A remark on the semi-classical measure from -ℎ²\over2Δ+𝑉 with a degenerate potential 𝑉

Yu, Yifeng

doi:10.1090/S0002-9939-06-08702-8

A remark on the semi-classical measure from ${-{h^2\over 2}\Delta +V}$ with a degenerate potential $V$
HTML articles powered by AMS MathViewer

by Yifeng Yu PDF

Proc. Amer. Math. Soc. 135 (2007), 1449-1454 Request permission

Abstract:

This note is motivated by Evans (2004) and Anantharaman (2004). We study the semiclassical measure arising from the operator $P(h)=-{h^2\over 2}\Delta +V(x)$ when the potential $V$ has degenerate minimum points. We will use the technique of integration by parts and some identities of Evans to derive information on the support of the measure.

References

Nalini Anantharaman, On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 207–276. MR 2055035, DOI 10.4171/jems/9
Mouez Dimassi and Johannes Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR 1735654, DOI 10.1017/CBO9780511662195
Lawrence C. Evans, Towards a quantum analog of weak KAM theory, Comm. Math. Phys. 244 (2004), no. 2, 311–334. MR 2031033, DOI 10.1007/s00220-003-0975-5
Lawrence C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159–177. MR 1986317, DOI 10.1007/s00526-002-0164-y
Lawrence C. Evans, Effective Hamiltonians and quantum states, Séminaire: Équations aux Dérivées Partielles, 2000–2001, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2001, pp. Exp. No. XXII, 13. MR 1860693
L.C. Evans, M. Zworski, Lectures on semiclassical analysis, Lecture note.
Bernard Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, Berlin, 1988. MR 960278, DOI 10.1007/BFb0078115
H. R. Jauslin, H. O. Kreiss, and J. Moser, On the forced Burgers equation with periodic boundary conditions, Differential equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 133–153. MR 1662751, DOI 10.1090/pspum/065/1662751
Y. Yu, $L^{\infty }$ variational problems, Aronsson equations and weak KAM theory, Ph.D. dissertation, U.C. Berkeley(2005).