数理解析研究所講究録
1119 巻 1999 年 83-97

83

The Spectral Structure and the WKB Approximation
on a Circle with Point Interaction1

IZUMI TSUTSUI (筒井泉)2
Institute of Particle and Nuclear Studies
High Energy Accelerator Research Organization $(KEK)$ , Tanashi BranchTokyo 188-8501, Japan(高エネルギー加速器研究機構素粒子原子核研究所田無分室)

Abstract. Application of the theory of self-adjoint extension reveals a $U(2)$ family of Hamiltonians to describe a quantum particle on a circle with point interaction.
We provide a classification of the family by introducing a number of subfamilies
and thereby analyze the spectral structure in detail. We find that the spectrum
depends on a subset of $U(2)$ parameters rather than the entire $U(2)$ , and that, inparticular, there exists a subfamily in $U(2)$ where the spectrum becomes parameterindependent. We also show that, in some specific cases, the WKB semiclassicalapproximation becomes exact (modulo phases) for the system.

$\mathrm{i}$This work was done in collaboration with Tam\’as F\"ul\"op of Roland E\"otv\"os University in Hungary.
izumi.tsutsui@kek.jp

2 E–mail:

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1. Introduction

Systems with point interaction form an important class of solvable models in quantum
mechanics, allowing for a variety of applications in physics (see [1] and references therein).
These systems are governed by Hamiltonian operators given by a point perturbation ofwith one point, that is, the self-adjoint Laplacian operators on
the Laplacian on
removed. In one dimension, it has been recognized $[2, 1]$ that the perturbation yields a, which implies that there is a $U(2)$ family$U(2)$ family of self-adjoint operators onof distinct point interactions possible there. The physical properties of the family has been$\mathrm{R}^{n}$$\mathrm{R}^{n}$$\mathrm{R}^{1}\backslash \{0\}$investigated in [3], and more fully in [4] including the spectra, time-dependent fundamental
solutions and scattering matrices. Recent technological advances in microscopic devices
have spurred a new interest in realizing these point interactions using regulated potentials[5, 6, 7, 8].
The aim of the present paper is twofold. First, we provide a study on point interaction
analogous to that made on a line . This is interesting for the reason thaton a circle–which are again characterized by
the difference in self-adjoint operators on$S^{1}$$\mathrm{R}^{1}$$S^{1}\backslash \{0\}$–arises in discrete energy spectrum rather than in the scattering amplitude as
whose positive energy spectrum is always continuous. We shall see that theonspectra of the $U(2)$ family exhibit remarkable characteristics; in particular, the spectral
space forms a subspace of $U(2)$ , rather than the whole $U(2)$ as apparently suggested by
the theory of self-adjoint extensions. In a subfamily of $U(2)$ (called tseparated subfamily’),
the circle becomes equivalent to a box with nontrivial boundary conditions, and hence ourresult is also relevant to a box if such boundary conditions are available. Secondly, we
admits the interpretation that theshow that in some specific cases our system on
WKB semicalssical approximation is exact up to a phase which is determined by the pointinteraction. In fact, the WKB exactness of a box system has been pointed out earlier inthe path-integral for perfectly reflecting walls [9], and in this respect our cases provide a$U(2)$$\mathrm{R}^{1}\backslash \{0\}$$S^{1}\backslash \{0\}$generalization of the particular case previously considered.
This paper is organized as follows. In sect. 2 we review briefly the system of a particle
confined to a half line where the basic properties mentioned above, such as the dependenceof the spectrum on the allowed self-adjoint extensions and the WKB exactness in thein sect. 3transition amplitude, can easily be seen. We then present our result on$S^{1}\backslash \{0\}$by considering a particle confined to a box under the most general self-adjoint extensionsof the Laplacian, a setup which is equivalent to a circle with point interaction. For ourconvenience we introduce a number of subfamilies defined within the $U(2)$ family. These

85

subfamilies are characterized by distinguished physical properties, wherein the spectra andthe WKB exactness are discussed separately.
2. Particle on a Half LineWe begin by discussing a particle restricted to move on the half line $x\geq 0$ . The system
will be given by placing an ‘infinite’ potential wall at $x=0$ on a line, and in quantum
mechanics one conventionally irnposes the vanishing boundary condition $\psi(0)=0$ forwavefunctions. This is based on the observation (see, $e.g.,$ $[10]$ ) that the boundary condition
that arises under a finite constant wall
at $x=0$ reduces to the conventional one in the. This, however, is too restrictive to specify the boundary condition, because
limit
other limits that also confine the particle in the half line can lead to boundary conditions
which are different from the conventional one [11]. It is therefore safe to say that the only
requirement for the particle to be confined on the half line is that the probability current$j(x)$ , not the wave function, vanish at the wall,$V_{0}$$V_{0}arrow\infty$$j(\mathrm{O})=0$

,

where

$j(x)=- \frac{i\hslash}{2m}((\psi^{*})^{\prime\psi_{-}\psi\psi)}*’(x)$

.

(2.1)

The condition (2.1) is then satisfied if$\psi(0)+L\psi’(0)=0$ ,(2.2)

being an arbitrary constant of length dimension, or $\psi’(0)=0$ which may bewiththought of the case $L=\infty$ in (2.2). The combined space of the solutions
to the condition (2.1) may thus be parametrized by $L:=L_{0}cot\phi$ with an angle $\phi\in[0, \pi)$. The same $U(1)$ family can be obtained from the theory
and a nonvanishing constantof self-adjoint extensions applied to the free Hamiltonian
restricted to thehalf line [12], because self-adjointness implies probability conservation in the system as a
$L\in \mathrm{R}$$\mathrm{R}\cup\{\infty\}\simeq U(1)$$L_{0}$$H=- \frac{\hslash^{2}}{2m}\frac{d^{2}}{dx^{2}}$whole.3
We then observe that, in addition to the continuum spectrum for positive energy$E_{k}=\hslash^{2}k^{2}/2m$ with $k>0$ formed by the states,$\psi_{k}^{L}(X)=\frac{1}{\sqrt{2\pi}}[e^{-ikx}+(\frac{1-ikL}{1+ikL})e^{ikx}]$

3

,

(2.3)

Here one adopts the local conservation $j(\infty)=0$ in addition to (2.1) on the ground that the moregeneral global conservation, $j(\mathrm{O})=j(\infty)$ , will never be realized physically in practice unless $j(\infty)=0$ . In
parallel, the possibility for a nonzero $j(\infty)$ is excluded by the results of self-adjoint extension theory, too.

86

the system admits a normalizable bound state$\psi_{b_{oun}d}^{L}(x)=\sqrt{\frac{2}{L}}e^{-\frac{x}{L}}$

(2.4)

,

for $L>0$ with energy $E_{B}=-h^{2}/2mL^{2}$ . The bound state (2.4) disappears either at $L=0$
or $L=\infty$ where the boundary condition becomes $\psi(0)--0$ or $\psi’(0)=0$ .
The two exceptional cases, $L=0$ and $L=\infty$ , which has no scale parameter areof particular interest, because one can then evaluate the transition amplitude (Feynman
kernel) in closed form. Indeed, from (2.3) the Feynman kernel $K(b, T;a, 0)$ , which describes
the transition from $x=a$ at $t=0$ to $x=b$ at $t=T$ , is found to be$K(b, T;a, \mathrm{o})=\int_{0}^{\infty}dke^{-}\frac{i}{\hslash}E_{k}\tau\psi^{L}k(b)(\psi_{k}^{L})*(a)$(2.5)$= \sqrt{\frac{m}{2\pi ihT}}(e^{\frac{l}{\hslash}\frac{m}{2T}(ba)^{2}}-\mp e^{\frac{i}{\hslash}}\frac{m}{2T}(b+a)^{2})$,where the $-,$ -sign in (2.5) is for $L=0$ whereas the $‘+$ ’-sign is for $L=\infty$ . In the pathintegral point of view, the result (2.5) may be interepreted as being a sum of contributions
from the two possible classical trajectories, the direct path from to and the reflected
path which starts from and hits the wall before reaching . We therefore see that, modulo
the phase shift by or acquired at the reflection, the WKB semiclassical approximation
becomes exact in the two cases of the present system. We remark that the phase shift may
(

$a$

$b$

$a$

$\pi$

$b$

$0$

also be explained by the corresponding shift at the reflection in the classical action, if one
considers a proper sequence of regulated potentials that leads to the infinite wall in thelimit. (The path-integral treatise on the half line has been discussed earlier in $[9, 13]$ for$L=0$ and in $[14, 15]$ for generic $L.$ )3. Particle in a Box or on a Circle with Point Interaction
We now consider the system of an infinite potential well, $i.e.$ , a free particle confinedin a box of infinite potential walls at $x=0$ and $x=l$ . It is known that there exists a$U(2)$ family of self-adjoint Hamiltonians for the system. Using a matrix $U\in U(2)$ and theidentity matrix , the different self-adjoint extensions in the family can be described by
the boundary conditions,$I$$(U-I)\Psi+iL_{0}(U+I)\Psi’=0$ ,$\Psi:=(_{\psi(l)}^{\psi()}0)$,$\Psi’:=(_{-\psi’}^{\psi’(}\mathrm{o}_{l)}())$

,

(3.1)

87

where
is again a constant of length dimension. As before, one can derive these boundary
conditions (3.1) by requiring obability conservation of the entire system. To see this,note first that withthe probability conservation,$L_{0}$$\mathrm{p}..\mathrm{r}$$\Psi^{\uparrow}=(\Psi^{*})^{t}$,$j(\mathrm{O})-j(l)=0$

is equivalent to$\Psi\dagger\Psi’-(\Psi\dagger)’\Psi=0$

(3.2)

, or

$|\Psi-iL_{0}\Psi’|^{2}-|\Psi+iL_{0}\Psi’|^{2}=0$

,

(3.3)

. Then fron this one deduces4 that $\Psi-iL_{0}\Psi/=U(\Psi+iL_{0^{\Psi’})}$ withfor anysome unitary matrix $U\in U(2)$ , arriving exactly at (3.1).
$L_{0}\in \mathrm{R}\backslash \{0\}$In practical settings of a box, it is natural to impose local probability conservation$j(\mathrm{O})=j(l)=0$ . However, it is perfectly legitimate to consider the generic boundary
conditions (3.1) which derive from global probability conservation (3.2). The physicalsettings for those generic cases arise when, $e.g.$ , the two ends of the box are combined
to form a circle, creating a possible singularity at the junction which induces a pointinteraction there. The singularity or the point interaction will then be characterized by
the boundary conditions (3.1). An analogous situation arises for the system on a linewith point interaction, where the family of self-adjoint extensions is also given by $U(2)$(although the local and incomplete parametrization $U(1)\cross SL(2,$instead of $U(2)$ has
been widely used in the literature; see [16] for the detail).
$\mathrm{R}$$\mathrm{R}\rangle$

Let us introduce a unique parametrization for $U(2)=U(1)\cross SU(2)$ by$U=e^{i\xi}$ ,(3.4)

with $\xi\in[0, \pi)$ andsatisfying $|\alpha|^{2}+|\beta|^{2}=1$ , and thereby consider the energy
spectra that arise under the boundary conditions (3.1). The general form of eigenfunctions
for positive energy is$\alpha,$$\beta\in \mathbb{C}$$\psi_{k}(x)=A_{k}e^{ikx}+B_{k}e^{-ikx}$

may depend on
where the coefficients
boundary conditions (3.1) become$A_{k},$($B_{k}$$eK+-(\beta^{*}+e-i\xi e^{i})kl+\alpha K+(\beta e-e-i\epsilon)iklKK_{-}$$k$,$k>0$ ,

(3.5)

. Then, in terms of $K_{\pm}:=1\pm kL_{0}$ the$\alpha^{*}e^{-}K_{-}^{+}ikl-(\alpha K_{+}(\beta e-ikl-e_{i}-i\xi)K\beta*+e-\xi e^{-i})klK-+$) $=0$ .
(3.6)

4

One might think thatshould be regarded as a free parameter as in (2.2), but it can be shown [11]that this freedom can be absorbed by adjusting the $U(2)$ parameters in .
$L_{0}$$U$

88

For nontrivial solutions forfulfilled,$A_{k},$$B_{k}$, the following condition for the momentum$k$should be

,

(3.7)

$2kL_{0}(\beta_{I}+sin\xi coskl)+[(cos\xi-\alpha_{R})+(cos\xi+\alpha_{R})(kL_{0})^{2}]sinkl=0$

being the imaginary part of and the real part of , respectively. Thusandwiththe positive spectrum consists of infinitely many discrete levels, and for large the allowed
approaches to the equidistant values $n\pi/l$ with integer.
$\beta$$\beta_{I}$$\alpha$$\alpha_{R}$

$k$

$n$

$k$

On the other hand, for negative energy states,$\psi_{\kappa}(x)=A_{\kappa}e^{\kappa x}+B_{\kappa}e^{-\kappa x}$

one proceeds analogously by replacing$2\kappa L_{0}$( $\beta_{I}+sin\xi$ cosh$\kappa l$$k\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}-i\kappa$,,$\kappa>0$(3.8)

in (3.7) to obtain) $+[(cos\xi-\alpha_{R})-(cos\xi+\alpha_{R})(\kappa L_{0})^{2}]$ sinh $\kappa l=0$ .

(3.9)

It is readily seen by inspection that the condition (3.9) has at most two solutions for . The
fact that the number of negative states is at most two is a norm for a system possessing a
as well$U(2)$ family of self-adjoint Hamiltonians, which can be seen in the system$\kappa$$\mathrm{R}^{1}\backslash \{0\}$$[3, 4]$. As for the existence condition for a zero energy state, one puts the form
(3.10)$\psi \mathrm{o}(x)=Ax+B$

in (3.1) and finds that the condition for the existence reads$( \beta_{I}+sin\xi)-\frac{l}{2L_{0}}(\alpha_{R}-cos\xi)=0$.

(3.11)

An important point to observe is that the entire energy spectrum (positive, negative andrather than on the full set of $U(2)$ pazero) depends only on the parameters
rameters in (3.4). This suggests that, even though all the distinct self-adjoint extensionsof Hamiltonians are furnished by the $U(2)$ family, some of them are unitarily related and$(\xi, \alpha_{R}, \beta_{I})$, forms a subspace of $U(2)$ . In otherthe true space of spectra, the spectral spacewords, different self-adjoint extensions may not correspond to different physics as opposed$\mathcal{X}_{SP}$to the common belief; see [12]. Within the $U(2)$ family there are a number of distinguished subfamilies (some of which are not disjoint), which we shall discuss in detail in thefollowing.

89: The first subfamilyof our concern is one given by theboundary conditions (3.1) with diagonal . The boundary conditions then split into twosets, one at the left wall $x=0$ and the other at the right wall $x=l$ . In our parametrization(3.4), this occurs for $\beta=0$ , leading to the subgroup $U(1)\cross U(1)\subset U(2)$ given by thetorus,
Separated subfamily$\mathcal{F}_{1}$$\mathcal{F}_{1}$$U$$\mathcal{F}_{1}=S^{1}\cross S^{1}=\{(\xi, \alpha_{R}, \alpha_{I}, \beta R, \beta_{I})|\xi\in[0, \pi), \alpha^{2}R+\alpha_{I}^{2}=1, \beta_{R}=0, \beta_{I}=0\}$

If we write$\alpha=e^{i\varphi}$$\phi_{\pm}=(\xi\pm\varphi)/2$. (3.12)with $\varphi\in[0,2\pi)$ and use the scale parameters $L_{\pm}:=L_{0}cot\phi_{\pm}$ with, we find that the boundary conditions (3.1) reduce to$\psi(0)+L_{+}\psi’(\mathrm{o})=0$,$\psi(l)+L_{-}\psi’(l)=0$.

(3.13)

Clearly, this subfamily arises when we require local probability conservation $j(\mathrm{O})=j(l)=$, and therefore it describes a box in its true sense of the word –the left and the rightwalls of the box are disconnected physically. The corresponding subfamily appearing in
the aforementioned context of a line with a point interaction is referred to as ‘separated’$0$[16], from which we adopt the name of the subfamily here.

are four special cases $(L_{+}, L_{-})=(\mathrm{O}, 0),$,Among this separated subfamily
in which the theory becomes free from scale parameters (apart from )and, consequently, solvable explicitly. Indeed, for the first two cases the eigenfunctions
for non-negative energies (no negative energy state is allowed)$(\infty, \infty)$$\mathcal{F}_{1}$$(0, \infty),$$(\infty, 0)$$l$$\psi_{n}^{(L,L_{-)}}+(x)$$E_{n}= \frac{h^{2}}{2ml^{2}}(n\pi)^{2}$,(3.14)are given, respectively, by$\psi_{n}^{(0,0)}(X)=\mathrm{v}^{\Gamma}\overline{\frac{2}{l}}sin\frac{n\pi}{l,}X$,

for$n=1,2,3,$for$n=1,2,3,$

for$n=0$ .

$\ldots$,(3.15)

and$\sqrt{\frac{2}{\iota}}cos\frac{n\pi}{l}x$,$\ldots$(3.16)$\psi_{n}^{(\infty,\infty)}(_{X)}=\{$$\sqrt{\frac{1}{l}}$,As for the Feynman kernel we find$K(b, \tau;a, \mathrm{o})=\frac{1}{2l}\sum_{n=-\infty}^{\infty}e^{-}\frac{i}{\hslash}EnT(e^{i\frac{n\pi}{l}(-a)}b\mp e^{i\frac{n\pi}{l}(ba}+))$,(3.17)

90. With thewhere the ‘-,-sign is for $(L_{+}, L_{-})=(0,0)$ whereas the $‘+$ ’-sign is forwhichhelp of the Poisson summation formula,( $i.e.$ , for $f(x)\in L_{1}(-\infty, \infty)$ being continuous andholds for well-behaved functionsof bounded variation), one can rewrite (3.17) into$(\infty, \infty)$$\sum_{n=-\infty}^{\infty}f(n)=\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}dpf(p)e2\pi ni$$f(x\rangle$(3.18)$K(b, \tau_{;a}, 0)=\sqrt{\frac{m}{2\pi ihT}}\sum_{n=-\infty}^{\infty}(e^{\frac{i}{\hslash}\frac{m}{2\mathrm{T}}\{(b}-a)+2nl\}^{2}\mp e^{\mathrm{F}^{\frac{m}{2T}\{(b}}i+a)+2nl\}2)$

As on the half line, the result (3.18) allows for the interpretation that the kernel is the sumof contributions from classical paths of two distinct classes, one hitting -times the rightand left walls equally while the other hitting -times the left and $(n\pm 1)$ -times the right,
where the signature of is determined according to whether the particle hits the right wall
, attached to the paths infirst or not. The phase factor, $-1$ for $(0,0)$ and +1 forthe latter class suggests that the phase shift occurs every time the particle hits the wall,and this is the only nontrivial factor for the exactness of the WKB approximation.
$n$

$n$

$n$

$(\infty, \infty)$

and for the fourth
One can proceed analogously for the third case$(L_{+}, L_{-})=(\infty, 0)$ , the latter being obtained from the former by the parity operation$x-\succ l-x$ , where one has the eigenfunctions,$(L_{+}, L_{-})=(\mathrm{O}, \infty)$,

for$n=0,1,2,$,

for$n=0,1,2,$$\psi_{n}^{(0,\infty)}(X)=\sqrt{\frac{2}{l}}sin\frac{(n+\frac{1}{2})\pi}{l}x$$\ldots$,(3.19),(3.20)

and$\psi_{n}^{(\infty,0)}(X)=\sqrt{\frac{2}{l}}cos\frac{(n+\frac{1}{2})\pi}{l}x$$\ldots$of energy(3.21)$E_{n}= \frac{h^{2}}{2ml^{2}}[(n+\frac{1}{2})\pi]^{2}$(Again, no negative energy state is allowed.) Then the Feynman kernel turns out to be$K(b, T;a, 0)= \sqrt{\frac{m}{2\pi i\hslash T}}\sum_{n=-\infty}^{\infty}(-1)^{n}(e^{\frac{i}{\hslash}\frac{m}{2T}\{()nl\}^{2}}ab-+2\mp e^{\frac{i}{\hslash}\frac{m}{2T}\{()2n})b+a+l\}^{2}$,(3.22). The status ofwhere the ‘-,-sign is for $(L_{+}, L_{-})=(0, \infty)$ and the $‘+$ ’-sign is for
the WKB exactness remains similar, where now one observes that the phase shift occurs. (The specor the right wall for
only when the particle hits the left wall fortrum decomposition of the Feynman kernel has been obtained in [17] for generic boundary
conditions, and the WKB exactness for the case $(L_{+}, L_{-})=(0,0)$ has been mentioned$(\infty, 0)$$(0, \infty)$$(\infty, 0)$91earlier $[9, 13]$ based on the ideas developed in $[18, 19]$ .) In passing we note that all of the
Feynman kernels obtained in closed form here can be cast into expressions in the thetafunction,. For example, the kernel (3.22) can be written$\theta_{3}(z, \tau)=\sum_{n=-\infty}^{\infty}e^{i+z}\pi\tau n^{2}i2n\pi$as$K(b, T;a, 0)=\sqrt{\frac{m}{2\pi ihT}}\{$$\theta_{3}$(a)$\frac{ml(b-}{\pi hT}$$\mp\theta_{3}$($- \frac{1}{2},$$\frac{2ml^{2}}{\pi hT})e^{\frac{i}{\hslash}\frac{m}{2T}}(b-a)2$(3.23)a)$\frac{ml(b+}{\pi\hslash T}$$- \frac{1}{2},$$\frac{2ml^{2}}{\pi\hslash T})e^{\frac{i}{\hslash}\frac{m}{2T}}(b+a)2\}$.

Scale independent subfamily: The second subfamilyarises when the derivative
terms become decoupled from non-derivative terms in (3.1). This occurs if and only if$\mathcal{F}_{2}$$\mathcal{F}_{2}$$det(U-I)=det(U+I)=0$ .

(3.24)

The conditions in (3.24) state that the two eigenvalues for the matrix, and hence
is symmetric, as can also be confirmed explicitly using (3.4). Then, from theunitarity of one deduces that each term in (3.1) must vanish separately,$U\mathrm{a}\mathrm{r}\mathrm{e}\pm 1$$U^{\uparrow}=U$

$U$

$U$

$(U-I)\Psi=0$ ,$(U+I)\Psi’=0$ .

(3.25)The set of satisfying (3.25) forms a $U(2)/(U(1)\cross U(1))\simeq S^{2}$ subspace in $U(2)$ , where
the second $U(1)$ is the Cartan subgroup of the $SU(2)\subset U(2)$ . Explicitly, the sphere isgiven by the constraint in the parameter space of $U(2)$ ,$U$$\mathcal{F}_{2}=S^{2}=\{(\xi, \alpha_{R}, \alpha_{I,\beta}R, \beta_{I})|\xi=\frac{\pi}{2},$$\alpha_{R}=0,$$\alpha_{I}^{2}+\beta_{R}^{2}+\beta_{I}^{2}=1\}$(3.26)

This subfamily is distinguished in that the dimensionful paraneter
drops out fromthe boundary conditions, leaving the width of the well as the only independent scaleparameter. One may therefore expect that the theory becomes scale invariant in the limit
where no scale parameter survives.
$L_{0}$$l$$larrow\infty$

The energy spectra in this subfamily are characteristic, due to the absence of the scaleparameter. First, the condition (3.9) does not admit any negative states, while (3.11)$L_{0}$

allows one zero energy state $\psi_{0}(X)=\sqrt{1/l}$ if and only if $\beta_{I}=-1$ (in which case $\alpha=0$ ,$\beta=-i)$ . Further, the positive energy spectra can be determined explicitly from (3.7),
which reduces to $\beta_{I}+coskl=0$ , yielding the two sets of solutions for the momenta $k=k_{s}$labeled by $s=\pm$ ,$k_{n}^{s}=sk_{n}$,$k_{n}:= \frac{1}{l}(\theta+2n\pi)$,$n=\{$$0,1,2,$$\ldots$,$-1,$ $-2,$ $-3,$$\ldots$,

for $s=+$ ,for $s=-,$(3.27)

92

where we have used$\theta:=arccos(-\beta_{I})$.

(3.28)

Thus the positive energy spectra are found to be$E_{n}= \frac{\hslash^{2}}{2ml^{2}}(\theta+2n\pi)^{2}$,$n\in \mathbb{Z}$.

(3.29)

It is also readily seen that normalized eigenfunctions are given by$\psi_{n}^{S}(X)=Asen-x\mathrm{A}ik^{s}-ik^{S}n-sex$(3.30),

with$A_{\pm}= \frac{(1+\alpha_{I})+(\beta I-i\beta R)e^{\mp}i\theta}{2\sqrt{l(1+\alpha_{I})(1+\beta Icos\theta)}}$for$\alpha_{I}\neq-1$

andintersection$\mathcal{F}_{1}\cap \mathcal{F}_{2}$$\mathcal{F}_{2}$$(\infty, 0)$.

(3.31)are identical to the special cases. The cases $\alpha_{I}=\pm 1$ in, which are the only two points of theof the subfamily
are undetermined,
We note that, for $\beta_{I}=\pm 1$ , the coefficients$\beta_{I}\neq\pm 1$$(L_{+}, L_{-})=(0, \infty),$,$\mathcal{F}_{1}$$A_{\pm}$

which implies that all the energy levels but one, $i.e.$ , the zero energy ground state forThis is also seen from (3.29) where the two sets of$\beta_{I}=-1$ , are doubly degenerated.
for $\theta=0$ and . Note also that the spectra are solely determined bylevels. In
as its image in the spectral spacehas
the angle , that is, the subfamily
contrast, the coefficients, and hence the eigenfunctions are dependent on the parameters
as well, even though the spectra are independent of them.
$s=\pm \mathrm{c}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{d}\mathrm{e}$$\pi$$\mathcal{F}_{2}$$\theta$$S^{1}$$\mathcal{X}_{SP}$$(\alpha_{I}, \beta_{R})$Having obtained the eigenfunctions in closed form, we shall now resort the sameprocedure used before to evaluate the Feynman kernel. To this end, let us denote by
for $s=-.$ We then haveforthe summation$\sum_{n}’$$\sum_{n=0}^{\infty}$$s=+ \mathrm{a}\mathrm{n}\mathrm{d}\sum_{n}^{-\infty}=-1$$K(b, \tau;a, \mathrm{o})=\sum_{s=\pm}\sum e-\frac{i}{\hslash}E_{n}\tau\psi sn(b)(\psi^{s}n(a)n/)^{*}$$= \sum_{n}e^{-\frac{i}{\hslash}}\{/AE_{n}\tau|+|^{2}(e^{ik_{n}^{+}}+(b-a)e^{-i}k^{-(}nb-a))+|A_{-}|^{2}(e^{i}+en-a)k_{\mathcal{R}}-(b-a)-ik^{+}(b)$$-A_{+}A_{-}^{*}(e^{ik^{+}}n(b+a)+e-ik^{-}(n)b+a)-A_{-}A_{+}^{*}(e^{ik_{n}^{-(+}}+ae^{-}n))b)ik^{+}(b+a\}$.

(3.32)
Recombining the terms in the summation, and using the Poisson summation formula, we

end up with$K(b, T;a, 0)= \sum_{\infty n=-}^{\infty}e^{-}\frac{i}{\hslash}En\tau(|\mathrm{A}+|2ikn(b-a)+e|A_{-}|^{2}e^{-})ik_{n}(b-a$$-A_{+^{\mathrm{A}_{-}^{*}e}-A}ikn(b+a)-A*+^{e}j-ikn(b+a))$(3.33)

93

with$C_{n}=|A+|^{2}e^{-i\theta n}+|A_{-}|^{2}e^{i\theta}n$,

The result (3.33) suggests that the WKB approximation yields thefactors
are properly interpreted.
$C_{n},$(3.34)$D_{n}=A_{+}A^{*}-e-i\theta n+A_{-^{A_{+^{e}}^{*i\theta n}}}$$\mathrm{e}_{\lrcorner}\mathrm{x}\mathrm{a}\mathrm{C}\mathrm{t}$kernel, if the$D_{n}$

Smooth subfamiIy$\mathcal{F}3$: Among$\mathcal{F}_{2}$is the $U(1)$ subfamily$\mathcal{F}_{3}=S^{1}=\{(\xi, \alpha_{R}, \alpha_{I}, \beta R, \beta_{I})|\xi=\frac{\pi}{2},$obtained by$\mathcal{F}_{3}\subset \mathcal{F}_{2}$$\alpha_{R}=0,$ $\alpha_{I}=0,$ $\beta_{R}^{2}+\beta_{I}^{2}=1\}$

This subfamily is distinguished in that with the remaining parameterthe boundary conditions (3.1) become$\psi(0)+e^{-i\theta}\psi(l)=0$,$\psi’(0)+e^{-i\theta}\psi’(l)=0$$\theta\in[0, \pi)$,(3.35)in (3.28)(3.36)

which are actually the boundary conditions familiar on a smooth circle ( $i.e.$ , one without
singularity; see ]) with representing possible phase change around arotation. Here
we find $A_{+}=\sqrt{1}/l$ and $A_{-}=0$ , and accordingly the Feynman kernel (3.33) takes thewell-known form on the circle,$2\pi$$\theta$$\lfloor\lceil 9$(3.37)$K(b, \tau_{;a}, \mathrm{o})=\sqrt{\frac{m}{2\pi ihT}}\sum_{n=-\infty}^{\infty}e^{-i}\theta ne^{\mathrm{F}^{\frac{m}{2T}\{(b}}i-a)+nl\}^{2}$

It is worth pointing out that the freedom expressed by the $U(1)\theta$ -parameter, which is a
prototype of the -parameter of the QCD vacua, is usually ascribed to the ambiguity inquantization on the circle which is topologically nontrivial (see, $e.g.,$ $[20,21]$ ). Here it arises
as part of the $U(2)$ family of systems allowed on a circle, where the $U(1)\subset U(2)$ subgroup
emerges upon demanding translational invariance ofsystem, that is, the smoothness
given by the boundary conditions (3.36). (The role of translational invariance among the$U(2)$ family has been remarked in [9].)$\theta$$\mathrm{t}\mathrm{h}\mathrm{e}|$: The energy spectra obtained explicitly for the precedingsubfamilies are solutions of the spectral condition (3.7) (and (3.11)). Conversely, one may
look for specific cases where other solutions can be found directly from (3.7). One thenfinds, for instance, that for $\xi=0$ andthe condition (3.7) simplifies to $sinkl=0$ andadmits equidistantfor the solutions. Hence, the subfamily
defined by
Isospectral subfamily$\mathcal{F}_{4}$$\beta_{I}=0$$k=n\pi/l_{\mathrm{W}}.\mathrm{i}\mathrm{t}\mathrm{h}n=1,2,3,$ $\ldots$$\mathcal{F}_{4}$$\mathcal{F}_{4}=S^{2}=\{(\xi, \alpha_{R}, \alpha_{I}, \beta_{R}, \beta I)|\xi=0, \alpha_{R}^{2}+\alpha_{I}^{2}+\beta_{R}^{2}=1, \beta_{I}=0\}$,(3.38)

94

possesses the unique spectrum given by (3.14), even though the sphere (3.38) retains
maps to a point in
as a free parameter for the spectra. In other words, the sphere.
is disjoint fromwhile
Note that$\mathcal{X}_{SP}$$\mathcal{F}_{4}$$\mathcal{F}_{4}\cap \mathcal{F}_{1}=S^{1}$$\alpha_{R}$.

$\mathcal{F}_{2}$$\mathcal{F}_{4}$Despite the triviality of the spectrum, the eigenfunctions turn out to be nontrivial inthat the coefficients appearing in the solution (3.5) become dependent on the level , inwheresharp contrast to the coefficients in (3.30) in the scale independent subfamily$n$$\mathcal{F}_{2}$). Because of this complication,
they become independent (except theit does not seem to be possible to proceed analogously to obtain the Feynman kernel in a$s=\pm \mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$form in which the WKB exactness can be examined.
: An extension of the isospectral subfamily
Semi-isospectral subfamilyby those cases with the property $sin\xi=\pm\beta_{I}$ , that is,$\mathcal{F}_{5}$$\mathcal{F}_{4}$is given(3.39)$\mathcal{F}_{5}=\{(\xi, \alpha_{R}, \alpha_{I,\beta R,\beta I})|sin\xi=\pm\beta I, \alpha_{R}^{2}+\alpha_{I^{+}}2\beta^{2}R+\beta_{I}^{2}=1\}$

the solutions of the spectral condition (3.7) consists of two infinite
sequences, one that is equidistant and parameter independent –like for the isospectralin–and another that is parameter dependent andsubfamily
which arises atgiven by transcendental roots. For example, on the positive branch $sin\xi=+\beta_{I}$ , the rootsof $coskl=-1,$ $k=n\pi/l$ with $n=1,3,5,$ , are the parameter independent solutions. By
inspection, one finds from (3.7) that the other roots are an infinite sequence of nontrivial,
Generically, in$\mathcal{F}_{5}$$\beta_{I}=0$$\mathcal{F}_{4}$$\mathcal{F}_{5}$$\ldots$transcendental solutions. The two sequences of roots are alternating, one transcendentalroot between any two succeeding equidistant roots and vice versa. For the negative branch$sin\xi=-\beta_{I}$ , the isospectral roots are now given by the solutions, $k=n\pi/l$ with $n=2$ ,4, 6, , of $coskl=+1$ and the transcendental roots are also different from the previous
ones, but the qualitative picture of the spectrum remains the same.
$\ldots$

at $\beta_{I}=0$ , where both
We observe that the two branches meet each other inare the twosequences become isospectral and equidistant. The other special points in, where the two sequences coincide and hence the energy levels become
points within the sense, are in fact exceptional in
doubly degenerate. These points, $\beta_{I}=0,$
that a single corresponds to the respective . Since there exist two for a genericis given
except the above three, we see from (3.39) that, topologically, the subfamily
sharing their equators $(\beta_{I}=0)$ and the North and South Poles $(\beta_{I}=\pm 1)$ ; seeby two$\mathcal{F}_{4}$$\mathcal{F}_{5}$$\beta_{I}=\pm 1$$\pm 1$$\mathcal{F}_{5}$$\beta_{I}$$\xi$$\beta_{I}$$\xi$$\mathcal{F}_{5}$$S^{3}$

Fig.1. We also note thatthe two special points$\mathcal{F}_{5}\cap \mathcal{F}_{1}=\mathcal{F}_{4}\cap \mathcal{F}_{1}=S^{1}$$\beta_{I}=\pm 1$.

, and that$\mathcal{F}_{5}\cap \mathcal{F}_{2}=\mathcal{F}_{5}\cap \mathcal{F}_{3}$consists of95

Figure 1. A schematical picture of Semi-isospectral subfamily
obtained byin (3.39). The North and South Poles are shared by thesetting
which are shown here by two circles. The four points marked by dotstwocorrespond to a spherewhich is the shared equator of the two .
$\mathcal{F}_{5}$$\alpha_{1}=\beta_{\mathrm{R}}=0$$S^{3}$$\bullet$$S^{2}$$S^{3}$

Finally, we stress that it is important to analyze the spectral space

thoroughly
in order to understand fully, $e.g.$ , the intriguing double spiral structure and a certain
recently reported $[22, 23]$ . We suspect that behind them
‘fermion-boson’ duality on
underlie certain symmetries, such as those associated with parity, time-reversal and scale. The analysis is also important with regard
transformations studied in [16] forto quantum mechanical symmetry breaking, in view of the fact that in two and three
dimensions the conformal SO $(2,1)$ symmetry is seen to be broken dynamically under thepresence of the delta-function interaction [24]. Our investigation on these issues, including
a fuller account of the result presented here, will be reported elsewhere [11].
$\mathcal{X}_{SP}$$X_{SP}$$\mathrm{R}^{1}\backslash \{0\}$Acknowledgenent: The authors wish to thank T. Cheon and P. \v{S}eba for helpful discussions. This work is supported in part by the Grant-in-Aid for Scientific Research (C)under Contract No. 11640301 provided by the Ministry of Education, Science, Sports andCulture of Japan.

96
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