b_{j}\cos\omega(x-O,\lambda)$ ),$\omega(x+O, \lambda)$$\lambda)$$c_{j}$(1.8)$a_{j}$$sgn(\sin w(x+O, \lambda))=sgn$ ($a_{j}$

sin $w(x-O,$$\lambda)+b_{j}$coo $\omega(x-O,$$\lambda)$

),

(1.9)

127

$sgn(\cos\omega(x+O, \lambda))=sgn$

for$x\in\Gamma_{j}$and $j=1,2,$$\ldots$( sin $\omega(x-O,$$c_{j}$$\lambda)+d_{j}$cos $\omega(x-O,$$\lambda)$(1.10)), , where$n$$sgn(x)=\{\begin{array}{ll}1 if x>0,0 if x=0,-1 if x<0.\end{array}$

To determine the principal value of $w(x+0,\lambda)$ by the boundary conditions (1.8), (1.9),
and (1.10), we must select a branch offor. We choose the branch ofas$x\in\Gamma$$\omega(x+0, \lambda)$$\omega(x+0, \lambda)$$w(x+O, \lambda)-\omega(x-O,\lambda)\in[-\pi,\pi)$for$x\in\Gamma$

.

(111)

Thanks to this selection,is uniquely determined. We pickbe the solution of $(1.7)-(1.10)$ subject to the initial condition$\omega(x+0, \lambda)$$w_{0}\in \mathbb{R}$. Let $w=$$\omega(x, \lambda,\omega_{0})$$w(+0,\lambda)=\omega_{0}$

.

(112)

We define the rotation number of $H$ as$\rho(\lambda)=\lim_{narrow\infty}\frac{\omega(2n\pi+0,\lambda,w_{0})-\omega_{0}}{2n\pi}$.

(1.13)We recall (1.3). In [14], we proved the following theorem which relatesof $H$ .

$\rho(\lambda)$

to the spectrum

Theorem 1.1. The following statements (a), (b), and (c) hold true.
(a) The limit on the right-hand side of (1.13) exists and is independent of the initial value$w_{0}$.

(b) The function(c) We put$\rho(\lambda)${

is non-decmasing on R.

$l=\#$ $j\in\{1,2,$ $\ldots,n\}|$$(b_{j}<0)$or

where $\# Astand_{8}$ for the number of the elements of
we have$(b_{j}=0,$$A$$d_{j}<0)$},(1.14)

for a finite set A. Then, for $j\in N$,$\lambda_{2j-2}=\max\{\lambda\in \mathbb{R}|$$\rho(\lambda)=\frac{j-1}{2}-\frac{l}{2}\}$$\lambda_{2j-1}=\min\{\lambda\in \mathbb{R}|$$\rho(\lambda)=\frac{j}{2}-\frac{l}{2}\}$,.

(1.15)(1.16)

We note that (1.15) and (1.16) critically depend on the choice of the branch of) for(see [14, Section 4]).
The rotation number has a close relationship to the density of states. In order to see
that, we introduce the density of states for $H$ . For $k\in N$ , we put. Let us$\omega(x+$$0,$$\lambda$$x\in\Gamma$$I_{k}=\Gamma^{c}\cap(0,2\pi k)$

128

introduce the generalized Kronig-Penney Hamiltonian in $L^{2}((0,2\pi k))$ with the Dirichletboundary conditions$y(+0)=y(2\pi k-0)=0$ .

We define the operator$H_{2\pi k,D}$

as$(H_{2\pi k,D}y)(x)=-y’’(x)$$Dom(H_{2\pi k,D})=\{y\in H^{2}(\mathbb{R}\backslash \Gamma)|$

For $n\in N\cup\{0\}$ , let$\lambda_{k,n}$for,$x\in I_{k}$,$(_{y(X+0)}y(x+0))=A_{j}(y(x-0).)x\in\Gamma_{j}\cap(0,2\pi k),j=1,2,,ny(+0)=y(2\pi k-0)=0y(x-0)..\}$

be the $(n+1)st$ eigenvalue of$H_{2\pi k,D}$$\nu(k, \lambda)=\#\{n\in N\cup\{0\}| \lambda_{k,n}\leq\lambda\}$

.

. Put
.

Then we have the following theorem.
Theorem 1.2. We have$\lim_{karrow\infty}\frac{\nu(k,\lambda)}{2\pi k}=\frac{\rho(\lambda)}{\pi}+\frac{l}{2\pi}$

.

(117)

In the physics literatures, the left-hand side of (1.17) is refered to as the density ofstates. We will give the outline of the proof of Theorem 1.2 in Section 2; the complete
proof is found in [14].. On the other hand, we did not describe Theorem 1.2 in [14]. So,
we give the complete proof of it in Section 2.
Our study [14] is also motivated by the works $[9, 12]$ , which we recal below. Johnson
and Moser found that the rotation number for the one-dimensional Schr\"odinger operatorswith almost periodic potentials has a close relation to its spectrum. They dealt with the
Schr\"odinger operator $L=-d^{2}/dx^{2}+q(x)$ , where is an almost periodic function with a$h\Re uency$ module. They proved that the rotation number
for exists and defines. Furthermore,is constant in an open
a continuous function infor. In the special case where isinterval in a spectral gap andofis expressed asperiodic of period , they found that the th band$q$$\mathcal{M}$$\alpha(\lambda)$$\{\lambda\in \mathbb{C}| {\rm Im}\lambda\leq 0\}$$I$

$2\alpha(\lambda)\in \mathcal{M}$$2\pi$$j$$L$$\alpha(\lambda)$$\lambda\in I$$\tilde{B}_{j}$$q$$\sigma(L)$(118)$\tilde{B}_{j}=\overline{\{\lambda|\frac{j-1}{2}<\alpha(\lambda)<\frac{j}{2}\}}$

for $j\in N$ . This means that$\tilde{\lambda}_{2j-2}=\max\{\lambda\in \mathbb{R}|$$\alpha(\lambda)=\frac{j-1}{2}\}$$\tilde{\lambda}_{2j-1}=\min\{\lambda\in \mathbb{R}|$$\alpha(\lambda)=\frac{j}{2}\}$

,

,

129
where
solution to. Let

be the number of the zeroes in $[0, x]$ of a nontrivial$(L\varphi)(x)=\lambda\varphi(x)$ . Then theydescribed that$\tilde{B}_{j}=[\tilde{\lambda}_{2j-2},\tilde{\lambda}_{2j-1}]$$N(x, \lambda)$$\lim_{xarrow\infty}\frac{N(x,\lambda)}{x}=\lim_{xarrow\infty}\frac{\nu(x,\lambda)}{x}=\frac{\alpha(\lambda)}{\pi}$

,

where
is the number of eigenvalues ofin $[0,x]$ with theboundary conditions $y(O)=y(x)=0$ . In contrast to these results, our theorems involve
the number of the interactions in the fundamental region.
Next, we introduce the results in $[13, 15]$ . The aim of those papers is to determine
the indices of the absent spectral gaps of. In [13], we dealt with the casewhere$\nu=\nu(x, \lambda)$$(Ly)(x, \lambda)=\lambda y(x, \lambda)$$H(\theta_{1},\theta_{2},A_{1},A_{2})$$A_{1},$$E$$A_{2}\in SO(2)\backslash \{E, -E\}$

,

(119)

being the 2 $x2$ unit matrix. We put$A_{j}=(\begin{array}{ll}cos\gamma_{j} -sin\gamma_{j}sin\gamma_{j} cos\gamma_{j}\end{array})$$\bm{t}d$$\gamma_{j}\in(0,\pi)\cup(\pi, 2\pi)$for $j=1,2$ . We define$\Lambda=\{m\in N| G_{m}=\emptyset\}$.

In [13], we have the following theorem.

Theorem 1.3. Adopt the assumption (1.19). Let(a) Suppose that
and$\Lambda=\emptyset$.
. Then we have$\kappa_{1}\neq\pi$$\gamma_{1}+\gamma_{2}\not\equiv 0(mod \pi)$$\gamma_{1}-\gamma_{2}\not\equiv 0$.

(b) Suppose that $\gamma_{1}+\gamma_{2}\equiv 0(mod \pi)$ . Then we have$\Lambda=\{${3}$j\in N$if$\{3\}\cup\{pk+1| k\in N\}$$(c)Assume$ that,

$ifif$$\gamma_{1}-\gamma_{2}\equiv 0$. Then it holds thatand$\lrcorner^{\kappa}=9\lrcorner^{\kappa}2\pi p\pi\not\in \mathbb{Q}$$(p, q)\in N^{2}$$\gamma_{1}+\gamma_{2}\not\equiv 0(mod \pi)$,. We putand$gcd(p, q)=1$ .

$\eta_{j}=\pi^{2}j^{2}/4(\pi-\kappa_{1})^{2}$$\bigcup_{k=1}^{\infty}B_{k}\cap B_{k+1}=\{\eta_{j}|-2(\sqrt{\eta_{j}}+\frac{1}{\sqrt{\eta_{j}}})^{-1}\cot\kappa_{1}\sqrt{\eta_{j}}=\tan\gamma i$forand $j\in N\}$ .

In [15], we dealt with the case where$A_{1}A_{2}=\pm E$

and$A_{1},A_{2}\in SL(2,\mathbb{R})\backslash \{E, -E\}$

For convenience we rewrite the elements of$A_{1}$

as$A_{1}=(\begin{array}{ll}a bc d\end{array})$.

Then we have the following theorem [15, Theorem 1.2].

.

(1.20)

130

Theorem 1.4. Adopt the assumption (1.20). Let.
(a) Assume that
. Then we have
if $d=a$ , $b\neq 0$ , $-c/b=k^{2}/4$$\Lambda=\{$otherwise.
$\kappa_{1}\neq\pi$$\kappa_{1}/\pi\not\in \mathbb{Q}$$\emptyset\{k+1\}$

for some $k\in N$ ,(b) Suppose that $\kappa_{1}/2\pi=q/p,$ $(p, q)\in N^{2}$ , and $gcd(p, q)=1$ . Then we have$\Lambda=\{\begin{array}{l}\{pj| j\in N\}b=0\{1+pj| j\in N\}\cup\{1+k\}d=ab\neq 0-c/b=k^{2}/4k\in Nk\not\equiv O(mod p)\{1+pj| j\in N\}\end{array}$Using Theorem 1.1, we can newly get a theorem. We discuss the spectral gaps of the
Schr\"odinger operator formally expressed as$L_{4}=- \frac{d^{2}}{dx^{2}}+\sum_{l=-\infty}^{\infty}(\beta_{1}\delta(x-\kappa_{1}-2\pi l)+\beta_{2}\delta’(x-2\pi l))$

,

where
and
are parameters. In our notations this operator isexpressed as $L_{4}=H(0,0, M_{1}, M_{2})$ , where$\kappa_{1}\in(0,2\pi)$$\beta_{1},\beta_{2}\in \mathbb{R}\backslash \{0\}$and$M_{1}=(\begin{array}{ll}1 0\beta_{1} 1\end{array})$$M_{2}=(\begin{array}{ll}1 \beta_{2}0 1\end{array})$

We have the following theorem for the operator
Theorem 1.5. We suppose that
and

$L_{4}$.

.

$\kappa_{1}\neq\pi$$(\beta_{1}, \ )$\not\in$\{(\frac{n\pi}{|\pi-\kappa_{1}|}t\bm{t}\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|},$ $- \frac{4|\pi-\kappa_{1}|}{n\pi}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|})|$$n\in N\}$.

Then we have the following statements (i) and (ii).
(i) If eitherorhol&, then$\kappa_{1}\not\in\{\pi/2,3\pi/2\}$$\beta_{1}\neq\beta_{2}$$\Lambda=\emptyset$(ii) If $\kappa_{1}\in\{\pi/2,3\pi/2\}$ and$\beta_{1}=\beta_{2}$.

, then$\Lambda=\{\begin{array}{ll}\{2\} if \beta_{1}>0,\{3\} if \beta_{1}<0.\end{array}$

The study ofis motivated by the work [17]. In [17], Yoshitomi investigated thespectral gaps of the operators$L_{4}$( $\beta_{1}\delta(x-\kappa-2\pi l)+$ 角\delta (x--2\pi l))in( $\beta_{1}\delta’(x-\kappa-2\pi l)+$ 角\delta ’(x-2\pi l))in$P_{0}=- \frac{d^{2}}{dx^{2}}+\sum_{l=-\infty}^{\infty}$$L^{2}(\mathbb{R})$,and$P_{1}=- \frac{d^{2}}{dx^{2}}+\sum_{l=-\infty}^{\infty}$$L^{2}(\mathbb{R})$,where $\kappa\in(0,2\pi)$ . For $j\in N$ and $k\in\{0,1\}$ , he described thathas an absent$\beta_{1}+\beta_{2}=0$
gap if and only if
andhold. Furthermore, his theorems say that if
and $\kappa/2\pi=m/n,$ $(n, m)\in N^{2}$ , and $gcd(m, n)=1$ , then the jth gap ofis absent if and only if $j-k\in nN$ . We prove Theorem 1.5 in Section 3.
$\sigma(P_{k})$$\kappa/\pi\in \mathbb{Q}$$\beta_{1}+\beta_{2}=0$$\sigma(P_{k})$

131

2.

Proof of Theorem 1.2 and 1.3

In this section, we describe the proof of Theorem 1.2 and 1.3. We recall (1.6). Let$q_{j}=\#${ $k\in\{1,2,$$q_{0}=0$$\ldots,j\}|$$(b_{k}<0)$or$(b_{k}=0,$$d_{k}<0)$

},

,

and

$\eta_{j}=\{\begin{array}{ll}Arc\tan(b_{j}/d_{j})-q_{j-1}\pi if b_{j}>0, d_{j}>0,Arc\tan(b_{j}/d_{j})+\pi-q_{j-1}\pi, if b_{j}>0, d_{j}<0,\pi/2-q_{j-1}\pi if b_{j}>0, d_{j}=0,Arc\tan(b_{j}/d_{j})-\pi-q_{j-1}\pi, if b_{j}<0, d_{j}<0,Arc\tan(b_{j}/d_{j})-q_{j-1}\pi if b_{j}<0, d_{j}>0,-\pi/2-q_{j-1}\pi if b_{j}<0, d_{j}=0,-q_{j-1}\pi if b_{j}=0, d_{j}>0,-\pi-q_{j-1}\pi if b_{j}=0, d_{j}<0\end{array}$for $j=1,2,$$\ldots$, , where $Arc\tan(x)\in(-\pi/2,\pi/2)$ for$n$$x\in \mathbb{R}$. Since$q_{j}=\{\begin{array}{ll}q_{j-1}+1 if (b_{j}<0) or (b_{j}=0, d_{j}<0),q_{j-1} therwise,\end{array}$we have.

$\eta_{j}\in[-q_{j}\pi, -q_{j}\pi+\pi)$

We pick a$\gamma\in(0,\pi)$(21)such that$\eta_{j}<-q_{j}\pi+\gamma$

for $j=1,2,$$,$$\ldots,n$.

Then we have the following lemma.

Lemma 2.1. There exists$\lambda_{0}\in \mathbb{R}$such that$-\pi(q_{j}+pq_{\mathfrak{n}})\leq\omega(\kappa_{j}+2\pi p+0, \lambda,\omega_{0})\leq-\pi(q_{j}+pq_{n})+\gamma$

for any $p\in N\cup\{0\},$ $j=1,2,$$\ldots,n,$$\lambda\leq\lambda_{0}$, and$\omega_{0}\in[0,\gamma]$.

To prove this lemma, we recall a fundamental fact on the Pr\"ufer transform from [3,Chapter 8, Theorem 2.1]. Let $c<d$ . For), letbe the $8olution$ to
the initial value problem$\beta\in[0,\pi$$\frac{d}{dx}\theta=\cos^{2}\theta+\lambda$

sin2$\theta|_{x=c}=\beta$$\theta$$\theta=\theta(x, \lambda, c,\beta)$on$\mathbb{R}$

,

.

(2.2)

(2.3)

Then, it holds that$\lim_{\lambdaarrow-\infty}\theta(d, \lambda,c,\beta)=0$.

Moreover, the function
is strictly monotone increasing on R.
We describe the outline of the proof of Lemma 2.1.
$\theta(d, \cdot, c,\beta)$

(2.4)

132

Outline of the proof of Lemma 2.1. We fix.
statements by induction on $j=1,2,$$\ldots,$

The limit$\beta_{j}$$\omega_{0}\in[0, \gamma]$. First, we shall show the following$n$$:= \lim_{\lambdaarrow-\infty}w(\kappa_{j}-0, \lambda, \omega_{0})\in \mathbb{R}$exists, and we have $\beta_{j}=-q_{j-1}\pi$ .

(2.5)

(2.6)
The function
is strictly monotone increasing on R.
It follows by (2.4) that (2.5) and (2.6) are valid for $j=1$ . We pick $m\in\{1,2, \ldots , n\}$ ,arbitrarily. Suppose that (2.5) and (2.6) hold for $j=m$ . Then we can show that the limit$\omega(\kappa_{j}-0, \cdot,\omega_{0})$$\alpha_{m}$$:= \lim_{\lambdaarrow-\infty}\omega(\kappa_{m}+0, \lambda,\omega_{0})$exists and$\alpha_{m}=\eta_{m}$

.

(2.7)

By (1.8), we havetan$w( \kappa_{m}+0, \lambda,\omega_{0})=\frac{a_{m}t\bm{t}\omega(\kappa_{m}-0,\lambda,\omega_{0})+b_{m}}{c_{m}\tan\omega(\kappa_{m}-0,\lambda,w_{0})+d_{m}}$

.

(2.8)

Combining the monotonicity ofand $a_{m}d_{m}-b_{m}c_{m}=1$ with (2.8), we find$w(\kappa_{m}+0,\cdot,w_{0})$strictly
increasing
monotone
on R.
is
that
$(2.6)$ is valid for $j=m+1$ .
Since, we infer that there exists
such that
Using the monotonicity of$w(\kappa_{m}-0, \cdot,\omega_{0})$$\omega(\kappa_{m+1}-0, \lambda,\omega_{0})=\theta(\kappa_{m+1}, \lambda, \kappa_{m},\omega(\kappa_{m}+0, \lambda,\omega_{0})),$$\omega(\kappa_{m}, \cdot,\omega_{0})$$\lambda_{m}\in \mathbb{R}$(2.9)$-q_{m}\pi\leq w(\kappa_{m}+0, \lambda,\omega_{0})\leq-q_{m}\pi+\gamma$

for$\lambda\leq\lambda_{m}$. By the comparison theorem [3, Chapter 8] and (2.9), we have$\theta(\kappa_{m+1}, \lambda, \kappa_{m}, -q_{m}\pi)\leq\omega(\kappa_{m+1}-0, \lambda,\omega_{0})<\theta(\kappa_{m+1}, \lambda, \kappa_{m}, -q_{m}\pi+\gamma)$

for$\lambda\leq\lambda_{m}$. Since the equation (2.2) is -periodic, we derive$\pi$$\lim_{\lambdaarrow-\infty}\theta(\kappa_{m+1}, \lambda, \kappa_{m}, -q_{m}\pi)=\lim_{\lambdaarrow-\infty}\theta(\kappa_{m+1}, \lambda, \kappa_{m}, -q_{m}\pi+\gamma)=-q_{m}\pi$so that$\beta_{m+1}=-q_{m}\pi$,.

So, we have proved (2.5) and (2.6) for $j=m+1$ . Therefore, (2.5) and (2.6) are valid for.
$j=1,2,$$\ldots,$

Put

$n$

$\lambda_{0}=m\dot{i}_{1<\leq n}\lrcorner\lambda_{j}$. We have(2.10)$-\pi q_{j}\leq w(\kappa_{j}+0, \lambda,\omega_{0})<-\pi q_{j}+\gamma$.
for $j=1,2,$ $\ldots,n$ , andUsing the comparison theorem and$\lambda\leq\lambda_{0}$$\omega_{0}\in[0,\gamma]$, we notice that$\omega(\kappa_{j}+0, \lambda, 0)\leq\omega(\kappa_{j}+0, \lambda,\omega_{0})\leq\omega(\kappa_{j}+0, \lambda,\gamma)$.

.
Therefore the estimate (2.10) is uniform with respect to-periodic$(1.6)-(1.9)$
respectiswithto , we have the desired
Since the equations
assertion from (2.10).
$w_{0}\in[0, \gamma]$$2\pi$$x$$\square$133
Proof of Theorem 1.1. By a similar way to the proof of [7, Theorem 2.1], it follows that(a) and (b) hold. So, we have only to show the statement (c). We recall (1.14). Then, we
notice that $q_{n}=l$ . By Lemma 2.1, we have$-\pi pl\leq\omega(2\pi p+0, \lambda,\omega_{0})\leq-\pi pl+\gamma$

for$0\leq\omega_{0}\leq\gamma,$$\lambda\leq\lambda_{0}$, and $p\in N$ . This together with (1.13) implies that$\lim_{\lambdaarrow-\infty}\rho(\lambda)=-\frac{l}{2}$.

(2.11)

with the discussion in the proof of [4, Proposition 2.1.], we get the$Combin\dot{i}g(2.11)$

assertion (c).

$\square$

Proof of Theorem 1.3. By (2.5) and (2.6), we have$\lim_{\lambdaarrow-\infty}\omega(\kappa_{j}-0, \lambda,w_{0})=-q_{j-1}\pi$,

and the function
is strictly monotone increasing on R. Since the equation$(1.7)-(1.10)$ is-periodic with respect to , we have$w(\kappa_{j}-0, \cdot,w_{0})$$2\pi$$x$$\lim_{\lambdaarrow-\infty}\omega(2\pi p-0, \lambda,\omega_{0})=-q_{n-1}\pi-\pi(p-1)l$and the function $w(2\pi p-0, \cdot,w_{0})$ is strictly monotone increasing on for $p\in N$ . Becauseof the monotonicity of $w(2\pi p-0, \cdot,w_{0})$ , there exists
satisfying$\mathbb{R}$$\lambda_{p,m}\in \mathbb{R}$$w(2\pi p-0, \lambda_{p,m}, \omega_{0})=-\pi\{q_{n-1}+(p-1)l\}+m\pi$

for each $m\in N$ . In a similar way to [3, Chapter 8, Theorem 2.1], we see that$(m+1)st$ eigenvalue of.
, arbitrarily. Define
We fix$\lambda_{p.m}$is the$H_{2\pi p,D}$$\lambda\in \mathbb{R}$$m_{p}^{*}=\#\{m\in N| \lambda_{p,m}\leq\lambda\}+1$.

Then we have$\lambda_{p,m_{p}^{*}}\leq\lambda<\lambda_{p,m_{\dot{p}}+1}$.

By the monotonicity of $\omega(2\pi p-0, \cdot,w_{0})$ , we have$-\pi\{q_{\mathfrak{n}-1}+(p-1)l\}+m_{p}^{*}\pi<w(2\pi p+0, \lambda,w_{0})<-\pi\{q_{n-1}+(p-1)l\}+(m_{p}^{*}+1)\pi$

This inequality reduces$m_{p}^{l}< \frac{\omega(2\pi p+0,\lambda,w_{0})}{\pi}+q_{n-1}+(p-1)l<m_{p}^{*}+1$

So we derive$m_{p}^{*}=[ \frac{\omega(2\pi p+0,\lambda,\omega_{0})}{\pi}]+q_{n-1}+$

(

$p$

一

$1$

) .
$l$

.

.

134

By the definition of$\gamma(p, \lambda)$and$m_{p}^{*}$, we have$\gamma(p, \lambda)=m_{p}^{*}=[\frac{\omega(2\pi p+0,\lambda,\omega_{0})}{\pi}]+q_{n-1}+(p-1)l$(2.12).

On the other hand, we notice that$\frac{w(2\pi p+0,\lambda,w_{0})/\pi}{2_{\Psi}}+\frac{q_{\mathfrak{n}-1}+(p-1)l-1}{2p\pi}$$\leq$$\frac{[w(2\pi p+0,\lambda,\omega_{0})/\pi]}{2p\pi}+\frac{q_{\mathfrak{n}-1}+(p-1)l}{2_{\Psi}}$$\leq$$\frac{w(2\pi p+0,\lambda,w_{0})/\pi}{2p\pi}+\frac{q_{n-1}+(p-1)l}{2p\pi}$(2.13)

Using (2.12), (2.13), and (1.11), we get (1.17).

3.

口Proof of Theorem 1.5

In this section, we prove Theorem 1.5. In the first place, we define the monodromy matrix.
For this purpose, we consider the equations
一$y”(x, \lambda)=\lambda y(x, \lambda)$,$x\in \mathbb{R}\backslash \Gamma$

,

(31)

$(y(x+0,\lambda)y(x+0,\lambda))=(\begin{array}{ll}l 0\beta_{1} 1\end{array})(_{y^{j}(x-0,\lambda)}y(x-0,\lambda))$$x\in\Gamma_{1}$

,

(3.2)

$(y(x+0,\lambda)y(x+0,\lambda))=(\begin{array}{ll}1 \beta_{2}0 1\end{array})(_{y(x-0,\lambda)}y(x-0,\lambda))$$x\in\Gamma_{2}$

,

(3.3)

where is real parameter. These equations have two solutions
are uniquely determined by the initial conditions$\lambda$$y_{1}(+0, \lambda)=1$,$y_{1}’(+0, \lambda)=0$,$y_{2}(+0, \lambda)=0$,$y_{2}’(+0, \lambda)=1$,

and$y_{1}(x, \lambda)$and$y_{2}(x, \lambda)$

whichrespectively. Then, the monodromy matrix of $(3.1)-(3.3)$ is defined as(3.4)$M(\lambda)=(_{y_{1}’(2\pi}y_{1}(2\pi\ddagger_{0,\lambda)}^{0,\lambda)}y_{2}(2\pi+0,\lambda)y_{2}’(2\pi+0,\lambda))$As described in [17, Lemma 4] (see also [13, 15]), we have$\mathcal{B}:=\bigcup_{k=1}^{\infty}B_{k}\cap B_{k+1}=${$\lambda\in \mathbb{R}|$$M(\lambda)=E$or$M(\lambda)=-E$

}.

135
Put
$\tau=2\pi-\kappa_{1}$.

By a direct calculation, we get$y_{1}(2\pi+0, \lambda)=(1+\beta_{1}oe)$$-\beta_{2}\sqrt{\lambda}$

cos

cos

$y_{1}’(2\pi+0, \lambda)=\beta_{1}$cos$\tau\sqrt{\lambda}$$y_{2}(2\pi+0, \lambda)=\hslash$cos$\tau\sqrt{\lambda}$$\tau^{\sqrt{\lambda}\cos\kappa_{1}\sqrt{\lambda}}+(\frac{\beta_{1}}{\sqrt{\lambda}}-\beta_{2}\sqrt{\lambda})$$\tau\sqrt{\lambda}$

cos
cos

sin

$\kappa_{1^{\sqrt{\lambda}}}$$\kappa_{1}\sqrt{\lambda}+\frac{1}{\sqrt{\lambda}}$

cos

sin

$\tau\sqrt{\lambda}$$\tau\sqrt{\lambda}$$\kappa_{1}\sqrt{\lambda}-\sqrt{\lambda}$$+ \frac{1+\beta_{1}\beta_{2}}{\sqrt{\lambda}}coe\tau\sqrt{\lambda}$$y_{2}’(2\pi+0, \lambda)=\cos\tau\sqrt{\lambda}$

-sin
sin
sin

sin

cos

$\kappa_{1^{\sqrt{\lambda}}}$sin$\tau\sqrt{\lambda}$

cos$\kappa_{1}\sqrt{\lambda}$

(3.5)
,
(3.6)

,

$\kappa_{1^{\sqrt{\lambda}-f_{\lambda}}}$cos$\tau\sqrt{\lambda}\sin\kappa_{1}\sqrt{\lambda}$$\tau\sqrt{\lambda}\cos\kappa_{1}\sqrt{\lambda}$$\kappa_{1^{\sqrt{\lambda}}}+(\frac{\beta_{1}}{\lambda}-\beta_{2})\sin\tau\sqrt{\lambda}$$\kappa_{1}\sqrt{\lambda}+\frac{\beta_{1}}{\sqrt{\lambda}}\cos\tau^{\sqrt{\lambda}}$sin$\kappa_{1}\sqrt{\lambda}$

-sin

sin

$\tau\sqrt{\lambda}$$\kappa_{1}\sqrt{\lambda}$

sin

(3.7)

,

$\kappa_{1}\sqrt{\lambda}$

.

(3.8)

.

(3.9)

In order to establish $Th\infty rem1.5$ , we show the following theorem.
Theorem 3.1. We suppose that($\beta_{1}$, 鳥)\not\inand$\kappa_{1}\neq\pi$$\{(\frac{n\pi}{|r,-\kappa_{1}|}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|},$$- \frac{4|\pi-\kappa_{1}|}{n\pi}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|})|$$n\in N\}$

Then we have the followzng statements (i) and (ii).
holds, then we haveor(i) If either$\kappa_{1}\not\in\{\pi/2,3\pi/2\}$$\beta_{1}\neq\beta_{2}$$\mathcal{B}=\emptyset$(ii) If $\kappa_{1}\in\{\pi/2,3\pi/2\}$ and$\beta_{1}=\beta_{2}$.

, then we have$\mathcal{B}=\{1\}$.

We prove this theorem by using the following lemma.

Lemma 3.2. Assume thatand $M(\lambda)=\pm E$ . Then we have the follo Utng statements.
(i) If, then.
and, then there exists $n\in N$ such that(ii) If$\kappa_{1}\neq\pi$$\lambda\neq-\beta_{1}/\$$\lambda=\ /\beta_{1}$$\infty s\kappa_{1}\sqrt{\lambda}=\cos\tau\sqrt{\lambda}=0$$\lambda=-\beta_{1}/\hslash$$\beta_{1}=\frac{n\pi}{|\pi-\kappa_{1}|}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|}$,and$\beta_{2}=-\frac{4|\pi-\kappa_{q}|}{n\pi}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|}$.

136
Proof. We suppose that $M(\lambda)=\pm E$ . We first show that$\lambda\neq 0$. We have$M(0)=(\begin{array}{ll}1+\beta_{l}\beta_{2}+\beta_{l}\tau \beta_{2}+\tau+(1+\beta_{1}\beta_{2})\kappa_{l}+\beta_{1}\kappa_{l}\tau 0 l+\beta_{1}\kappa_{1}\end{array})$This means $M(O)\neq\pm E$ because ofSince $M(\lambda)=\pm E$ , we have

$1+\beta_{1}\kappa_{1}\neq 1$. This is why$\lambda\neq 0$.

.

$y_{1}(2\pi+0, \lambda)-y_{2}’(2\pi+0, \lambda)=y_{1}’(2\pi+0,\lambda)=y_{2}(2\pi+0, \lambda)=0$

By$\{y_{1}’(2\pi+0, \lambda)/\lambda+y_{2}(2\pi+0, \lambda)\}\sqrt{\lambda}$$( \frac{\beta_{1}}{\sqrt{\lambda}}+\beta_{2}\sqrt{\lambda})$cos$\tau\sqrt{\lambda}$

cos2

cos

$\kappa_{1}\sqrt{\lambda}=0$$\kappa_{1}\sqrt{\lambda}+\beta_{1}\beta_{2}$$+( \frac{\beta_{1}}{\sqrt{\lambda}}-\beta_{2}\sqrt{\lambda})$, it turns out that

cos

sin

$\tau\sqrt{\lambda}$$\tau\sqrt{\lambda}$

cos

cos

$\kappa_{1}\sqrt{\lambda}\sin\kappa_{1}\sqrt{\lambda}$$\kappa_{1^{\sqrt{\lambda}}}$On the other hand, it follows by $(y_{1}(2\pi+0, \lambda)-y_{2}’(2\pi+0, \lambda))$ sin$\beta_{1}\hslash$cos$\tau^{\sqrt{\lambda}\sqrt{\lambda}}\cos\kappa_{1}$sin.

$\kappa_{1}\sqrt{\lambda}+(\frac{\beta_{1}}{\sqrt{\lambda}}-\hslash^{\sqrt{\lambda})}$sin$\tau\sqrt{\lambda}$

sin$\kappa_{1^{\sqrt{\lambda}}}=0$$\kappa_{1}\sqrt{\lambda}=0$

cos. $(3.10)$

that$\kappa_{1^{\sqrt{\lambda}\sin\kappa_{1}\sqrt{\lambda}}}$.

$-( \beta_{2}\sqrt{\lambda}+\frac{\beta_{1}}{\sqrt{\lambda}})\cos\tau\sqrt{\lambda}\sin^{2}\kappa_{1}\sqrt{\lambda}=0$(3.11)

Substituting (3.11) $hom(3.10)$ , we have$( \frac{\beta_{1}}{\sqrt{\lambda}}+h^{\sqrt{\lambda}}I^{\cos\tau\sqrt{\lambda}=0}$,

namelyor cos$\frac{\beta_{1}}{\sqrt{\lambda}}+\beta_{2}\sqrt{\lambda}=0$$\tau\sqrt{\lambda}=0$.

(3.12)$that\cos\tau\sqrt{\lambda}=0.Thicombinedwith\lambda\neq 0andy_{1}’(2\pi+0,\lambda)=0means\cos\kappa_{1}w_{eshowthestate_{S}ment(i).We\sup posethat\lambda\neq-\beta_{1}/\beta_{2}.Thenitf_{0}g_{owsby}}P_{\lambda=0}^{3.12)}$Substituting cos$\kappa_{1}\sqrt{\lambda}=\cos\tau\sqrt{\lambda}=0$

for$y_{2}(2\pi+0, \lambda)=0$, we have$\lambda=\beta_{2}/\beta_{1}$.

. Therefore

we get (i).
Next, we show the statement (ii). We suppose that
. Then we have
. Substituting
for $(y_{1}(2\pi+0, \lambda)-y_{2}(2\pi+0, \lambda))/\beta_{2}=0$ , wehave$\beta_{1}/\sqrt{\lambda}+$$\lambda=-\beta_{1}/\beta_{2}$$h\sqrt{\lambda}=0$$\beta_{1}/\sqrt{\lambda}=-\beta_{2}\sqrt{\lambda}$sin$\tau\sqrt{\lambda}$

cos$\kappa_{1}\sqrt{\lambda}-\frac{\beta_{1}}{2\sqrt{\lambda}}coe\tau\sqrt{\lambda}\cos\kappa_{1}\sqrt{\lambda}=0$.

(3.13)

by contradiction. Seeking a contradiction, we assume cos
We prove cos. Then it follows by $y_{1}’(2\pi+0, \lambda)=0$ and. Substitutingthat cos$y_{2}(2\pi+0,\lambda)=0$
,
for
we
cos
cos
. This contradicts
have. Therefore we have cos.
$\kappa_{1}\sqrt{\lambda}\neq 0$$\kappa_{1}\sqrt{\lambda}=$$\lambda\neq 0$$0$$\kappa_{1}\sqrt{\lambda}=$$\lambda=-\beta_{1}/\hslash$$\tau\sqrt{\lambda}=0$$\tau\sqrt{\lambda}=0$$\lambda=\beta_{1}/\$$\kappa_{1}\sqrt{\lambda}\neq 0$

137
By (3.13) and cos$\kappa_{1}\sqrt{\lambda}\neq 0$, it follows thatsin

Inserting$\beta_{1}/\lambda=-\beta_{2}$$\tau^{\sqrt{\lambda}}=\frac{\beta_{1}}{2\sqrt{\lambda}}$cos$\tau^{\sqrt{\lambda}}$(3.14).

and (3.14) into (3.6), we have$\sin\kappa_{1^{\sqrt{\lambda}=}}\frac{\beta_{1}}{2\sqrt{\lambda}}$CO8$\kappa_{1}\sqrt{\lambda}$

By (3.14) and (3.15), it turns out that
because ofand. Substituting, we obtain.

(3.15)$\sin(\tau-\kappa_{1})\sqrt{\lambda}=0$$\lambda=-\beta_{1}/\beta_{2}$$\tau-\kappa_{1}\neq 0$. This implies that$\lambda=-\beta_{1}/\beta_{2}$$\beta_{1}/\ <0$

and $\tau=2\pi-\kappa_{1}$ for$8in(\tau-\kappa_{1})\sqrt{\lambda}=0$$\sin 2(\pi-\kappa_{1})\sqrt{-\frac{\beta_{1}}{\hslash}}=0$.

Namely, there exists $n\in N$ such that$- \frac{\beta_{1}}{\beta_{2}}=\frac{n^{2}}{4(\pi-\kappa_{1})^{2}}$.

(3.16)On the other hand, Equation (3.15) means$\beta_{1}=\frac{n\pi}{|\pi-\kappa_{1}|}$tm$\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|}$.

This combined with (3.16) implies.

$\beta_{2}=-\frac{4|\pi-\kappa_{1}|}{n\pi}\tan\frac{\kappa_{1}n\pi}{2|\pi-\kappa_{1}|}$口

Next, we show Theorem 3.1.

Prvof of Theorem 3.1. We suppose$\kappa_{1}\neq\pi$and (3.9). We define

ifotherwise.

$\cos\kappa_{1}\sqrt{\beta_{2}/\beta_{1}}=\cos\tau\sqrt{\beta_{2}/\beta_{1}}=0$$S=\{$$\emptyset\{\ /\beta_{1}\}$

Then, Lemma 3.2 says
$S\subset \mathcal{B}$, we have
Since
have $S=\{\xi\}$ , where$S\supset \mathcal{B}$if$\xi=\ /\beta_{1}$ . Since$\mathcal{B}=\emptyset$$M(\xi)=$-sin$S=\emptyset$,.

. Next we consider the case where$\kappa_{1}\sqrt{\xi}\sin\tau\sqrt{\xi}(\begin{array}{lll}1 \beta_{2}- A1\lambda 0 1 \end{array})$,$S\neq\emptyset$

. We

138
and
一

$\beta_{2}-\frac{\beta_{1}}{\lambda}=\beta_{2}-\frac{\beta_{1}}{\ ,\beta_{1}}$$\frac{(\beta_{2}-\beta_{1})(\beta_{2}+\beta_{1})}{\beta_{2}}$$M(\xi)=\pm E$ is equivalent to$\beta_{2}-\beta_{1}=0$

whence$\xi\in \mathcal{B}$$\beta_{1}+\beta_{2}=0$$\beta_{2}+\beta_{1}=0$,(3.17)

if and only if (3.17) holds. This together with$\mathcal{B}=\{$

Ifor$if\ -\beta_{1}=0h$$\emptyset\{\xi\}$, then we haveor$\{\xi\}=S\supset \mathcal{B}$$\beta_{2}+\beta_{1}=0$

implies that,

otherwise.

$S=\emptyset$, so that$\mathcal{B}=\emptyset$. If$\beta_{2}-\beta_{1}=0$

ifotherwise.

$\kappa_{1}=\frac{\pi}{2},$$\mathcal{B}=S=\{$$\emptyset\{1\}$$\frac{3}{2}\pi$, then we obtain,
口Finally, we prove Theorem 1.5.

Proof of Theorem 1.5. Theorem 3.1 (i) directly follows Theorem 1.5 (i). So, our last work.
. Then, Theorem 3.1 (ii) reads
andis to prove (ii). We suppose$\kappa_{1}-\pi/2$

We calculate the rotation number$\mathcal{B}=\{1\}$$\beta_{1}=\beta_{2}$$\rho(1)$. Substituting$\frac{d}{dx}w(x, \lambda)=1$,$x\in \mathbb{R}\backslash \Gamma$$\lambda=1$

for (1.7), we have.

(3.18)

Since the rotation number is independent of the initial value , we may put $w_{0}=0$ .
Equation (3.18) means $\omega(\kappa_{1}-0,1,0)=\pi/2$ . It folows $kom(1.8)-(1.11)$ that$w_{0}$$\omega(\kappa_{1}+0,1,0)=\{\begin{array}{ll}Arc\tan(\frac{1}{\beta_{1}}) if \beta_{1}>0,\pi+Arc\tan(\frac{1}{\beta_{1}}) if \beta_{1}<0.\end{array}$Using Equation (3.18) again, we have$\omega(2\pi-0,1,0)=\{\begin{array}{ll}Arc\tan(\frac{1}{\beta_{1}})+(2\pi-\kappa_{1}) if \beta_{1}>0,\pi+Arct\bm{t}(\frac{1}{\beta_{1}})+(2\pi-\kappa_{1}) if \beta_{1}<0.\end{array}$Using $(1.8)-(1.11)$ in the case where $x=2\pi-0$ , we have $w(2\pi+0,1,0)=2\pi$ . Sinoe theequation (1.7) is -periodic in , we have $\omega(2\pi t+0,1,0)=2\pi t$ for $t\in N$ . Therefore wehave $\rho(1)=1$ .
We recall (1.14). Since$\pi$$\omega$$l=\{\begin{array}{ll}1 if \beta_{1}>0,0 if \beta_{1}<0,\end{array}$then we arrive at the goal owing to Theorem 1.1.
In a similar way, we obtain (ii) in the case where $\kappa_{1}=3\pi/2$ and$\beta_{1}=\beta_{2}$

.

口

139

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140
Hiroaki Niikuni
Department of Mathematics and Information Sciences
Tokyo Metropolitan University
Minami-Ohsawa 1-1
Hachioji Tokyo 192-0397
Japan
email: dreamsphere@infoseek.jp