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/

[C-I] Global minimizers of autonomous Lagrangians

[E-G1] Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis, Tome 157 (2001), pp. 1-33 | MR 1822413 | Zbl 0986.37056

[E-G2] Effective Hamiltonians and averaging for Hamiltonian dynamics II | MR 1891169 | Zbl 1100.37039

[EW] Aubry–Mather theory and periodic solutions of the forced Burgers equation, Comm Pure and Appl Math, Tome 52 (1999), pp. 811-828 | MR 1682812 | Zbl 0916.35099

[F1] Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., Tome 324 (1997), pp. 1043-1046 | MR 1451248 | Zbl 0885.58022

[F2] Weak KAM theory in Lagrangian Dynamics, Preliminary Version (2001)

[G1] Hamilton–Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, University of California, Berkeley (2000) (Ph. D. Thesis)

[G2] Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems | MR 1899451 | Zbl 1005.49027

[G3] Regularity theory for Hamilton-Jacobi equations | Zbl 1023.35028

[L-P-V] Homogenization of Hamilton–Jacobi equations

[M-F] Action minimizing orbits in Hamiltonian systems, Transition to Chaos in Classical and Quantum Mechanics, Sringer (Lecture Notes in Math.) (1994) no. 1589 | MR 998855 | Zbl 0689.58025

[M1] Minimal measures, Comment. Math Helvetici, Tome 64 (1989), pp. 375-394 | MR 1109661 | Zbl 0696.58027

[M2] Action minimizing invariant measures for positive definite Lagrangian systems, Math. Zeitschrift, Tome 207 (1991), pp. 169-207 | MR 1323222 | Zbl 0822.70011