Bounds on eigenfunctions of semiclassical operators with double characteristics

We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along the double characteristics are elliptic

Magnetic WKB Constructions

This paper is devoted to the semiclassical magnetic Laplacian. Until now WKB expansions for the eigenfunctions were only established in presence of a non-zero electric potential. Here we tackle the pure magnetic case. Thanks to Feynman-Hellmann type formulas and coherent states decomposition, we develop here a magnetic Born-Oppenheimer theory. Exploiting the multiple scales of the problem, we are led to solve an effective eikonal equation in pure magnetic cases and to obtain WKB expansions. We also investigate explicit examples for which we can improve our general theorem: global WKB expansions, quasi-optimal estimates of Agmon and upper bound of the tunelling effect (in symmetric cases). We also apply our strategy to get more accurate descriptions of the eigenvalues and eigenfunctions in a wide range of situations analyzed in the last two decades

Spectral projections and resolvent bounds for partially elliptic quadratic differential operators

We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.Comment: 60 pages, 3 figures. J. Pseudo-Differ. Oper. Appl., to appear. Revised according to referee report, including minor changes to Corollary 1.8. The final publication will be available at link.springer.co

Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators

Non-elliptic quadratic forms and semiclassical estimates for non-selfadjoint operators

We consider a class of pseudodifferential operators with a doubly characteristic point, where the quadratic part of the symbol fails to be elliptic but obeys an averaging assumption. Under suitable additional assumptions, semiclassical resolvent estimates are established, where the modulus of the spectral parameter is allowed to grow slightly more rapidly than the semiclassical parameter.Comment: 41 pages, 1 figur

Magnetic edge states

Magnetic edge states are responsible for various phenomena of magneto-transport. Their importance is due to the fact that, unlike the bulk of the eigenstates in a magnetic system, they carry electric current along the boundary of a confined domain. Edge states can exist both as interior (quantum dot) and exterior (anti-dot) states. In the present report we develop a consistent and practical spectral theory for the edge states encountered in magnetic billiards. It provides an objective definition for the notion of edge states, is applicable for interior and exterior problems, facilitates efficient quantization schemes, and forms a convenient starting point for both the semiclassical description and the statistical analysis. After elaborating these topics we use the semiclassical spectral theory to uncover nontrivial spectral correlations between the interior and the exterior edge states. We show that they are the quantum manifestation of a classical duality between the trajectories in an interior and an exterior magnetic billiard.Comment: 170 pages, 48 figures (high quality version available at http://www.klaus-hornberger.de