Timeline for Will the eigenvalue of the dirac operater tend to negative infinity?
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| Aug 27, 2014 at 18:52 | comment | added | Urs Schreiber | BTW, a good account of these facts about the eta function is given by Ken Richardson in his note "Introduction to the eta invariant" ncatlab.org/nlab/show/eta+invariant#Richardson | |
| Oct 29, 2011 at 22:57 | comment | added | Tim Perutz | In the larger ball, define $\phi$ by cutting off $\psi$, but slowly, so that the behaviour of $\psi$ dominates. This reduces the problem to the case of constant coefficient operators over the torus, for which Fourier series are available. | |
| Oct 29, 2011 at 22:56 | comment | added | Tim Perutz | "There are probably elementary ways". I think the following should work: It suffices to show that the Rayleigh quotient $\langle \phi, D\phi\rangle/|\phi|^2$ is doubly unbounded. Take $\phi$ to be supported in a ball where $D$ is well approximated by a constant coefficient operator. In a much smaller domain, take $\phi$ to be an eigenvector $\psi$ for the latter operator thought of as an operator with periodic boundary condition (i.e. an operator on a torus). | |
| Oct 29, 2011 at 17:19 | history | answered | Paul Siegel | CC BY-SA 3.0 |