Anisotropic Shubin operators and eigenfunction expansions in Gelfand-Shilov spaces

Abstract

We derive new results on the characterization of Gelfand-Shilov spaces $\mathcal{S}_{\nu}^{\mu}(\mathbb{R}^n)$, μ, ν > 0, μ + ν ≥ 1 byGevrey estimates of the L² norms of iterates of (m, k) anisotropic globally elliptic Shubin (or Γ) type operators, (- Δ)^m/2 + |x>^k with m, k ∈ 2ℕ being a model operator, and on the decay of the Fourier coefficients in the related eigenfunction expansions. Similar results are obtained for the spaces $\Sigma_{\nu}^{\mu}(\mathbb{R}^n)$, μ, ν > 0, μ + ν > 1, cf. (1.2). 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 $\mathcal{S}_{\nu}^{\mu}(\mathbb{R}^n)$ and $\Sigma_{\nu}^{\mu}(\mathbb{R}^n)$ in the case μ = kt/(k + m), ν = mt/(k + m), t ≥ 1, that is, when κ = k/m ∈ ℚ.