Please forgive me for my inability to pose a mathematics question properly.
In one dimension the eigenfunctions of Laplacian (simply double derivative) are also eigenfunctions of its square root (single derivative). Is anything known regarding a possible generalisation to an arbitrary Riemannian manifold? Square root of the Laplacian is essentially a single derivative. Therefore, if such eigenfunctions exists or are known, then they will be exponential in some sense.
Even if it's not known in a fully general context, are there special situations where such eigenfunctions can be constructed?
Thanks in advance.