Shigeki Matsutani

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations satisfied by the \wp-functions, a proof that the coefficients of the power series expansion of the sigma-function are polynomials of moduli parameters, and the derivation of two addition formulae.

Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations are constructed for a given curve y2 = f(x) whose genus is three. This paper is based upon the fact that about one hundred years ago (Baker H F 1903 Acta Math. 27 135-56), Baker essentially derived KdV hierarchy and KP equations by using a bilinear differential operator D, identities of Pfaffians, symmetric functions, the hyperelliptic σ-function and -functions; µν = -∂µ∂νlog σ = -(DµDνσσ)/2σ2. The connection between his theory and the modern soliton theory is also discussed.

Reflectionless Quantum Wire

When a quantum wire is not straight, an effective potential appears in the Schrodinger equation according to its curvature. Generally due to this geometrical potential the electron along the wire cannot change its direction of the motion without reflection. Here, the new curve is proposed in which the electron traverses changing its direction without reflection. The curve is determined under the condition that the effective potential should be reflectionless one which is well-known in the scattered problem.

Statistical mechanics of elastica on a plane: origin of the MKdV hierarchy

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 bath and summed the Boltzmann weight with the thermal fluctuation over the moduli. Due to the bilinearity of the energy functional, its exact partition function has been obtained. By investigation of the system, it is conjectured that an expectation value at a critical point of this system obeys the Painleve equation of the first kind and its related equations are extended by the Korteweg-de Vries hierarchy. Furthermore, we also comment on the relation between the MKdV hierarchy and the Becchi-Rouet-Stora transformation in this system.

Hyperelliptic loop solitons with genus g: investigations of a quantized elastica

Abstract 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. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al-function for a given hyperelliptic curve y 2 = f ( x ) with genus g .

Wave-Particle Complementarity and Reciprocity of Gauss Sums on Talbot Effects

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. However function forms of these two computational results are apparently different. We show that the reciprocity law of Gauss sums connects these results and both completely agree with. The Helmholtz equation can be regarded as an equation based upon wavy nature in optics whereas the Fresnel integral is defined by a sum over the paths based upon a particle picture in optics. Thus the agreement of these two computational results could be interpreted in terms of the concept of the wave-particle complementarity, though the concept is for quantum mechanical phenomenon. This interpretation leads us to a relation between the reciprocity of Gauss sums in number theory and the wave-particle complementarity in wave physics. We discuss it in detail.

The physical meaning of the embedded effect in the quantum submanifold system

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 anomalous correspondence between the quantum and the classical mechanics. In this paper, we show the physical meaning of the origin of it through the polar representation and then the results of Ikegami et al. are naturally understood.

Path Integral Formulation of Curved Low Dimensional Space

When a low dimensional space has a curvature, there is an effective potential in the Schrodinger equation as a geometrical correction. In this paper, we have shown that a confined space in three dimensional space can be regarded as a curved low (one or two) dimensional space when the thickness of the space multiplied by the Weingarten map of each space is sufficiently smaller than unity. Under the condition, we have also evaluated the effective potential using the path integral method.

Recursion relation of hyperelliptic PSI-functions of genus two

A recursion relation of hyperelliptic ψ functions of genus two, which was derived by D. G. Cantor (J. reine angew. Math. 447 (1994) 91–145), is studied. As Cantors approach is algebraic, another derivation is presented as a natural extension of the analytic derivation of the recursion relation of the elliptic ψ function in a sense of study of special functions.

Statistical mechanics of non-stretching elastica in three-dimensional space

Recently by using path integral method and theory of soliton, a new calculation scheme of a partition function of an immersion object has been proposed [J. Phys. A 31 (1998) 2705–2725]. In this paper, the scheme to elastica (space curve with the Bernoulli-Euler functional) immersed in three-dimensional space R3 as a physical model in polymer science is applied. It is shown that the nonlinear Schrodinger and complex modified Korteweg-de Vries hierarchies naturally appear to express the functional space of the partition function. In other words, it is shown that the configuration space of an elastica immersed in R3 can be classified by these equations. Then the partition function is reduced to an ordinary integral over the orbit space of these hierarchies.