scholarly journals DARBOUX TRANSFORMATION OF THE GREEN FUNCTION FOR THE DIRAC EQUATION WITH THE GENERAL POTENTIAL

2008 ◽  
Vol 23 (02) ◽  
pp. 247-258 ◽  
Author(s):  
EKATERINA POZDEEVA
Keyword(s):  
Green Function ◽  
Dirac Equation ◽  
Boundary Problem ◽  
Darboux Transformation ◽  
Green Functions ◽  
Regular Boundary ◽  
One Dimensional ◽  
The Green Function ◽  
The One ◽  
Stationary Dirac Equation

We consider the Darboux transformation of the Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We obtained the Green functions of the transformed Dirac equation with the initial regular boundary conditions. We also obtain the formula for the full trace of the difference of the transformed and initial Green functions of the regular boundary problem of the one-dimensional stationary Dirac equation. We illustrate our findings by the consideration of the Darboux transformation of the Green function on an interval.

DARBOUX TRANSFORMATION FOR THE ONE-DIMENSIONAL STATIONARY DIRAC EQUATION WITH NON-HERMITIAN INTERACTION

2006 ◽  
Vol 21 (28n29) ◽  
pp. 5807-5822 ◽  
Author(s):  
A. SINHA ◽  
P. ROY
Keyword(s):  
Dirac Equation ◽  
Darboux Transformation ◽  
One Dimensional ◽  
Exactly Solvable ◽  
The One ◽  
Stationary Dirac Equation

The Darboux algorithm is applied to an exactly solvable one-dimensional stationary Dirac equation, with non-Hermitian, pseudoscalar interaction V0(x). This generates a hierarchy of exactly solvable Dirac Hamiltonians, [Formula: see text], defined by new non-Hermitian interactions V1(x), which are also pseudoscalar. It is shown that [Formula: see text] are isospectral to the initial Hamiltonian h0, except for certain missing states.

scholarly journals Intertwining technique for the one-dimensional stationary Dirac equation

2003 ◽  
Vol 305 (2) ◽  
pp. 151-189 ◽  
Author(s):  
L.M. Nieto ◽  
A.A. Pecheritsin ◽  
Boris F. Samsonov
Keyword(s):  
Dirac Equation ◽  
One Dimensional ◽  
The One ◽  
Stationary Dirac Equation

scholarly journals Simplifying the one-loop five graviton amplitude in type IIB string theory

2017 ◽  
Vol 32 (14) ◽  
pp. 1750074 ◽  
Author(s):  
Anirban Basu
Keyword(s):  
Green Function ◽  
String Theory ◽  
Fundamental Domain ◽  
Green Functions ◽  
Considerable Simplification ◽  
The Green Function ◽  
Type Iib String Theory ◽  
The One

We consider the [Formula: see text] and [Formula: see text] terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of [Formula: see text]. Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the [Formula: see text] and [Formula: see text] terms where each of these two links are given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.

Darboux transformations of the one-dimensional stationary Dirac equation

2002 ◽  
Vol 35 (14) ◽  
pp. 3279-3287 ◽  
Author(s):  
N Debergh ◽  
A A Pecheritsin ◽  
B F Samsonov ◽  
B Van den Bossche
Keyword(s):  
Dirac Equation ◽  
Darboux Transformations ◽  
One Dimensional ◽  
The One ◽  
Stationary Dirac Equation

scholarly journals Evaluation of the One-Dimensional Lp Sobolev Type Inequality

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 296
Author(s):  
Kazuo Takemura ◽  
Yoshinori Kametaka
Keyword(s):  
Differential Equation ◽  
Green Function ◽  
Type Inequality ◽  
Sobolev Type ◽  
Sharp Constant ◽  
Order Ordinary Differential Equation ◽  
One Dimensional ◽  
The Green Function ◽  
The One ◽  
Sobolev Type Inequality

This study applies the extended L 2 Sobolev type inequality to the L p Sobolev type inequality using Hölder’s inequality. The sharp constant and best function of the L p Sobolev type inequality are found using a Green function for the nth order ordinary differential equation. The sharp constant is shown to be equal to the L p norm of the Green function and to the pth root of the value of the origin of the best function.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

Unilateral Global Bifurcation from Infinity in Nonlinearizable One-Dimensional Dirac Problems

2021 ◽  
Vol 31 (01) ◽  
pp. 2150005
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim A. Neymatov ◽  
Humay Sh. Rzayeva
Keyword(s):  
Dirac Equation ◽  
Eigenvalue Problems ◽  
Global Bifurcation ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Bifurcation From Infinity ◽  
The One ◽  
Nodal Properties

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

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