Asymptotics of singular values for quantum derivatives

Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy

doi:10.1090/tran/8827

Asymptotics of singular values for quantum derivatives
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by Rupert L. Frank, Fedor Sukochev and Dmitriy Zanin HTML | PDF

Trans. Amer. Math. Soc. 376 (2023), 2047-2088

Abstract:

We obtain Weyl type asymptotics for the quantised derivative $\dj \mkern 1muf$ of a function $f$ from the homgeneous Sobolev space $\dot {W}^1_d(\mathbb {R}^d)$ on $\mathbb {R}^d.$ The asymptotic coefficient $\|\nabla f\|_{L_d(\mathbb R^d)}$ is equivalent to the norm of $\dj \mkern 1muf$ in the principal ideal $\mathcal {L}_{d,\infty },$ thus, providing a non-asymptotic, uniform bound on the spectrum of $\dj \mkern 1muf.$ Our methods are based on the $C^{\ast }$-algebraic notion of the principal symbol mapping on $\mathbb {R}^d$, as developed recently by the last two authors and collaborators.

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