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Let $\mathbb{T}^d$ be the $d$-dimensional torus. Consider the operator

$$ \Delta^{(\alpha/2)}u(x):= \int_{\mathbb{T}^d} \frac{u(x+y)+u(x-y)-2u(x)}{(d_{\mathbb{T}^d}(x,y))^{d+\alpha}} dy$$

Where $u$ is a test function, $d_{\mathbb{T}^d}(x,y)$ is the canonical Riemannian metric in $\mathbb{T}^d$ and $\alpha \in (0,2)$.

I would like to know:

  • What is the precise asymptotic behaviour for the eigenvalues of such operator.
  • Does it hold that the kernel of such operator is the set of the constant functions in $\mathbb{T}^d$?

I would appreciate any references as this is not my first area. Thank you very much.

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  • $\begingroup$ Is this the same operator as $\Delta^\alpha$ defined by the functional calculus? Then it is just Hörmander's theorem (Weyl law with precise remainder estimate). Same if it is an elliptic pseudodifferential operator of order $\alpha$. $\endgroup$
    – mcd
    Apr 17, 2018 at 14:45
  • $\begingroup$ I was wondering if there were any results along the lines: $\lambda_k^{(\alpha)}=c_{d,\alpha}\|k\|^\alpha + o(1)$, which is stronger than the Weyl law, no? $\endgroup$
    – Kernel
    Apr 26, 2018 at 12:34
  • $\begingroup$ I believe the operator $\Delta^{(\alpha/2)}$ has not been studied in detail. In particular, $\Delta^{(\alpha/2)}$ is not a fractional power of the Laplace operator on $\mathbb{T}^d$. I suppose the best you can get is what follows from Ivrii's asymptotic results. You may wish to consult his monster book. Is there any motivation for this particular definition? $\endgroup$
    – Mateusz Kwaśnicki
    May 15, 2018 at 11:50
  • $\begingroup$ @MateuszKwaśnicki I am looking at long-range random walks in the torus, depending on how you define the walk, this would be the scaling limit of the of its semigroup. I believe that my estimative in the above comment is actually false. $\endgroup$
    – Kernel
    May 15, 2018 at 13:48
  • $\begingroup$ @Kernel: Wouldn't the scaling limit involve the periodization of the Euclidean kernel rather than the power of the geodesic metric on the torus? $\endgroup$
    – Mateusz Kwaśnicki
    May 15, 2018 at 19:09

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