The Fourier transform interchanges the Schrödinger representation with a dual representation, yielding dual equivalence bimodules and a conditional correspondence between self-dual and anti-self-dual solitons.
Solitons of general topological charge over noncommutative tori
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abstract
We continue the study of solitons over noncommutative tori from the perspective of time-frequency analysis and treat the case of a general topological charge. Solutions are associated with vector bundles of higher rank over noncommutative tori. We express these vector bundles in terms of vector-valued Gabor frames and apply the duality theory of Gabor analysis to show that Gaussians are solitons of general topological charge over noncommutative tori. An energy functional for projections over noncommutative tori is the basis for the self and anti-self duality equations of the solitons which turns out to have a reformulation in terms of Gabor atoms and we prove that projections generated by Gaussians minimize this energy functional. Finally, we comment on the case of the Moyal plane and the associated continuous vector-valued Gabor frames and show that Gaussians are the only class of solitons.
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math.OA 1years
2019 1verdicts
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Fourier transform, Schr\"odinger representation, and Heisenberg modules
The Fourier transform interchanges the Schrödinger representation with a dual representation, yielding dual equivalence bimodules and a conditional correspondence between self-dual and anti-self-dual solitons.