Choice of the eigenbasis for the Dirac operator on $S^d$

Question

This question is a simplified version of my previous one. I think that adding a gauge potential complicates the problem too much.

Let us consider the Dirac operator $D$ on the $d$-sphere $S^d$ with respect to the spin $1/2$ structure. I have $d=4$ in mind, but would like to think in more generality. An exact definition of the Dirac operator seems to be summarized in this note.

Then, I have two questions:

Is it possible to find a collection of smooth eigenspinors $\{ \psi_n\}_{n \in \mathbb{N}}$ on $S^d$ for $D$ such that there exists a point $a \in S^d$ at which all orders of derivatives of all $\psi_n$ vanish.

At the same time, is it possible for such $\{ \psi_n\}_{n \in \mathbb{N}}$ to be complete? That is, $$ \sum_{n \in \mathbb{N}} \psi_n(x) \psi_n^\dagger(y) = \delta(x-y)I $$ where $I$ is the identity matrix on the representation space of the spinors and this equality should be understood in the sense of distributions on $S^d$.

The first question is equivalent to the possibility of having Schwartz function components for each $\psi_n$, as shown in this ME post.

Could anyone please help me? My impression is that the Dirac operator has been studied extensively from geometric perspectives, but somewhat less in the context of spectral analysis. Perhaps I am missing something here?

Answer 1

This is a bit too long to fit as a comment.

For $d\geq 3$ there is only one spin structure on $S^d$ thus a unique spin Dirac operator $D: C^\infty(S^+)\to C^\infty(S^-)$ where $S^\pm$ denotes the bundle of even/odd spinors.

If you choose any complete orthonormal system $(\psi_n)$ of $L^2(S^+)$ consisting of smooth spinors, then the identity

$$\sum_n \psi_n(x)\otimes \psi_n(y)^\dagger=\delta(x-y) $$

is tautologically satisfied, but you need to recall that the left hand side is an operator from $S^+_y\to S^+_x$ so $\delta(x-y)$ should be understood as an operator valued distribution.

Let me define a section to be flat at $a\in S^d$ if all its partial derivatives vanish at that point. One can show that the space of smooth functions that are flat at a point $a$ is dense in $L^2(S^d)$. Thus, using Gram-Schmidt, you can choose the above sections $\psi_n$ to be flat at a point $a$.

If you want eigen-spnors you need to consider the operator $$ \widehat{D}=\left[\begin{array}{cc} 0 & D^*\\ D & 0 \end{array}\right] :\begin{array}{c} C^\infty(S^+)\\ \oplus \\ C^\infty(S^-) \end{array} \to \begin{array}{c} C^\infty(S^+)\\ \oplus \\ C^\infty(S^-) \end{array} $$ Then you can choose an orthonormal basis of $L^2(S^+\oplus S^-)$ consisting of eigen-spinors of $\widehat{D}$. They cannot be flat at any point since they are real analytic.

For more about Dirac operators, check Chapter 12 of these notes.