287 research outputs found
The Dyson equation for -positive maps and H\"older bounds for the L\'evy distance of densities of states
The so-called density of states is a Borel probability measure on the real
line associated with the solution of the Dyson equation which we set up, on any
fixed -probability space, for a selfadjoint offset and a -positive
linear map. Using techniques from free noncommutative function theory, we prove
explicit H\"older bounds for the L\'evy distance of two such measures when any
of the two parameters varies. As the main tools for the proof, which are also
of independent interest, we show that solutions of the Dyson equation have
strong analytic properties and evolve along any -path of -positive
linear maps according to an operator-valued version of the inviscid Burgers
equation.Comment: 27 page
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