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Quantum Simulation

Symmetries of the pseudo-diffusion equation, and its unconventional 2-sided kernel

arXiv

Authors: Jamil Daboul, Faruk Gungor, Dongsheng Liu, David McAnally

Year

2015

Paper ID

26443

Status

Preprint

Abstract Read

~2 min

Abstract Words

221

Citations

N/A

Abstract

We determine by two related methods the invariance algebra g of the `pseudo-diffusion equation' (PSDE) $L Q equiv left\[frac {partial}{partial t} -frac 1 4 left$frac {partial²}{partial x²} -frac 1 {t²} frac {partial²}{partial p²}right$right\] Q(x,p,t)=0,which describes the behavior of theQfunctions in the(x,p)-phase space as a function of a squeeze parametery, wheret=e^{2y}. The algebra turns out to be isomorphic to that of its constant coefficient version. Relying on this isomorphism we construct a local point transformation which maps the factort^{-2}to 1. We show that any generalized versionu_t-u_{xx}+ b(t) u_{yy}=0of PSDE has a smaller symmetry algebra than\g, except forb(t)equals to a constant or it is proportional tot^{-2}. We apply the group elementsG_iga := \exp\[\ga A_i\]and obtain new solutions of the PSDE from simple ones, and interpret the physics of some of the results. We make use of the `factorization property' of the PSDE to construct its textit{`2-sided kernel'}, because it has to depend on two times,t_0 < t < t_1. We include a detailed discussion of the identification of the Lie algebraic structure of the symmetry algebra\g, and its contraction from\su(1,1)\oplus\so(3,1)$.

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We determine by two related methods the invariance algebra g of the `pseudo-diffusion equation' (PSDE) L Q equiv left[frac partialpartial t -frac 1 4 left(frac partial^2partial...

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References & Citation Signals

[1] DOI https://doi.org/arXiv:1510.08643 [2] arXiv https://arxiv.org/abs/1510.08643 [3] Publisher https://arxiv.org/abs/1510.08643

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