Effective Hamiltonians and Quantum States
Séminaire Équations aux dérivées partielles (Polytechnique) (2000-2001), Talk no. 23, 13 p.

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.

@article{SEDP_2000-2001____A23_0,
author = {Evans, Lawrence C.},
title = {Effective Hamiltonians and Quantum States},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2000-2001},
note = {talk:23},
mrnumber = {1860693},
zbl = {1055.81524},
language = {en},
url = {http://www.numdam.org/item/SEDP_2000-2001____A23_0}
}

Evans, Lawrence C. Effective Hamiltonians and Quantum States. Séminaire Équations aux dérivées partielles (Polytechnique) (2000-2001), Talk no. 23, 13 p. http://www.numdam.org/item/SEDP_2000-2001____A23_0/

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