New solvable singular potentials

Abstract

We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, Pöschl-Teller I and Pöschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse square singularity at the origin. The regularization procedure gives rise to a delta-function behavior at the origin.Our new systems possess underlying non-linear potential algebras, which can also be used to determine their spectra analytically.

On the Dirac-Infeld-Plebański Delta Function

“…Similar results are obtained for other singular potentials defined on the complete real line, including the cases of discrete spectrum (see e.g. Dutt et al [36], and Negro et al [59]). Furthermore, it has been shown that this procedure does not change the results of a supersymmetric transformation.…”

“…In two and higher dimensions it provides a pedagogical introduction to the techniques of regularization in quantum field theory [31] (in one dimension, the quantum system needs no regularization.) It has been also studied in the context of supersymmetric quantum mechanics [32][33][34][35][36] where the susy partner of the attractive δ-function is the purely repulsive δ-function [32,33,35]. Similar results are obtained for potentials made up of additive δ-function terms [34,36].…”

Untitled

“…In the factorization of the Hamiltonian, Eqs. (4), (8) and (9) are used respectively. Hence we obtain…”

“…So far, SUSYQM has been extensively used to explore different aspects of nonrelativistic Quantum Mechanics systems [7]. This algebraic method was successful to study analytically solvable [7,8], the partially solvable [9,10], the isospectral, the periodic and exponential-type potential. Levai and Williams suggested a simple method for constructing potentials for which the Schrödinger equation can be solved exactly in terms of special functions [11] and showed relationship between the introduced formalism and Supersymmetric Quantum Mechanics [1].…”

Pauli isotonic oscillator with an anomalous magnetic moment in the presence of the Aharonov-Bohm effect: Laplace transform approach

“…Влияние аномального магнитного момента фермионов на его взаимодействие с сильным внешним магнитным полем составляет предмет физики высоких энергий. Анализ точных энергетических спектров нерелятивистских и релятивистских частиц может быть очень важным во многих областях теоретической физики, но его можно провести только с использованием небольшого числа потенциалов, таких как псевдогармонический потенциал [6], кулоновский потенциал [7], потенциал гармонического осциллятора [8] и т. д. Следовательно, поиски новых семейств таких потенциалов бесспорно важны. Наиболее интересной системой в этой категории является гармонический осциллятор, который в классической физике представляет собой изохронное колебание, а в квантовой -равноотстоящие друг от друга энергетические уровни [9].…”