# Shigeki MatsutaniKanazawa University | Kindai · Graduate School of Natural Science & Technology

Shigeki Matsutani

PhD

## About

116

Publications

8,538

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925

Citations

Introduction

Additional affiliations

April 2019 - present

July 2015 - March 2019

**Institute of Mathematics for Industry, Kyushu University**

Position

- Professor

April 2015 - March 2019

**National Institute of Technology Sasebo college**

Position

- Professor (Full)

Education

December 1995 - December 1995

April 1986 - March 1988

April 1982 - March 1986

**Shizuoka University**

Field of study

- Physics

## Publications

Publications (116)

The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r is a positive integer, and each Aj is a polynomial in x with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve X, which is birational to the surface. The form pro...

In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves Xs defined by the equation y3=x(x−s)(x−b1)(x−b2) in the affine (x, y) plane, for s ∈ Dε:= {s ∈ ℂ∥s∣ < ε}. We compare the sigma function over the punctured disc Dµ*:= Dε ∖ {0} with the extension over s = 0 that specializes to the sigma function...

The Weierstrass curve $X$ is a normalized curve of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $\leq j s/r$ for certain coprime positive integers $r$ and $s$ ($r<s$); the Weierstrass non-gap sequence at $\infty\in...

The Weierstrass curve is a pointed curve $(X,\infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $\leq j s/r$ for certain coprime positive integers $r...

This article proposes a model including thermal effects for closed supercoiled DNA. Existing models include an elastic rod. Euler’s elastica, ideal elastic rods on a plane, have only two kinds of closed shapes, the circle and a figure-eight, realized as minima of the Euler–Bernoulli energy. Even considering three dimensional effects, this elastica...

Euler's elastica, ideal elastic rods on a plane, have only two kinds of closed shapes, the circle and a figure-eight, realized as minima of the Euler-Bernoulli energy. Although closed supercoiled DNA has been modeled by the elastic rod model, even considering 3D effects this model (which reduces to elastica) provides much simpler shapes than observ...

In this article, we give the trigonal Toda lattice equation,
for a lattice point as a directed 6-regular graph where , and its elliptic solution for the curve y(y–s) = x ³, (s ≠ 0).

Integrable Systems and Algebraic Geometry - edited by Ron Donagi April 2020

In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves $X_s$ defined by the equation $y^3 =x(x-s)(x-b_1)(x-b_2)$ in the affine $(x,y)$ plane, for $s\in D_\varepsilon:=\{s \in \mathbb{C} | |s|<\varepsilon\}$. We compare the sigma function over the punctured disc $D_\varepsilon^*:=D_\varepsilon\setmi...

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $\phi_0$; we refer the piece-wised elastica $\Lambda_{\phi_0}$-elastica or $\Lambda$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly due the ruptur...

We consider the second weighted Bartholdi zeta function of a graph G, and present weighted versions for the results of Li and Hou’s on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of G. Furthermore, we give a formula for the weighted Kirchhoff index of a regular covering of G in terms...

In this article, we give the trigonal Toda lattice equation, $$ -\frac{1}{2}\frac{d^3}{du^3} q_{\ell}(u) = e^{q_{\ell+1}(u)} +e^{q_{\ell+\zeta_3}(u)} +e^{q_{\ell-1-\zeta_3}(u)}-3e^{q_\ell(u)}, $$ for a lattice points $\ell \in \mathbb{Z}[\zeta_3]$ of a 3-regular graph, and its elliptic solution for the curve $y(y-s)=x^3$, ($s\neq 0$) and $\zeta_3=e...

In this paper, we proposed a novel method using the elementary number theory to investigate the discrete nature of the screw dislocations in crystal lattices, simple cubic (SC) lattice and body centered cubic (BCC) lattice, by developing the algebraic description of the dislocations in the previous report (Hamada, Matsutani, Nakagawa, Saeki, Uesaka...

We consider the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the result of Li and Hou's on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$. Furthermore, we give a formula for the weighted Kirchhoff index of a regular covering of $G$ in t...

A recent generalization of the "Kleinian sigma function" involves the choice of a point $P$ of a Riemann surface $X$, namely a "pointed curve" $(X, P)$. This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at $P$. We exhibit the Riemann constant for a Weierstrass semigroup at $P$ with minimal set of generat...

We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y+ A_{r}(x)$ where $x$ is an affine coordinate on $\mathbb{P}^1$, the point $\infty$ on $X$ is mapped to $x=\infty$, and each $A_j$ is a polynomial in $x$ of degree $\le j s/r$ for a certain coprime posi...

In the previous report [Phys. Rev. B {\bf{62}} 13812 (2000)], by proposing the mechanism under which electric conductivity is caused by the activational hopping conduction with the Wigner surmise of the level statistics, the temperature-dependent of electronic conductivity of a highly disordered carbon system was evaluated including apparent metal-...

The Abelian function theory was constructed by Abel, Jacobi, Weierstrass, Rie-mann and so on in XIX and developed as an abstract theory in XX. Now we in XXI should mixed up both theories to reconstruct the Abelian function theory. In this note, I show its history and its beyond.

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate...

We propose a new theory on a relation between diffusive and coherent nature in one dimensional wave mechanics based on a quantum walk. It is known that the quantum walk in homogeneous matrices provides the coherent property of wave mechanics. Using the recent result of a localization phenomenon in a one-dimensional quantum walk (Konno, Quantum Inf....

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the energy of a screw dislocation based on the spring model is expressed by the Epstein zeta function approximately.

It is a English translation version of Japanese article in Japanese journal, Gendai-Sugaku.

The zero divisor of the theta function of a compact Riemann surface $X$ of genus $g$ is the canonical theta divisor of Pic${}^{(g-1)}$ up to translation by the Riemann constant $\Delta$ for a base point $P$ of $X$. The complement of the Weierstrass gaps at the base point $P$ given as a numerical semigroup plays an important role, which is called th...

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing
isometry and isoenergy conditions on a moduli space of plane loops. The
conditions are compared to the constraints that define Euler's elastica.
Moreover, the conditions are shown to be constraints on the curvature and other
invariants of the loops which appear as coefficients o...

A cyclic trigonal curve of genus three is a $\mathbb{Z}_3$ Galois cover of
$\mathbb{P}^1$, therefore can be written as a smooth plane curve with equation
$y^3 = f(x) =(x - b_1) (x - b_2) (x - b_3) (x - b_4)$. Following Weierstrass
for the hyperelliptic case, we define an ``$\mathrm{al}$'' function for this
curve and $\mathrm{al}^{(c)}_r$, $c=0,1,2$...

In the previous article (S. Matsutani and Y. Shimosako and Y. Wang, Physica A
\bf{391} (2012) 5802-5809) we numerically investigated an electric potential
problem with high contrast local conductivities ($\gamma_0$ and $\gamma_1$,
$0<\gamma_0 \ll \gamma_1$) for a two-dimensional continuum percolation model
(CPM). As numerical results, we showed the...

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular forms have pointed to the relevance of τ-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) σ-function. Klein originally generali...

In this paper, we study some cyclic (r, s) curves X given by
We give an expression for the prime form , where (P, Q ∈ X), in terms of the sigma function for some such curves, specifically any hyperelliptic curve (r, s) = (2, 2g + 1) as well as the cyclic trigonal curve (r, s) = (3, 4),
where r is a certain multi-index of differentials. Here u1 an...

Compact Riemann surfaces and their abelian functions are instrumental to
solve integrable equations; more recently the representation theory of the
Monster and related modular form have pointed to the relevance of
$\tau$-functions, which are in turn connected with a specific type of abelian
function, the (Kleinian) $\sigma$-function. This paper pro...

In this article, we introduce a novel geometrical index $\delta_{agg}$, which
is associated with the Euler number and is obtained by an image processing
procedure for a given digital picture of aggregated particles such that
$\delta_{agg}$ exhibits the degree of the agglomerations of the particles. In
the previous work (Matsutani, Shimosako, Wang,...

In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r =x^s +
\lambda_{1} x^{s-1} +...+ \lambda_{s-1} x + \lambda_s$. We give an expression
for the prime form $\cE(P,Q)$, where $(P, Q \in X)$, in terms of the sigma
function for some such curves, specifically any hyperelliptic curve $(r,s) =
(2, 2g+1)$ as well as the cyclic trigonal...

In this article, a generalized Kleinian sigma function for an affine (3,4,5)
space curve of genus 2 was constructed as the simplest example of the sigma
function for an affine space curve, and in terms of the sigma function, the
Jacobi inversion formulae for the curve are obtained. An interesting relation
between a space curve with a semigroup gene...

We numerically investigate the electric potential distribution over a
two-dimensional continuum percolation model between the electrodes. The model
consists of overlapped conductive particles on the background with an
infinitesimal conductivity. Using the finite difference method, we solve the
generalized Laplace equation and show that in the poten...

We explore a computational model of an incompressible fluid with a
multi-phase field in three-dimensional Euclidean space. By investigating an
incompressible fluid with a two-phase field geometrically, we reformulate the
expression of the surface tension for the two-phase field found by Lafaurie,
Nardone, Scardovelli, Zaleski and Zanetti (J. Comp....

In order to clarify how the percolation theory governs the conductivities in
real materials which consist of small conductive particles, e.g.,
nanoparticles, with random configurations in an insulator, we numerically
investigate the conductivities of continuum percolation models consisting of
overlapped particles using the finite difference method...

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of cur...

The CIP-method is a numerical computational method in the computational fluid dynamics discovered by Takewaki, Nishiguchi and Yabe [H. Takewaki, A. Nishiguchi, T. Yabe, The cubic-interpolated pseudo-particle (CIP) method for solving hyperbolic-type equations, J. Comput. Phys. 61 (1985) 261–268], which shows nice properties as a numerical computatio...

M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23})
considered a lattice model with exponential interaction and proved, as
suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact
periodic and soliton solutions. The Toda lattice, as it came to be known, was
then extensively studied as one of the complete...

Previous work by the authors (this journal, \vol{60} (2008), 1009-1044)
produced equations that hold on certain loci of the Jacobian of a cyclic
$C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the
equations in question are given in terms of (Klein's) generalization of
Weierstrass' $\sigma$-function. The key tool is a matrix wi...

By numerically solving the generalized Laplace equations by means of the finite difference method, we investigated isotropic electric conductivity of a three-dimensional continuum percolation model consisting of overlapped spheroids of revolution in continuum. Since the computational results strongly depend upon parameters in the discretization met...

In 1691, James (Jacob) Bernoulli proposed a problem called elastica problem: What shape of elástica, an ideal thin elastic rod on a plane, is allowed? Daniel Bernoulli discovered its energy functional, Euler-Bernoulli energy function, and the minimal principle of the elastica. Using it, Euler essentially solved the problem in 1744 by developing the...

In this article, I propose a concept of the $p$-on which is modelled on the multi-photon absorptions in quantum optics. It provides a commutative ring structure in quantum mechanics. Using it, I will give an operator representation of the Riemann $\zeta$ function.

We construct a class of (complex-valued) solutions to the dispersionless KP equation using a meromorphic function on a plane algebraic curve as a variable dependent on suitable Abelian integrals.

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing
parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface
which, outside from one...

By using the generalized sigma function of a Crs
curve yr=f(x)
, we give a solution to the Jacobi inversion problem over the stratification in the Jacobian given by the Abel image of the symmetric products of the curve. We show that determinants consisting of algebraic functions on the curve, whose zeros give the Abelian pre-image of the strata, ar...

Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705–742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata Wk, the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations fo...

This article shows that the Neumann dynamical system is described well in terms of the Weierstrass hyperelliptic al functions. The descriptions are very primitive; their proofs are provided only by residual computations but don't require any theta functions.

In the previous article (Found Phys. Lett. {\bf{16}} 325-341), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a...

In the previous article (Matsutani S 2002 J. Geom. Phys. 43 146–62), we showed the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. Using the results, this paper studies relations between the quantized elastica and integrals of its Schwarz derivative, the winding effects in the quantized elastica problems and some other...

We develop the theory of generalized Weierstrass σ- and ℘-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘-functions, a proof that the coefficients of the power series expansion of the σ-function are polynomials of coefficients of the defi...

This article is devoted to an investigation of a reality condition of a hyperelliptic loop soliton of higher genus. In the investigation, we have a natural extension of Jacobi am-function for an elliptic curves to that for a hyperelliptic curve. We also compute winding numbers of loop solitons.

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold $S$ immersed in a higher dimensional spin manifold $M$ from viewpoint of study of submanifold quantum mechanics. We show that kernel of a certain Dirac operator defined over $S$, which we call submanifold Dirac operator, gives the data of...

This article shows explicit relation between fractional expressions of Schottky-Klein type for hyperelliptic $\sigma$-functions and a product of differences of the algebraic coordinates on each stratum of natural stratification in a hyperelliptic Jacobian.

This article shows that the Neumann dynamical system is described well in terms of the Weierestrass hyperelliptic al functions.

This article extends relations of Mumford's UVW-expressions to those in subvarieties in a hyperelliptic Jacobian using Baker's method.

The submanifold quantum mechanics was opened by Jensen and Koppe and has been studied for more than three decades. This article gives its more algebraic definition and show the essential aspects of the submanifold quantum mechanics from an algebraic viewpoint.

In the previous article (J. Geom. Phys. {\bf 43} (2002) 146), we show the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. This article gives some functional relations in a loop soliton as a quantized elastica.

The submanifold quantum mechanics was opened by Jensen and Koppe (Ann. Phys. {\bf 63} (1971) 586-591) and has been studied for these three decades. This article gives its more algebraic definition and show what is the essential of the submanifold quantum mechanics from an algebraic viewpoint.

The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in $\EE^4$ with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of im...

We study the Toda equations in the continuous level, discrete level and
ultradiscrete level in terms of elliptic and hyperelliptic $\sigma$ and $\psi$
functions of genera one and two. The ultradiscrete Toda equation appears as a
discrete-valuation of recursion relations of $\psi$ functions.

In the previous work [J. Geom. Phys. 39 (2001) 50], the closed loop solitons in a plane, i.e., loops whose curvatures obey the modified Korteweg–de Vries equations, were investigated for the case related to algebraic curves with genera 1 and 2. This paper is a generalization of the previous paper to those of hyperelliptic curves with general genera...

Berry and Klein [J. Mod. Opt.
43, 2139-2164 (1997)] showed that the Talbot effects in classical optics are naturally expressed by Gauss sums in number theory. Their result was obtained by a computation of Helmholtz equation. In this article, we calculate the effects using Fresnel integral and show that the result is also represented by Gauss sums....

The solutions of the discrete Painlevé equation I were constructed in terms of elliptic and hyperelliptic ψ functions for algebraic curves of genera one and two. For the case of genus two, there appear higher order difference equations which naturally contain the discrete Painlevé equation I as a special case.

The sine-Gordon equation has hyperelliptic al function solutions over a hyperelliptic Jacobian for $y^2 = f(x)$ of arbitrary genus $g$. This article gives an extension of the sine-Gordon equation to that over subvarieties of the hyperelliptic Jacobian. We also obtain the condition that the sine-Gordon equation in a proper subvariety of the Jacobian...

We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$ as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gordon equation had already obtain...

The discrete Lotka-Volterra equation over p-adic space was
constructed since p-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking
the ultra-discrete limit studied in soliton theory should be
regarded as a space with the non-Archimedean valuations given in my
previous paper (Matsutani S 2001 Int. J. Mat...

Motion of electrons in a half space with cylindrical electro-static field, which is associated with electron emission devices,
is investigated. The equation of motion is given as a non-linear ordinary differential equation of the second order. Statistical
behavior of electrons exhibits a dilatational invariance property of the system. An approximat...

Explicit hyperelliptic solutions of the modified Korteweg-de
Vries equations without any ambiguous parameters were
constructed in terms of only the hyperelliptic al-functions
over the non-degenerate hyperelliptic curve y2 = f(x) of
arbitrary genus g. In the derivation, any θ-functions
or Baker-Akhiezer functions were not essentially used. Then the...

Closed loop solitons in a plane, whose curvatures obey the modified Korteweg–de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker’s sigma functions for the genus two case. This study is closely related to classical and quantized e...

A recursion relation of hyperelliptic psi functions of genus two, which was derived by D.G. Cantor (J. reine angew. Math. 447 (1994) 91-145), is studied. As Cantor's approach is algebraic, another derivation is presented as a natural extension of the analytic derivation of the recursion relation of the elliptic psi function.

Soliton Solutions of Korteweg-de Vries (KdV) were constructed for given degenerate curves $y^2 = (x-c)P(x)^2$ in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were also presented.

The hopping conductivity in a disordered system which is composed of small (semi-) metallic granules is presented. Due to the irregular shape of each granule, the level statistics of a free electron in a granule is expressed by a random matrix, and a formula for the temperature-dependent conductivity (TDC) is obtained for such a disordered system....

Using the submanifold quantum mechanical scheme, the restricted Dirac operator in a submanifold is defined. Then it is shown that the zero mode of the Dirac operator expresses the local properties of the submanifold, such as the Frenet-Serret and generalized Weierstrass relations. In other words this article gives a representation of a further gene...

In this paper we give natural generalization of the formula of Kiepert (see (1.1) below) for all hyperelliptic curves.

In a previous report [Phys. Lett. A 216, 178 (1996)] we phenomenologically considered the conduction mechanism of a highly disordered carbon system. In this article a metal-insulator transition is investigated based on our theory for the disordered system. Our results for the temperature-dependent conductivity show an apparent metal-insulator trans...

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy...

In the previous report (J. Phys. A30 (1997) 4019-4029), I showed that the Dirac operator defined over a conformaL surface immersed in ℝ3 by means of confinement procedure is identified with the differential operator of the generalized Weierstrass equation and the Lax operator of the modified Novikov-Veselov (MNV) equation. In this article, using th...

For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operators in the submanifold have bee...

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in $n$-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret relation for a space curve case...

We study the difference-difference Lotka-Volterra equations in
p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the
ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a
classical limit of a quantum object, it implies...

In previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac operator confined in a surface immersed in R3 by means of a mass type potential completely exhibits the surface itself and is identified with that of the generalized Weierstrass equation. In this article, I quantized the Dirac field and calculated the gauge transformation...

Recently the submanifold quantum system has been studied. In this article, after we confine a Dirac field in a thin curved rod, we find an anomalous term in the fermionic field theory related to the extrinsic curvature. In other words, we find a new Atiyah-Singer-type index theorem related to the geometry or the submanifold. As the anomaly is assoc...

In this paper, the statistical mechanics of a non-stretching elastica in two-dimensional space using the path integral method is investigated. In the calculation, the modified Korteweg-de Vries (MKdV) hierarchy naturally appeared in the equations including the temperature fluctuation. We have classified the moduli of the closed elastica in a heat b...

In quantum mechanics on a submanifold, it is known that when the submanifold bas an extrinsic curvature, an effective potential appears in the Schrodinger equation even if it does not curve intrinsically. Recently Ikegami et al. (1992) applied the Dirac quantization scheme for a constrained system to submanifold physics and found that there is an a...

The shape of a surface with constant mean curvature (CMC) has been studied in mathematics and physics related to nonlinear integrable theory and harmonic map (-model) theory. In the study a fictitious (linear) Dirac-type operator appears as a tool of the calculus (Konopelchenko B G and Taimanov I A 1996 J. Phys. A: Math. Gen. 29 1261 - 5).
In this...

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli [Formula: see text] of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that it...

Electron trajectories in surface conduction electron emitter displays (SEDs) are analyzed based on the multiple scattering model. Besides calculating beam spot patterns on the phosphor by the ray tracing method, simple formulae for calculating beam spot size and electron emission e#ciency are proposed. It is shown that these calculations reproduce...

In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system...

In the previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac operator defined over a conformal surface immersed in R^3 is identified with the Dirac operator which is generalized the Weierstrass- Enneper equation and Lax operator of the modified Novikov-Veselov (MNV) equation. In this article, I determine the Dirac operator defin...

Recently I proposed a new calculation scheme of a partition function of an immersion object using path integral method and theory of soliton (to appear in J.Phys.A). I applied the scheme to problem of elastica in two-dimensional space and Willmore surface in three dimensional space. In this article, I will apply the scheme to elastica in three dime...

In this article, I have precisely considered the time development of a quantum particle (of excited states) in the quartic potential (x2-a2)2/2g by means of the semiclassical path integral method. Using the elliptic functions, I have evaluated the tunneling phenomena and the quasi-quantum fluctuation around the quasi-classical paths. I found that t...

In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system...

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state o...

We consider a one-dimensional classical hard core chain with different alternating masses m and M. For a certain mass ratio , there exists a localized state which consists of three adjacent particles and propagates. Then its mass ratio is given by a polynomial with integer coefficients, which turns out to be the cyclotomic polynomial. We can derive...

In this article, I have investigated statistical mechanics of a
non-stretched elastica in two dimensional space using path integral
method. In the calculation, the MKdV hierarchy naturally appeared as the
equations including the temperature fluctuation.I have classified the
moduli of the closed elastica in heat bath and summed the Boltzmann
weight...

We have phenomenologically investigated the conduction mechanism related to the activation energy in a highly disordered carbon system, which was proposed by Kuriyama [Phys. Rev. B 47 (1993) 12415], and found an alternative expression for the temperature-dependent conductivity of the system from a more microscopic view point. Our theoretical result...

We present in this paper a quantum-mechanical problem of a particle in a Corbino ring (i.e. a ring-shaped quantum wire inR
2) with finite-potential barriers, and consider i) the mathematical aspects associated with a quantum mechanics of the particle for a Corbino geometry and ii) the confinement (or escape) problem of the particle in the ring. It...

Recently we found that the Dirac operator on a thin elastic rod is identical with the Lax operator of the modified Korteweg-de Vries (MKdV) equation while the thin elastic rod is governed by the MKdV equation. In this article, we will show the physical relation between the Hirota bilinear method and the Dirac field in a thin rod on two-dimensional...