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Effective Evolution Equations in Quantum Physics

Benjamin Schlein — 2011

Journées Équations aux dérivées partielles

In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand,...

The local relaxation flow approach to universality of the local statistics for random matrices

László Erdős;  Benjamin Schlein;  Horng-Tzer Yau;  Jun Yin — 2012

Annales de l'I.H.P. Probabilités et statistiques

We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues { }=1 are close to their classical location { }=1 determined by the limiting density...

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