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Anisotropic Shubin operators and eigenfunctions expansions in Gelfand-Shilov spaces

We derive new results on the characterization of Gelfand--Shilov spaces $\mathcal{S}^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν\geq 1$ by Gevrey estimates of the $L^2$ norms of iterates of $(m,k)$ anisotropic globally elliptic Shubin (or $Γ$) type operators, $(-Δ)^{m/2} +| x |^k$ 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 $Σ^μ_ν(\R^n)$, $μ,ν>0$, $μ+ν> 1$, cf. \eqref{GSdef}. In contrast to the symmetric case $μ= ν$ and $k=m$ (classical Shubin operators) we encounter resonance type phenomena involving the ratio $κ:=μ/ν$; namely we obtain a characterization of $\mathcal{S}^μ_ν(\R^n)$ and $Σ^μ_ν(\R^n)$ in the case $μ=kt/(k+m), ν= mt/(k+m), t \geq 1$, that is, when $κ=k/m \in \Q$.