On the microlocal phase-space concentration of Wigner distributions associated with Schrödinger evolutions

In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

💡 Research Summary

The paper investigates the microlocal behavior of Schrödinger evolutions through the lens of metaplectic Wigner distributions, extending previous results that were limited to special subclasses of such distributions. The authors first define an “A‑Wigner distribution” (W_{\mathcal A}) associated with any metaplectic operator (\widehat{\mathcal A}\in Mp(2n,\mathbb R)) and prove the existence of an associated A‑Wigner kernel (k_{\mathcal A}) for any continuous linear operator (T:\mathcal S(\mathbb R^n)\to\mathcal S’(\mathbb R^n)). This kernel satisfies the intertwining relation
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