Fundamental Gap conjecture and log-concavity

Speaker: 

Daniel Hauer

Affiliation: 

University of Sydney

Date: 

Tue, 23/07/2019 - 12:00pm to 1:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

The fundamental gap conjecture states that the difference of the 2nd and 1st eigenvalue of the Schroedinger operator equipped with a bounded potential in a convex bounded region in $\R^{d}$ with diameter $D$ is bounded from below by the difference of the 2nd and 1st Laplace eigenvalue on the interval $(-D/2,D/2)$. This conjecture was confirmed in 2011 by Andrews and Clutterbuck in the case that the Schroedinger Operator is equipped with homogeneous Dirichlet boundary conditions. Their proof uses the fact that the first ground state is log-concave, and it was natural to assume that this proof could be adapted to other boundary conditions. However, recently Andrews, Clutterbuck, and Hauer constructed examples of convex domains on which the first Robin ground state is not log-concave. 
 
In this talk, I will review briefly these counter-examples and present an alternative way to circumvent the property of log-concavity for proving the fundamental gap conjecture.
 
This is joint work with Ben Andrews (Australian National University) and Julie Clutterbuck (Monash University).

School Seminar Series: