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How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space. We recall $$(-\Delta)^\alpha u: \sum_{k=1}^\infty \sqrt 2 c_n\cos(\pi n x) \mapsto \sum_{k=1}^\infty \sqrt 2 (\pi n)^{2s} c_n\cos(\pi n x),$$ where $c_n = \int_{0}^{1} \cos(\pi n x) u(x) dx$.

Note that in Computing the fractional Laplacian of power function the formula is given in $\mathbb R^n$ instead of in the interval $(0,1)$.

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  • $\begingroup$ the fact that the basis function $\cos(\pi nx)$ do not vanish at the end points of your interval creates difficulties in the definition of the fractional Laplacian; see page 18 of arXiv:1810.07028 --- Dirichlet boundary conditions are preferred to avoid these difficulties. $\endgroup$
    – Carlo Beenakker
    Commented Mar 16, 2021 at 12:02
  • $\begingroup$ @CarloBeenakker Thanks! But (unfortunately) I need to consider the Neumann fractional Laplacian $\endgroup$
    – user173196
    Commented Mar 16, 2021 at 13:39
  • $\begingroup$ the point is that it may not exist (at least not for $\alpha>1$) $\endgroup$
    – Carlo Beenakker
    Commented Mar 16, 2021 at 13:44

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