Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

TL;DR

This paper characterizes Gelfand-Shilov spaces using Gevrey estimates of elliptic anisotropic Shubin operators and explores eigenfunction expansions, revealing resonance phenomena related to the ratio of parameters.

Contribution

It provides new characterizations of Gelfand-Shilov spaces via Gevrey estimates and eigenfunction decay for anisotropic elliptic operators, including resonance effects for rational parameter ratios.

Findings

Characterization of Gelfand-Shilov spaces through Gevrey estimates.

Analysis of eigenfunction expansion decay in these spaces.

Identification of resonance phenomena when parameter ratios are rational.

Abstract

We derive new results on the characterization of Gelfand--Shilov spaces $\mathcal{S}^\mu_\nu (\R^n)$, $\mu,\nu >0$, $\mu+\nu \geq 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $\Gamma$) type operators, $(-\Delta)^{m/2} +| x |^k$ with $m,k\in 2\N$ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces $\Sigma^\mu_\nu (\R^n)$, $\mu,\nu >0$, $\mu+\nu > 1$, cf. \eqref{GSdef}. In contrast to the symmetric case $\mu = \nu$ and $k=m$ (classical Shubin operators) we encounter resonance type phenomena involving the ratio $\kappa:=\mu/\nu$; namely we obtain a characterization of $\mathcal{S}^\mu_\nu(\R^n)$ and $\Sigma^\mu_\nu(\R^n)$ in the case $\mu=kt/(k+m), \nu= mt/(k+m), t \geq 1$, that is, when $\kappa=k/m \in \Q$.