Effective Hamiltonians and Quantum States

Lawrence C. Evans[1]

  • [1] Department of Mathematics, University of California, Berkeley

Séminaire Équations aux dérivées partielles (2000-2001)

  • Volume: 2000-2001, page 1-13

Abstract

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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.

How to cite

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Evans, Lawrence C.. "Effective Hamiltonians and Quantum States." Séminaire Équations aux dérivées partielles 2000-2001 (2000-2001): 1-13. <http://eudml.org/doc/11019>.

@article{Evans2000-2001,
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.},
affiliation = {Department of Mathematics, University of California, Berkeley},
author = {Evans, Lawrence C.},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
pages = {1-13},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Effective Hamiltonians and Quantum States},
url = {http://eudml.org/doc/11019},
volume = {2000-2001},
year = {2000-2001},
}

TY - JOUR
AU - Evans, Lawrence C.
TI - Effective Hamiltonians and Quantum States
JO - Séminaire Équations aux dérivées partielles
PY - 2000-2001
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2000-2001
SP - 1
EP - 13
AB - 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.
LA - eng
UR - http://eudml.org/doc/11019
ER -

References

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  1. G. Contreras, R. Iturriaga, Global minimizers of autonomous Lagrangians Zbl0957.37065
  2. L. C. Evans, D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I, Archive Rational Mech and Analysis 157 (2001), 1-33 Zbl0986.37056MR1822413
  3. L. C. Evans, D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics II Zbl1100.37039MR1891169
  4. Weinan E, Aubry–Mather theory and periodic solutions of the forced Burgers equation, Comm Pure and Appl Math 52 (1999), 811-828 Zbl0916.35099MR1682812
  5. A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 1043-1046 Zbl0885.58022MR1451248
  6. A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version, (2001) 
  7. D. Gomes, Hamilton–Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, (2000) 
  8. D. Gomes, Viscosity solutions of Hamilton-Jacobi equations and asymptotics for Hamiltonian systems Zbl1005.49027MR1899451
  9. D. Gomes, Regularity theory for Hamilton-Jacobi equations Zbl1023.35028
  10. P.-L. Lions, G. Papanicolaou, S. R. S. Varadhan, Homogenization of Hamilton–Jacobi equations 
  11. J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989), 375-394 Zbl0689.58025MR998855
  12. J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Zeitschrift 207 (1991), 169-207 Zbl0696.58027MR1109661
  13. J. Mather, G. Forni, Action minimizing orbits in Hamiltonian systems, Transition to Chaos in Classical and Quantum Mechanics (1994), GraffiS.S. Zbl0822.70011MR1323222

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