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Journal Article

A calculus for magnetic pseudodifferential super operators

MPS-Authors
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Lee,  Gihyun
Max Planck Institute for Mathematics, Max Planck Society;

External Resource

https://doi.org/10.1063/5.0090191
(Publisher version)

https://doi.org/10.48550/arXiv.2201.11487
(Preprint)

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Citation

Lee, G., & Lein, M. (2022). A calculus for magnetic pseudodifferential super operators. Journal of mathematical physics, 63(10): 103506. doi:10.1063/5.0090191.


Cite as: https://hdl.handle.net/21.11116/0000-000B-699D-F
Abstract
This work develops a magnetic pseudodifferential calculus for super operators
OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq
functions). Here, F could be a tempered distribution or a H\"ormander symbol.
An important example is Liouville super operators defined in terms of a
magnetic pseudodifferential operator. Our work combines ideas from magnetic
Weyl calculus developed in [MP04, IMP07, Lei11] and the pseudodifferential
calculus on the non-commutative torus from [HLP18a, HLP18b]. Thus, our calculus
is inherently gauge-covariant, which means all essential properties of OpA(F)
are determined by properties of the magnetic field B = dA rather than the
vector potential A.
There are conceptual differences to ordinary pseudodifferential theory. For
example, in addition to an analog of the (magnetic) Weyl product that emulates
the composition of two magnetic pseudodifferential super operators on the level
of functions, the so-called semi-super product describes the action of a
pseudodifferential super operator on a pseudodifferential operator.