Submanifold Differential Operators in \Cal D-Module Theory I : Schr\"odinger Operators

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 been investigated in such a scheme , which are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case. These Dirac operators are concerned well in the differential geometry, since they completely represent the submanifolds. In this and a future series of articles, we will give mathematical construction of the differential operators on a submanifold in \EE^n in terms of \DMod-module theory and rewrite recent results of the Dirac operators mathematically. In this article, we will formulate Schr\"odinger operators in a low-dimensional submanifold in \EE^n.Comment: AMS-Tex Use 24 page

Submanifold Dirac Operator with Torsion

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 immersion of the surface in \EE^4.sComment: 16 page

Relations in a Loop Soliton as a Quantized Elastica

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.Comment: 9 page

Lotka-Volterra Equation over a Finite Ring Z/p^N Z

Discrete Lotka-Volterra equation over $p$-adic space was constructed since $p$-adic space is a prototype of spaces with the non-Archimedean valuations and the space given by taking ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations in the previous report (solv-int/9906011). In this article, using the natural projection from $p$-adic integer to a ring $Z/p^n Z$, a soliton equation is defined over the ring. Numerical computations shows that it behaves regularly.Comment: LaTex, 10 page

On a commutative ring structure in quantum mechanics

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

On Relations of Hyperelliptic Weierstrass al Functions

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 obtained, our derivations of them are simple; they need only residual computations over the curve and primitive matrix computations.Comment: AMS-Tex, 10 page

Reality Conditions of Loop Solitons Genus g: Hyperelliptic am Functions

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.Comment: 13 pages, 3 figure

Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian

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 is defined.Comment: 9pages. to appear in Cubo Journa

Soliton Solutions of Kortweg-de Vries Equations and Hyperelliptic Sigma Functions

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.Comment: AMS-TeX, 12 page

Hyperelliptic Solutions of Modified Korteweg-de Vries Equation of Genus g: Essentials of Miura Transformation

Explicit hyperelliptic solutions of the modified Korteweg-de Vries equations without any ambiguous parameters were constructed in terms only of the hyperelliptic \al-functions over non-degenerated hyperelliptic curve $y^2 = f(x)$ of arbitrary genus $g$. In the derivation, any $\theta$-functions or Baker-Akhiezer functions were not essentially used. Then the Miura transformation naturally appears as the connections between the hyperelliptic $\wp$-functions and hyperelliptic \al-functions.Comment: AMS-Tex, 13 page