I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-forms. Let $S(M,g) \subset \Omega^1(M)$ denote the set of $1$-forms $\theta$ that satisfy $$ \Delta \theta = f \theta \qquad\text{for some smooth function}\qquad f:M \to \mathbb{R} $$ for some smooth function $f:M \to \mathbb{R}$.

Question 1. How large is the space $S(M,g)$? Is it significantly larger than the space of, say, eigen-forms of $\Delta$?

Question 2. Is there a simple way to construct examples of $1$-forms in $S(M,g)$ that are not eigen-forms?

Thanks to anyone with insight!

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-forms. Let $S(M,g) \subset \Omega^1(M)$ denote the set of $1$-forms $\theta$ that satisfy $$ \Delta \theta = f \theta $$ for some smooth function $f:M \to \mathbb{R}$.

Question 1. How large is the space $S(M,g)$? Is it significantly larger than the space of, say, eigen-forms of $\Delta$?

Question 2. Is there a simple way to construct examples of $1$-forms in $S(M,g)$ that are not eigen-forms?

Thanks to anyone with insight!