GENERATING SHAPE INVARIANT POTENTIALS

2008 ◽  
Vol 23 (31) ◽  
pp. 4959-4978 ◽  
Author(s):  
ASIM GANGOPADHYAYA ◽  
JEFFRY V. MALLOW
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Quantum Mechanics ◽  
Supersymmetric Quantum Mechanics ◽  
New Method ◽  
Invariance Condition ◽  
Partial Differential ◽  
Shape Invariance ◽  
Shape Invariant

We transform the shape invariance condition, a difference-differential equation of supersymmetric quantum mechanics, into a local partial differential equation. We develop a new method for generating translationally shape invariant potentials from this equation. We generate precisely all the known shape invariant potentials, and argue that there are unlikely to be others.

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 PDE Surface-Represented Facial Blendshapes

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2905
Author(s):  
Haibin Fu ◽  
Shaojun Bian ◽  
Ehtzaz Chaudhry ◽  
Shuangbu Wang ◽  
Lihua You ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fourier Series ◽  
Mathematical Expression ◽  
Second Order ◽  
New Method ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Boundary Curves ◽  
Pde Surfaces

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

A New Method for Finding Solutions of Linear Partial Differential Equation

2011 ◽  
Vol 219-220 ◽  
pp. 675-679
Author(s):  
Yan Tang ◽  
Mao Chang Qin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Linear Differential Equation ◽  
Variable Coefficient ◽  
Linear Partial Differential Equation ◽  
New Method ◽  
Partial Differential ◽  
Special Solutions ◽  
Linear Differential

A useful technique is adopted to study special solutions of a general Black and Scholes equation in this letter. Several kinds of new special solutions are obtained. This method is effective for finding special solutions of linear differential equation with variable coefficient.

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