Effective Mass Hamiltonians With Linear Terms in the Momentum: Darboux Transformations and Form-Preserving Transformations

Abstract

We define form-preserving transformations and Darboux transformations for time-dependent, effective mass Hamiltonians with additional linear terms. We give reality conditions for both transformations, guaranteeing the transformed potential to be real-valued. We further show that our form-preserving transformation preserves normalizability of the Schrödinger wave function. Our results generalize all former results on form-preserving transformations and Darboux transformations for the time-dependent Schrödinger equation. This paper is a sequel of Refs. 16–18.

Supersymmetry of time-dependent Schrödinger equations with effective mass

“…We omitted to state concrete applications of our supersymmetry formalism here, because such applications can already be found in the context of the effective mass Darboux transformation [13,15]. Applications of the effective mass supersymmetry formalism to physically interesting cases, e.g.…”

“…It is well known that the stationary Schrödinger equation with effective mass admits the supersymmetry formalism, but this has not yet been shown for the timedependent case. It is also unknown whether this supersymmetry formalism and the recently introduced effective mass Darboux transformation [13][14][15] are equivalent, as they are for constant mass. The remainder of this note is organized as follows: in Section 2 we review the effective mass TDSE and its Darboux transformation, while Section 3 is devoted to the derivation of the supersymmetry formalism and its equivalence with the effective mass Darboux transformation.…”

Точно Решаемые Потенциалы И Решения Для Связанных Состояний Уравнения Шредингера В $D$-Мерном Пространстве С Зависящей От Координат Массой

“…Для решения уравнения Шредин-гера с массой, зависящей от координат, применяются различные методы, в ли-тературе можно найти каноническое точечное преобразование [9], [18]- [22], метод Никифорова-Уварова [23]- [25], суперсимметричный подход [26], [27], метод квадра-тичной алгебры [28], аналитический метод [29], поиск решений в виде ряда [30], ме-тоды преобразований Дарбу [31], [32], сплетающих операторов [33], восстановления волновых пакетов [34], -разложение [35], метод расширенного преобразования [12] и т. д. Во всех этих случаях волновые функции выражаются через классические ортогональные полиномы (КОП). Фактически оказывается, что КОП играют важ-ную роль с самого зарождения квантовой механики, поскольку через эти полиномы выражаются собственные функции связанных состояний.…”

Time dependent potentials associated with exceptional orthogonal polynomials

“…In particular, FPTs have been used to derive exact solutions for quadratic potentials, 34 including harmonic oscillators 57 linear as well as noncentral potentials, 58 and harmonic oscillators with damping. 59 Recently, FPTs were also defined for TDSEs with position dependent mass, [60][61][62][63] in higher dimensions. 64 Properties of FPTs like reality conditions for the transformed potential 64 and interplay with Darboux transformations 56 were also studied.…”