Timeline for General questions on the eigenfunctions of Laplacian and Dirac operators
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| Jun 3, 2017 at 17:37 | comment | added | Z. Ye | @BranimirĆaćić: Sorry, I don't know much about the noncommutative geometry. Just wonder, is there any similar results formulate in the classical language or is it really necessary to go deep into the noncommutative geometry? Thank you. | |
| May 21, 2017 at 5:44 | comment | added | Bernd Ammann | @Carlo Beenakker: However, I do not know any reference where Dirac-isospectrals pairs buidling on Heisenberg manifolds are worked out. | |
| May 21, 2017 at 5:34 | comment | added | Bernd Ammann | @Carlo Beenakker. The examples of 4-dimensional tori follow from [8] Schiemann A. Ein Beispiel positiv definiter quadratischer Formen der Dimension 4 mit gleichen Darstellungszahlen // Arch. Math. 1990. V. 54. P. 372–375 and Conway J. H., Sloane N. J. A. Four-dimensional lattices with the same theta series // International Mathematics Research Notices. 1992. V. 4. P. 93–96. Every pair of Laplace-isospectral tori qith the trivial spin structure is also | |
| May 13, 2017 at 17:58 | history | edited | Carlo Beenakker | CC BY-SA 3.0 |
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| May 13, 2017 at 17:38 | comment | added | Branimir Ćaćić | (+1) That said, by a result of Connes, if one knows not only the spectrum of the Dirac operator $D$ as an operator on the separable Hilbert space $H$ of $L^2$ spinor fields, but also the relative position within the von Neumann algebra $B(H)$ of bounded operators on $H$ of the von Neumann subalgebra of $B(H)$ generated by the functional calculus of $D$ (i.e., the datum of the actual diagonalisation of $D$) and $L^\infty(M)$, viewed as multiplication operators on $B(H)$, then you can indeed pin down your Riemannian geometry. | |
| May 13, 2017 at 17:30 | history | answered | Carlo Beenakker | CC BY-SA 3.0 |