SUPERSYMMETRY APPROACHES TO THE BOUND STATES OF THE GENERALIZED WOODS–SAXON POTENTIAL

2004 ◽  
Vol 19 (08) ◽  
pp. 615-625 ◽  
Author(s):  
H. FAKHRI ◽  
J. SADEGHI
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Bound States ◽  
Differential Operators ◽  
Energy Levels ◽  
Negative Energy ◽  
Saxon Potential ◽  
First Order ◽  
Jacobi Differential Equation ◽  
Algebraic Solutions

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

scholarly journals Algebraic solutions of differential equations over ℙ1 −{0,1,∞}

2018 ◽  
Vol 14 (05) ◽  
pp. 1427-1457
Author(s):  
Yunqing Tang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Elliptic Curve ◽  
Convergence Condition ◽  
Differential Galois Group ◽  
Reduction Modulo ◽  
Local Monodromy ◽  
Monodromy Groups ◽  
Almost All ◽  
Algebraic Solutions

The Grothendieck–Katz [Formula: see text]-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo [Formula: see text] has vanishing [Formula: see text]-curvatures for almost all [Formula: see text] has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on [Formula: see text] We prove a variant of this conjecture for [Formula: see text] which asserts that if the equation satisfies a certain convergence condition for all [Formula: see text] then its monodromy is trivial. For those [Formula: see text] for which the [Formula: see text]-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of [Formula: see text]-curvatures and certain local monodromy groups. We also prove similar variants of the [Formula: see text]-curvature conjecture for an elliptic curve with [Formula: see text]-invariant [Formula: see text] minus its identity and for [Formula: see text].

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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