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 σ solving a related transport equation.

We present some elementary formal identities relating certain quantum states ψ and u,σ. We show also how to build out of u,σ 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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