Approximate commutativity for a decaying potential and a function of an elliptic operator

Sloushch, V.

doi:10.1090/spmj/1362

Approximate commutativity for a decaying potential and a function of an elliptic operator
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by V. A. Sloushch
Translated by: A. V. Kiselev

St. Petersburg Math. J. 26 (2015), 849-857

DOI: https://doi.org/10.1090/spmj/1362

Published electronically: July 27, 2015

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Abstract:

For a continuous function $\varphi (\lambda )$, $\lambda \in \mathbb {R}$, with compact support, a bounded function $W(x)$, $x\in \mathbb {R}^{d}$, with power-like asymptotics at infinity, and a suitable selfadjoint operator $H$ in $L_{2}({\mathbb R}^{d})$, estimates for the singular values of the operator $\varphi (H)W-W\varphi (H)$ are considered. It is proved that the singular values of $\varphi (H)W-W\varphi (H)$ decay faster than those of $\varphi (H)W$. A relationship between the singular values asymptotics for the operators $\varphi (H)W$ and $\varphi ^{n}(H)W^{n}$ is also established.

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