Title: Nodal deficiency via equipartition energy functionals and the Dirichlet-to-Neumann map

Abstract: A classic result in differential equations is that the nth eigenfunction
of a Sturm-Liouville boundary value problem has precisely n-1 zeros.
Courant’s nodal domain theorem provides a natural generalization of this
result to higher dimensions, but it is generally not sharp. The lack of
sharpness is measured by the “nodal deficiency” of an eigenfunction.
Despite over a century of intensive study, this quantity is still not
very well understood.

The first explicit formula for the nodal deficiency was obtained in 2012
by Berkolaiko, Kuchment and Smilansky, using an energy functional
defined on the space of equipartitions. More recently, with Jones and
Marzuola, I obtained another formula for the nodal deficiency, in terms
of Dirichlet-to-Neumann operators defined on the eigenfunction’s nodal
domains. While originally derived using symplectic methods, this result
can also be understood using the spectral flow generated by a family of
boundary conditions imposed on the nodal set. In this talk I will
describe this flow, and explain how it provides a concrete mechanism by
which low energy eigenfunctions do or do not contribute to the nodal
deficiency. I will also describe recent progress relating these two
formulas for the nodal deficiency, and hint at some applications to the
theory of spectral minimal partitions.

This talk represents joint work with Thomas Beck, Gregory Berkolaiko,
Isabel Bors, Yaiza Canzani, Grace Conte, Christopher Jones and Jeremy
Marzuola.

The Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 – 5 pm Atlantic Time over Zoom.  If you would like to attend, please email the organizers for connection details.