Wave packet analysis of semigroups generated by quadratic differential operators

2024/08/20 by S. Ivan Trapasso, Trapasso, S. Ivan · 1 citation
Physics and Astronomy · Mathematics · Engineering · #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2408.11130

Abstract

We perform a phase space analysis of evolution equations associated with the Weyl quantization q^w of a complex quadratic form q on ℝ^2d with non-positive real part. In particular, we obtain pointwise bounds for the matrix coefficients of the Gabor wave packet decomposition of the generated semigroup e^tq^w if Re (q) ≤ 0 and the companion singular space associated is trivial. This result is then leveraged to achieve a comprehensive analysis of the phase regularity of e^tq^w with Re (q) ≤ 0, thereby extending the L² analysis of quadratic semigroups initiated by Hitrik and Pravda-Starov to general modulation spaces M^p(ℝ^d), 1 ≤ p ≤ ∞, with optimal explicit bounds.

Cited by

• Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement

Related

• L^p-bounds for semigroups generated by non-elliptic quadratic differential operators
• Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects
• Spectra and semigroup smoothing for non-elliptic quadratic operators
• Gains of integrability and local smoothing effects for quadratic\n evolution equations
• Phase space analysis of the Hermite semigroup and applications to\n nonlinear global well-posedness