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## Some elementary observations about spectra of Dirac operators on S4(1/h)

The spectrum of the Dirac operator are $\pm(2+k)h$ for $k=1,2,3,...$ so the first observation is that if consider a perturbation by a multiplication operator by a function $F$, i.e. the spectrum of the operator $D + F$.  Consider the general problem of a self-adjoint operator $L$ with spectrum $\lambda_i$ which have the good properties of the Dirac or Laplacian such as compact resolvent.  We want to understand the spectrum of a perturbation by some multiplication operator $F$, i.e. the operator $L + F$.  Suppose $\mu$ is in the spectrum of $L+F$ and suppose $\psi$ is a corresponding eigenfunction.  What we would like to do is understand the expansion of $\psi$ in terms of the eigenfunctions of $L$.  Suppose $\phi_i$ are the eigenfunctions of $L$ and

$\psi = \sum_i \alpha_i \phi_i$

We specialize to the $i$-th eigenspace by taking inner product with $\phi_i$ which gives us:

$\alpha_i( \lambda_i + \beta_i - \mu) = 0$

where $\beta_i = \langle F, \phi_i\rangle$.

From this, either factor has to be zero.  When $\alpha_i=0$ there are no contributions from the $\lambda_i$-the eigenfunctions of $L$ to $\psi$, and this will happen whenever $\lambda_i + \beta_i = \mu$ does not have a solution.

If we fix a particular value $\mu$ of the spectrum of $L + F$ we can thus read off the possible contributions of $L$-eigenspaces to the $mu$-eigenspace of $L+F$ in this manner.  Now if we are interested in the spectrum of the operator $G= L+F - c$ for a some constant $c$ that is a slightly different problem and in this case, we can first calculate the spectrum of $L+F$ and then check to see whether $c$ belongs in the spectrum; if so, then the spectrum of $G$ is simply a shift from the spectrum of $L+F$ and otherwise there are no solutions.

In the case of the problem with $L$ the Dirac operator on $S^4(1/h)$ and $F = Ze/\sin(t)$ for example, where we approximate the Coulomb potential using sine near zero we want the spectrum for $G - m_0c^2$.  The spectrum will be the same as $G$ but shifted if $m_0c^2$ belongs to the spectrum of $G$ and no solution otherwise.  In particular the spectrum of $G$ will tell us the restriction of possible masses that admit solutions for the Dirac equation and at the same time tell us the behavior of the spectrum.

Now in terms of $\beta_i$, the spectrum of $G$ is simply $\lambda_i + \beta_i$ with the same eigenfunctions as $L$.