Symmetries of the pseudo-diffusion equation and related topics |
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Authors: | J. Daboul |
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Affiliation: | 1.Physics Department,Ben Gurion University of the Negev,Beer Sheva,Israel |
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Abstract: | We show in details how to determine and identify the algebra g = {Ai} of the infinitesimal symmetry operators of the following pseudo-diffusion equation (PSDE) LQ ≡ (left[ {frac{partial }{{partial t}} - frac{1}{4}left( {frac{{{partial ^2}}}{{partial {x^2}}} - frac{1}{{{t^2}}}frac{{{partial ^2}}}{{partial {p^2}}}} right)} right]) Q(x, p, t) = 0. This equation describes the behavior of the Q functions in the (x, p) phase space as a function of a squeeze parameter y, where t = e 2y. We illustrate how G i(λ) ≡ exp[λA i] can be used to obtain interesting solutions. We show that one of the symmetry generators, A 4, acts in the (x, p) plane like the Lorentz boost in (x, t) plane. We construct the Anti-de-Sitter algebra so(3, 2) from quadratic products of 4 of the A i, which makes it the invariance algebra of the PSDE. We also discuss the unusual contraction of so(3, 1) to so(1, 1)? h2. We show that the spherical Bessel functions I 0(z) and K 0(z) yield solutions of the PSDE, where z is scaling and “twist” invariant. |
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