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Max Lein (Tohoku University, Sendai, Japan)

An Analytic-Algebraic Approach to Linear Response Theory

Linear response theory allows us to study systems that are driven out of equilibrium by external perturbations. The idea here is to Taylor expand the expectation value of a current observable with respect to the state of the system in terms of external parameters that quantify the perturbation; the linear response is captured by the “conductivity coefficients”, which are the proportionality factors in the linear term of the expansion. The main task of linear response theory is to rigorously justify this Taylor expansion and find explicit expressions for the conductivity coefficients, e. g. via the Kubo and the Kubo-Strěda formulas.

Most previous works on linear response theory rigorous either concerned one particular system or the framework would apply only to a very restrictive class of systems. A common restriction is to consider only systems on the discrete. In a recent book Giuseppe De Nittis and I have developed a novel analytic-algebraic approach to make linear response theory rigorous. This unified and thoroughly modern framework applies to discrete and continuous operators alike, can deal with disorder and is not tailored to a specific model. It relies on the theory of von Neumann (rather than C*-) algebras and we work with non-commutative $L^p$- and Sobolev spaces.

This is joint work with Giuseppe De Nittis.