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The so-called density of states is a Borel probability measure on the realline associated with the solution of the Dyson equation which we set up, on anyfixed $C^\ast$-probability space, for a selfadjoint offset and a $2$-positivelinear map. Using techniques from free noncommutative function theory, we proveexplicit H\"older bounds for the L\'evy distance of two such measures when anyof the two parameters varies. As the main tools for the proof, which are alsoof independent interest, we show that solutions of the Dyson equation havestrong analytic properties and evolve along any $C^1$-path of $2$-positivelinear maps according to an operator-valued version of the inviscid Burgersequation.

The Dyson equation for $2$-positive maps and H\"older bounds for the L\'evy distance of densities of states