Computing Nodal Deficiency with a Refined Spectral Flow

Explore a comprehensive lecture on computing nodal deficiency using a refined spectral flow, delivered by Bernard Helffer at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into recent advancements connecting the Dirichlet-to-Neumann map, spectral flow of self-adjoint operators, and nodal deficiency of Laplacian eigenfunctions. Discover how a refined construction of the Dirichlet-to-Neumann map strengthens previous results and provides improved bounds on nodal deficiency for degenerate eigenfunctions. Examine the broad applicability of this framework, which accommodates non-bipartite partitions, non-simple eigenvalues, and non-smooth nodal sets. Learn how these findings contribute to the general study of spectral minimal partitions, extending beyond nodal partitions of generic Laplacian eigenfunctions. Gain insights from this 44-minute talk, which was part of the Workshop on "Spectral Theory of Differential Operators in Quantum Theory" held at the ESI in November 2022.