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 Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of ℏ

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1853
Author(s):  
Christiane Quesne
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Supersymmetric Quantum Mechanics ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant ◽  
Infinite Set

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translational) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, which, by an infinite set of partial differential equations, can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to ℏ-independent superpotentials and to a superpotential explicitly depending on ℏ.

scholarly journals First-order modal logic in the necessary framework of objects

2016 ◽  
Vol 46 (4-5) ◽  
pp. 584-609 ◽  
Author(s):  
Peter Fritz
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Consequence Relations ◽  
Possible World ◽  
Logical Constants ◽  
First Order ◽  
Timothy Williamson ◽  
Infinite Sets ◽  
Infinite Set ◽  
First Order Modal Logic

AbstractI consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

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 On Accuracy and Coherence with Infinite Opinion Sets

2022 ◽  
pp. 1-37
Author(s):  
Mikayla Kelley
Keyword(s):  
De Finetti ◽  
Infinite Sets ◽  
Countably Infinite ◽  
Infinite Set

Abstract There is a well-known equivalence between avoiding accuracy dominance and having probabilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009, Pettigrew 2016). However, this equivalence has been established only when the set of propositions on which credence functions are defined is finite. In this paper, I establish connections between accuracy dominance and coherence when credence functions are defined on an infinite set of propositions. In particular, I establish the necessary results to extend the classic accuracy argument for probabilism to certain classes of infinite sets of propositions including countably infinite partitions.

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