Analysis/Mathematical Physics Seminar

In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when $M$ is a principle $S^1$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically $2$, independent of the eigenvalues. Note that principle $S^1$-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when $(M,g)$ is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to $+\infty$. I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. In particular, this tells us that the number of nodal domain could be uniformly bounded independent of the eigenvalue.