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St. Petersburg Mathematical Journal

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

Author: V. A. Sloushch
Translated by: A. V. Kiselev
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 849-857
MSC (2010): Primary 35P20
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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Additional Information

V. A. Sloushch
Affiliation: Physics Department, St. Petersburg State university, Ul′anovskaya 3, Petrodvorets, St. Petersburg 198504, Russia

Keywords: Elliptic differential operators, integral operators, estimates for singular values, classes of compact operators
Received by editor(s): March 20, 2014
Published electronically: July 27, 2015
Additional Notes: Supported by RFBR (grant no. 14-01-00760) and by St.Petersburg State University (grant no.
Article copyright: © Copyright 2015 American Mathematical Society