Approximation of the fractional Schrödinger propagator on compact manifolds. (English) Zbl 1483.35317
Summary: Let \(\mathcal{L}\) be a second order positive, elliptic differential operator that is self-adjoint with respect to some \(C^\infty\) density \(dx\) on a compact connected manifold \(\mathbb{M}\). We proved that if \(0<\alpha<1\), \(\alpha/2<s<\alpha\) and \(f\in H^s(\mathbb{M})\) then the fractional Schrödinger propagator \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}\) on \(\mathbb{M}\) satisfies \(\mathrm{e}^{\mathrm{i}t\mathcal{L}^{\alpha/2}}f(x)-f(x)=o(t^{s/\alpha-\varepsilon})\) almost everywhere as \(t\rightarrow 0^+\), for any \(\varepsilon>0\).
MSC:
| 35R11 | Fractional partial differential equations |
| 35R01 | PDEs on manifolds |
| 42B15 | Multipliers for harmonic analysis in several variables |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 58J05 | Elliptic equations on manifolds, general theory |
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