DIRAC OPERATOR ON A CONFORMAL SURFACE IMMERSED IN ℝ4: A WAY TO FURTHER GENERALIZED WEIERSTRASS EQUATION

2000 ◽  
Vol 12 (03) ◽  
pp. 431-444 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Differential Operator ◽  
Dirac Operator ◽  
Dirac Field ◽  
Nonlinear Schrödinger ◽  
Lax Operator ◽  
Weierstrass Equation

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 the same procedure, I determine the Dirac operator defined over a conformal surface immersed in ℝ4, which is for a Dirac field confined in the surface. Then it is reduced to the Lax operators of the nonlinear Schrödinger and the MNV equations by taking appropriate limits. It means that the Dirac operator is related to the further generalized Weierstrass equation for a surface in ℝ4.

scholarly journals IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ3

1999 ◽  
Vol 11 (02) ◽  
pp. 171-186 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Dirac Operator ◽  
Gauge Transformation ◽  
Dirac Field ◽  
Willmore Functional ◽  
Gauge Freedom ◽  
Expectation Value ◽  
Type Potential ◽  
Mass Type ◽  
Weierstrass Equation

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 which exhibits the gauge freedom of the parameterization of the surface. Then using the Ward–Takahashi identity, I showed that the expectation value of the action of the Dirac field is expressed by the Willmore functional and area of the surface, or the action of Polyakov's extrinsic string.

PHYSICAL REALIZATION OF THE JIMBO-MIWA THEORY OF THE MODIFIED KORTEWEG-DE VRIES EQUATION ON A THIN ELASTIC ROD: FERMIONIC THEORY

1995 ◽  
Vol 10 (22) ◽  
pp. 3091-3107 ◽  
Author(s):  
SHIGEKI MATSUTANI
Keyword(s):  
Dimensional Space ◽  
Mkdv Equation ◽  
Dirac Field ◽  
Elastic Rod ◽  
Bilinear Method ◽  
Physical Relation ◽  
De Vries ◽  
Lax Operator ◽  
Thin Elastic Rod ◽  
Hirota Bilinear

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 space, along the lines of the Jimbo-Miwa construction of the MKdV soliton.

scholarly journals Stationary waves of Schrödinger-type equations with variable exponent

2015 ◽  
Vol 13 (06) ◽  
pp. 645-661 ◽  
Author(s):  
Dušan Repovš
Keyword(s):  
Differential Operator ◽  
Variational Methods ◽  
Variable Exponent ◽  
Existence Results ◽  
Stationary Waves ◽  
Nonlinear Schrödinger ◽  
Reaction Term

We are concerned with a class of nonlinear Schrödinger-type equations with a reaction term and a differential operator that involves a variable exponent. By using related variational methods, we establish several existence results.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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