Semiclassical L p Estimates

Semiclassical L Estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize L estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. To mo...

Simplified Semiclassical Propagation Estimates

Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time–dependent Schrödinger equation i h̄ ∂ψ ∂t = − h̄ 2 2 ∆ψ + V ψ for small values of h̄. If V satisfies appropriate analyticity and growth hypotheses and |t| ≤ T , these solutions agree with exact solutions up to errors whose norms are bounded by C exp {− γ/h̄ } , for some C and γ > 0. Under more restrictive hypotheses, we prove that for sufficiently smal...

Universal bounds and semiclassical estimates for eigenvalues

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues {λk} that are analogous to those known for Schrödinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These...

Semiclassical Estimates in Asymptotically Euclidean Scattering

The purpose of this note is to obtain semiclassical resolvent estimates for long range perturbations of the Laplacian on asymptotically Euclidean manifolds. For an estimate which is uniform in the Planck constant h we need to assume that the energy level is non-trapping. In the high energy limit (that is, when we consider ∆ − λ, as λ → ∞, which is equivalent to h∆ − 1, h → 0), this corresponds ...