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Séminaire Équations aux dérivées partielles (Polytechnique)

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Lawrence C. Evans
Effective Hamiltonians and Quantum States
Séminaire Équations aux dérivées partielles (Polytechnique) (2000-2001), Exp. No. 23, 13 p.
Article PDF | Reviews MR 1860693 | Zbl 1055.81524

Résumé - Abstract

We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function $u$ solving the eikonal equation aėȧnd a probability measure $\sigma $ solving a related transport equation.

We present some elementary formal identities relating certain quantum states $\psi $ and $u, \sigma $. We show also how to build out of $u, \sigma $ an approximate solution of the stationary Schrödinger eigenvalue problem, although the error estimates for this construction are not very good.

Bibliography

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