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Refined Weyl law for homogeneous perturbations of the harmonic oscillator
- Publication Year :
- 2017
-
Abstract
- Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schr\"odinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 \pi \mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 \pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(\lambda^{d-1})$, improving in these cases the $O(\lambda^{d-1})$ remainder previously established by Helffer--Robert.<br />Comment: 28 pages; new section added on propagation of singularities
- Subjects :
- Mathematics - Analysis of PDEs
Mathematical Physics
Mathematics - Spectral Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1708.06825
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00220-018-3100-5