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.

pseudodifferentialoperator. It has the same eigenfunctions as the Laplacian and the eigenvalues are the square roots of the eigenvalue s of the Laplacian. To get a differential operator you need to give up working with scalar functions and instead work with sections of (certain) vector bundles. – Liviu Nicolaescu Nov 14 at 21:12Functional calculusis a good search term. – Neal Nov 14 at 21:25