The compound equation related to the Bessel-Helmholtz equation and the Bessel-Klein-Gordon equation

2013 ◽  
Vol 7 ◽  
pp. 4521-4530
Author(s):  
Chalermpon Bunpog
Keyword(s):  
Helmholtz Equation ◽  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

scholarly journals Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1302
Author(s):  
Francisco F. López-Ruiz ◽  
Julio Guerrero ◽  
Victor Aldaya
Keyword(s):  
Helmholtz Equation ◽  
Scalar Product ◽  
Gordon Equation ◽  
Invariance Group ◽  
Unified Framework ◽  
Property A ◽  
Oscillatory Solutions ◽  
Klein Gordon ◽  
Scalar Products ◽  
Klein Gordon Equation

Although describing very different physical systems, both the Klein–Gordon equation for tachyons (m2<0) and the Helmholtz equation share a remarkable property: a unitary and irreducible representation of the corresponding invariance group on a suitable subspace of solutions is only achieved if a non-local scalar product is defined. Then, a subset of oscillatory solutions of the Helmholtz equation supports a unirrep of the Euclidean group, and a subset of oscillatory solutions of the Klein–Gordon equation with m2<0 supports the scalar tachyonic representation of the Poincaré group. As a consequence, these systems also share similar structures, such as certain singularized solutions and projectors on the representation spaces, but they must be treated carefully in each case. We analyze differences and analogies, compare both equations with the conventional m2>0 Klein–Gordon equation, and provide a unified framework for the scalar products of the three equations.

scholarly journals Fractional Klein-Gordon equation with singular mass

2021 ◽  
Vol 143 ◽  
pp. 110579
Author(s):  
Arshyn Altybay ◽  
Michael Ruzhansky ◽  
Mohammed Elamine Sebih ◽  
Niyaz Tokmagambetov
Keyword(s):  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

scholarly journals Bound states of a Klein–Gordon particle in the presence of a smooth potential well

2020 ◽  
Vol 35 (23) ◽  
pp. 2050140
Author(s):  
Eduardo López ◽  
Clara Rojas
Keyword(s):  
Bound States ◽  
Gordon Equation ◽  
Bound State ◽  
Potential Well ◽  
One Dimensional ◽  
Smooth Potential ◽  
Klein Gordon ◽  
The One ◽  
Potential Parameters ◽  
Klein Gordon Equation

We solve the one-dimensional time-independent Klein–Gordon equation in the presence of a smooth potential well. The bound state solutions are given in terms of the Whittaker [Formula: see text] function, and the antiparticle bound state is discussed in terms of potential parameters.

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