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

Author: Yifeng Yu
Journal: Proc. Amer. Math. Soc. 135 (2007), 1449-1454
MSC (2000): Primary 35P20
DOI: https://doi.org/10.1090/S0002-9939-06-08702-8
Published electronically: November 13, 2006
MathSciNet review: 2276654
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 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.

[A] 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
[D-S] 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
[E1] Lawrence C. Evans, Towards a quantum analog of weak KAM theory, Comm. Math. Phys. 244 (2004), no. 2, 311–334. MR 2031033, https://doi.org/10.1007/s00220-003-0975-5
[E2] Lawrence C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159–177. MR 1986317, https://doi.org/10.1007/s00526-002-0164-y
[E3] 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
[E-Z] L.C. Evans, M. Zworski, Lectures on semiclassical analysis, Lecture note.
[H] Bernard Helffer, Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, vol. 1336, Springer-Verlag, Berlin, 1988. MR 960278
[J-K-M] 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, https://doi.org/10.1090/pspum/065/1662751
[Y] Y. Yu, $L^{\infty}$ variational problems, Aronsson equations and weak KAM theory, Ph.D. dissertation, U.C. Berkeley(2005).