For a (compact) spin manifold, we know that the eigenvalues of the Laplace = Dirac$^2$ operator tend to infinity, since it is an elliptic differential operator. I am wonder for the space, or gap, between the eigenvalues as we tend to infinity - will they become as large as we want, or at least is there a minimum distance between succesive eigenvalues.

(Honestly, I care most about Hermitian manifolds that are spin, so if it is easier in this case, please let me know!)

For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy $$ |\lambda_n| \to \infty, ~~~ \text{ as } n \to \infty. $$ This can be concluded, for example, from the fact that they have compact resolvent, as established in Friedrich's book on Dirac operators in Chapter 4.2.

I am wondering for the gap between the eigenvalues, as we tend to infinity, will it become as large as we want, or at least is there a minimum distance between succesive eigenvalues.

(Honestly, I care most about Hermitian manifolds that are spin, so if it is easier in this case, please let me know!)