New implicitly solvable potential produced by second order shape invariance

Abstract

The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallow [1] to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.Recently, the general investigation of additive form of shape invariance for superpotentials without explicit dependence on parameters was performed by J.Bougie et al [1], [11]. It was demonstrated that no additional shape invariant models in one-dimensional case can be constructed. This result was obtained in the framework of standard SUSY QM with supercharges of first order in derivatives. Meanwhile, it is known [18], [19], [20] that such standard SUSY QM does not exhaust all opportunities to fulfill the generalized SUSY algebra for SuperhamiltonianĤ and superchargesQ ± . Higher order supercharges are also possible, and for example supercharges of second order in momentum lead to some new results [20].

Некоторые Свойства Форминвариантной Двумерной Модели Скарф II

“…Как правило, это свойство обеспечивает элегантный чисто алгебраический метод решения задачи, а именно позволяет аналитически находить и энергетический спектр, и соответствующие волновые функции. К классу форминвариантных потенциалов принадлежат [8] все известные точно решаемые одномерные потенциалы (некоторые дополнительные, неизвестные, точно решаемые системы были построены [9] как раз суперсимметричным методом). Понятие форминвариантности было также обобщено [10]- [13] и на двумерную суперсимметричную квантовую механику.…”

Shape invariance and SUSY separation of variables

“…The Schrödinger equation with this potential was solved analytically, it turned out [11] to be an implicitly solvable model.…”

N=2 supersymmetry and anisotropic scale invariance

“…These models would be an interesting starting point for tackling field theoretic questions about anisotropic scale invariance. Moreover, the supercharges will be higher derivative and the corresponding systems could be relevant for investigations into polynomial SUSY [53][54][55].…”