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
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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] N. 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 (2005i:82004)
[D-S] M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, 1999. MR 1735654 (2001b:35237)
[E1] L.C. Evans, Toward a quantum analogue of weak KAM theory, Comm. Math. Phys. 244 (2004), no. 2, 311-334. MR 2031033 (2005c:81063)
[E2] L.C. Evans, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations 17 (2003), no. 2, 159-177. MR 1986317 (2004e:37097)
[E3] L.C. Evans, Effective Hamiltonians and quantum states, Seminaire Equations aux Dérivées Partielles, Ecole Polytechnique (2000-2001). MR 1860693 (2002j:81049)
[E-Z] L.C. Evans, M. Zworski, Lectures on semiclassical analysis, Lecture note.
[H] B. Helffer, Semi-classical analysis for the Schr $\ddot o$ dinger operator and applications, Springer Lect. Notes in Math., 1336 (1988). MR 0960278 (90c:81043)
[J-K-M] H.R. Jauslin, H.O. Kreiss, J. Moser, On the forced Burgers equation with periodic boundary conditions, Proc. Symposia in Pure Math 65, 133-153, 1999. MR 1662751 (99m:35208)
[Y] Y. Yu, $L^{\infty}$ variational problems, Aronsson equations and weak KAM theory, Ph.D. dissertation, U.C. Berkeley(2005).