Date:
Wed, 09/11/202214:00-15:00
Title: The generalized nodal deficiency on metric graphs
Abstract:
Sturm's oscillation theorem says that the n-th eigenfunction of a Sturm-Liouville operator on an interval [a,b] has exactly n-1 zeros within (a,b). Equivalently, it says that the zeros of the n-th eigenfunction partition the interval [a,b] into exactly connected components, called nodal domains.
Sturm's theorem is no longer true when moving to Laplacian eigenfunctions on metric graphs, and the amount by which a given eigenfunction violates Sturm's theorem is known as the nodal deficiency.
What if instead of partitioning the graph at the zeros of an eigenfunction we choose to partition it at the eigenfunction's extreme points? More generally, what if we choose to partition the graph at all points such that the eigenfunction satisfies f'(x)/f(x) = s for some fixed value of s? These partitions give a generalized notion of nodal domains, which can be used to define a generalized nodal deficiency.
We show that this generalized nodal deficiency can be expressed via the Morse index of a certain linear map, which may be thought of as a generalization of the Dirichlet to Neumann map. This result generalizes an analogous index formula proven for domains in [1,2].
The talk is based on a joint work with Ram Band and Marina Prokhorova.
[1] G. Berkolaiko, G. Cox, J. Marzuola, Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann
map, Letters in Mathematical Physics (2019)
[2] G. Cox, C. Jones, J. Marzuola, Manifold decompositions and indices of Schrödinger operators,
Indiana University Mathematics Journal (2017)
Abstract:
Sturm's oscillation theorem says that the n-th eigenfunction of a Sturm-Liouville operator on an interval [a,b] has exactly n-1 zeros within (a,b). Equivalently, it says that the zeros of the n-th eigenfunction partition the interval [a,b] into exactly connected components, called nodal domains.
Sturm's theorem is no longer true when moving to Laplacian eigenfunctions on metric graphs, and the amount by which a given eigenfunction violates Sturm's theorem is known as the nodal deficiency.
What if instead of partitioning the graph at the zeros of an eigenfunction we choose to partition it at the eigenfunction's extreme points? More generally, what if we choose to partition the graph at all points such that the eigenfunction satisfies f'(x)/f(x) = s for some fixed value of s? These partitions give a generalized notion of nodal domains, which can be used to define a generalized nodal deficiency.
We show that this generalized nodal deficiency can be expressed via the Morse index of a certain linear map, which may be thought of as a generalization of the Dirichlet to Neumann map. This result generalizes an analogous index formula proven for domains in [1,2].
The talk is based on a joint work with Ram Band and Marina Prokhorova.
[1] G. Berkolaiko, G. Cox, J. Marzuola, Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann
map, Letters in Mathematical Physics (2019)
[2] G. Cox, C. Jones, J. Marzuola, Manifold decompositions and indices of Schrödinger operators,
Indiana University Mathematics Journal (2017)