Intertwining technique for the one-dimensional stationary Dirac equation

Abstract

The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples

Darboux operators for linear first-order multi-component equations in arbitrary dimensions

“…In the next step we consider (11). In particular, we show that the latter equation is equivalent to a matrix equation, the form of which resembles (1).…”

“…Hence, we have now shown that if equation (18) holds for some matrix , then condition (17) is fulfilled. The latter condition is equivalent to (11), which emerged directly from our intertwining relation (4). Hence, in order to perform a Darboux transformation by means of our intertwiner (3), we must provide a solution of (18), which for that reason will be referred to as auxiliary equation.…”

“…The Darboux transformation has also been made applicable to multi-component equations, in particular to the Dirac equation. The first work on the topic [7] was followed by applications [8,9] and generalizations [10][11][12][13][14]. While in the latter references at most two dimensions are considered, the present study introduces a Darboux transformation applicable to arbitrary dimensional, multicomponent equations that are linear and of first order.…”

Новое Двухпараметрическое Семейство Точно Решаемых Дираковских Гамильтонианов

“…В явном виде систематическая разработка метода преобразований Дарбу-Крума для одномерного двухкомпонентного уравнения Дирака представлена в работе [15] (см. обзор литературы в [7], [15]).…”

“…В явном виде систематическая разработка метода преобразований Дарбу-Крума для одномерного двухкомпонентного уравнения Дирака представлена в работе [15] (см. обзор литературы в [7], [15]). Согласно работе [7] форма преобразований Дарбу для четырехкомпонентного ста-ционарного уравнения Дирака может быть аналогична форме преобразования для двухкомпонентного стационарного уравнения Дирака, рассмотренной в работе [15].…”