Semi-classical trace formulas and heat expansions

in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schr\"odinger operator with magnetic field. We show how to recover their formula by using the methods developped by the geometers in the seventies for the heat expansions.Comment: To appear in "Analysis of Partial Differential Equations

Transversally Elliptic Operators

We construct certain spectral triples in the sense of A. ~Connes and H. \~Moscovici (``The local index formula in noncommutative geometry'' {\it Geom. Funct. Anal.}, 5(2):174--243, 1995) that is transversally elliptic but not necessarily elliptic. We prove that these spectral triples satisfie the conditions which ensure the Connes-Moscovici local index formula applies. We show that such a spectral triple has discrete dimensional spectrum. A notable feature of the spectral triple is that its corresponding zeta functions have multiple poles, while in the classical elliptic cases only simple poles appear for the zeta functions. We show that the multiplicities of the poles of the zeta functions have an upper bound, which is the sum of dimensions of the base manifold and the acting compact Lie group. Moreover for our spectral triple the Connes-Moscovici local index formula involves only local transverse symbol of the operator.Comment: Updated 11/25/2003 with corrected format, and in 12pt fonts Updated 5/20/2004, major reorganizatio

Spectral asymptotics via the semiclassical Birkhoff normal form

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters (E,\h), and gives improvements and new proofs for many of the results in the field. In the completely resonant case we show that the pseudo-differential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained.Comment: 44 pages, 2 figure

Projective Pseudodifferential Analysis and Harmonic Analysis

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space $G_n/H_n=SL(n+1,\R)/GL(n,\R)$, and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space $P_n(\R)$ : these spaces are the representation spaces of the maximal degenerate series $(\pi_{i\lambda,\epsilon})$ of $G_n$ . This new approach to the quantization of $G_n/H_n$, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of $L^2(G_n/H_n)$ under the quasiregular action of $G_n$ . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation $\pi_{i\lambda,\epsilon}$

Asymptotic boundary forms for tight Gabor frames and lattice localization domains

We consider Gabor localization operators $G_{\phi,\Omega}$ defined by two parameters, the generating function $\phi$ of a tight Gabor frame $\{\phi_\lambda\}_{\lambda \in \Lambda}$, parametrized by the elements of a given lattice $\Lambda \subset \Bbb{R}^2$, i.e. a discrete cocompact subgroup of $\Bbb{R}^2$, and a lattice localization domain $\Omega \subset \Bbb{R}^2$ with its boundary consisting of line segments connecting points of $\Lambda$. We find an explicit formula for the boundary form $BF(\phi,\Omega)=\text{A}_\Lambda \lim_{R\rightarrow \infty}\frac{PF(G_{\phi,R\Omega})}{R}$, the normalized limit of the projection functional $PF(G_{\phi,\Omega})=\sum_{i=0}^{\infty}\lambda_i(G_{\phi,\Omega})(1-\lambda_i(G_{\phi,\Omega}))$, where $\lambda_i(G_{\phi,\Omega})$ are the eigenvalues of the localization operators $G_{\phi,\Omega}$ applied to dilated domains $R\Omega$, $R$ is an integer and $\text{A}_\Lambda$ is the area of the fundamental domain of the lattice $\Lambda$.Comment: 35 page

Semiclassical analysis for the Kramers-Fokker-Planck equation

We study some accurate semiclassical resolvent estimates for operators that are neither selfadjoint nor elliptic, and applications to the Cauchy problem. In particular we get a precise description of the spectrum near the imaginary axis and precise resolvent estimates inside the pseudo-spectrum. We apply our results to the Kramers-Fokker-Planck operator