Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand–Shilov spaces ${\mathcal{S}}_{\nu }^{\mu }({\mathbb{R}}^n)$, $\mu ,\nu >0$, $\mu +\nu \ge 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $\mathrm{\Gamma }$) type operators, $(-\mathrm{\Delta }{)}^{m\mathrm{/}2}+\mathrm{\mid }x{\mathrm{\mid }}^k$ with $m,k\in 2\mathbb{N}$ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces ${\mathrm{\Sigma }}_{\nu }^{\mu }({\mathbb{R}}^n)$, $\mu ,\nu >0$, $\mu +\nu >1$, 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 $\kappa :=\mu \mathrm{/}\nu$; namely we obtain a characterization of ${\mathcal{S}}_{\nu }^{\mu }({\mathbb{R}}^n)$ and ${\mathrm{\Sigma }}_{\nu }^{\mu }({\mathbb{R}}^n)$ in the case $\mu =kt\mathrm{/}(k+m),\nu =mt\mathrm{/}(k+m),t\ge 1$, that is, when $\kappa =k\mathrm{/}m\in \text{Q}$.