Fractional elliptic operators from a generalized Glaeske-Kilbas-Saigo-Mellin transform. (English) Zbl 1353.58014
Summary: We show that the deformation of the canonical spectral triples over the \(n\)-dimensional torus which is characterized by a conjectured elliptic operator \(\mathrm D_\beta={\kern-0.8pt\not\kern0.8pt D}(1+|{\kern-0.8pt\not\kern0.8pt D}|^2)^{-\beta}=\frac{1}{\Gamma(\beta)}\int^\infty_0 \tau^{\beta-1} e^{-\tau(1+{\kern-1.8pt\not\kern2.8pt D}^2)} {\kern-0.8pt\not\kern0.8pt D} d\tau\) with \(\beta \geq 0\) and by a discrete dimension spectrum with fractional values less than \(n\) may be obtained if the elliptic operator is defined by means of the fractional Glaeske-Kilbas-Saigo-Mellin transform.
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 26A33 | Fractional derivatives and integrals |
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