A New Type of Shape-Invariant Potential

1998 ◽  
Vol 30 (2) ◽  
pp. 305-308 ◽  
Author(s):  
Jia Chunsheng ◽  
Wang Xiaoguo ◽  
Yang Qiubo ◽  
He Su ◽  
Li Hao
Keyword(s):  
Shape Invariant ◽  
Invariant Potential ◽  
New Type

scholarly journals Shape invariant rational extensions and potentials related to exceptional polynomials

2015 ◽  
Vol 30 (24) ◽  
pp. 1550146 ◽  
Author(s):  
S. Sree Ranjani ◽  
R. Sandhya ◽  
A. K. Kapoor
Keyword(s):  
Shape Invariant ◽  
Invariant Potential ◽  
Exceptional Polynomials ◽  
Infinite Sets ◽  
Infinite Set ◽  
Explicit Expressions

In this paper, we show that an attempt to construct shape invariant extensions of a known shape invariant potential leads to, apart from a shift by a constant, the well known technique of isospectral shift deformation. Using this, we construct infinite sets of generalized potentials with [Formula: see text] exceptional polynomials as solutions. The method is simple and transparent and is elucidated using the radial oscillator and the trigonometric Pöschl–Teller potentials. For the case of radial oscillator, in addition to the known rational extensions, we construct two infinite sets of rational extensions, which seem to be less studied. Explicit expressions of the generalized infinite set of potentials and the corresponding solutions are presented. For the trigonometric Pöschl–Teller potential, our analysis points to the possibility of several rational extensions beyond those known in literature.

scholarly journals A New Type of Shape-Invariant Beams with Structured Coherence: Laguerre-Christoffel-Darboux Beams

Photonics ◽  
2021 ◽  
Vol 8 (4) ◽  
pp. 134
Author(s):  
Rosario Martínez-Herrero ◽  
Massimo Santarsiero ◽  
Gemma Piquero ◽  
Juan Carlos González de Sande
Keyword(s):  
Spectral Density ◽  
Spatial Coherence ◽  
Intensity Profile ◽  
Paraxial Approximation ◽  
Simple Analytical Expression ◽  
Analytical Expression ◽  
New Class ◽  
Shape Invariant ◽  
Cross Spectral Density ◽  
New Type

A new class of sources presenting structured coherence properties is introduced and analyzed. They are obtained as the incoherent superposition of coherent Laguerre-Gaussian modes with suitable coefficients. This ensures that the shape of the intensity profile and the spatial coherence features of the propagated beams are invariant during paraxial approximation. A simple analytical expression is obtained for the cross-spectral density of the sources of this class, regardless of the number of superposed modes. Properties of these sources are analyzed and described by several examples.

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