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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.


[C-I] G. Contreras & R. Iturriaga, “Global minimizers of autonomous Lagrangians
[E-G1] L. C. Evans & D. Gomes, “Effective Hamiltonians and averaging for Hamiltonian dynamics I”, Archive Rational Mech and Analysis 157 (2001), p. 1-33  MR 1822413 |  Zbl 0986.37056
[E-G2] L. C. Evans & D. Gomes, “Effective Hamiltonians and averaging for Hamiltonian dynamics II”  MR 1891169 |  Zbl 1100.37039
[EW] Weinan E, “Aubry–Mather theory and periodic solutions of the forced Burgers equation”, Comm Pure and Appl Math 52 (1999), p. 811-828  MR 1682812 |  Zbl 0916.35099
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[F2] A. Fathi, “Weak KAM theory in Lagrangian Dynamics, Preliminary Version”, 2001
[G1] D. Gomes, Hamilton–Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, Ph. D. Thesis, University of California, Berkeley, 2000
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[G3] D. Gomes, “Regularity theory for Hamilton-Jacobi equations”  Zbl 1023.35028
[L-P-V] P.-L. Lions, G. Papanicolaou & S. R. S. Varadhan, “Homogenization of Hamilton–Jacobi equations
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[M2] J. Mather, “Action minimizing invariant measures for positive definite Lagrangian systems”, Math. Zeitschrift 207 (1991), p. 169-207 Article |  MR 1109661 |  Zbl 0696.58027
[M-F] J. Mather & G. Forni, Action minimizing orbits in Hamiltonian systems, in S. Graffi, ed., Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math., Sringer, 1994  MR 1323222 |  Zbl 0822.70011
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