Take the 2-minute tour ×  Question: If M is a spin manifold.
Condider the dirac operator on a spinor bundle. Can the
eigenvalue of this operater tend to negative infinity? If it
can, can we choose a riemann matrix such that under this matrix,
the eigenvalue of dirac operator does not tend to negative
infinity??  share|improve this question  2 Answers 2  Given any
first order, graded, self-adjoint, elliptic operator on a
manifold the spectrum is unbounded in both the positive and
negative directions. There are probably elementary ways to see
this, but I'll give an argument which is overly sophisticated
but a bit more quantitative.  Given $D$ as above, consider the
eta function associated to $D$:  $$\eta(s) = \sum_{\lambda}
sign(\lambda) |\lambda|^{-s}$$  (the sum ranges over the
eigenvalues of $D$)  By a variation on Weyl's asymptotic formula
for the eigenvalues of the Laplacian together with some basic
complex analysis, this function extends to a meromorphic
function on the complex plane. It is a very deep to