Women in Math Colloquium Series

Guofang Wei (UCSB)
Fundamental Gap Estimate for Convex Domains

The fundamental (or mass) gap refers to the difference between the first two eigenvalues of the Laplacian or more generally for Schr\"{o}dinger operators. It is a very interesting quantity both in mathematics and physics as the eigenvalues are possible allowed energy values in quantum physics.  In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture  that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to $3\pi^2/D^2$.   In several joint works with  X. Dai, Z. He,  S. Seto, L. Wang (in various subsets)  the estimate is generalized, showing the same lower bound holds for convex domains  in the unit sphere. In sharp contrast,  in recent joint work with T. Bourni, J. Clutterbuck,  X. Nguyen, A. Stancu and V. Wheeler (a group of women mathematicians),  we prove that there is no lower bound at all  for the fundamental gap  of convex  domains   in  hyperbolic space in terms of  the diameter. Very recently, jointed with X. Nguyen,  A. Stancu,  we show that even  for horoconvex (which is much stronger than convex) domains in the hyperbolic space, the product of their fundamental gap with the square of their diameter has no positive lower bound.  All necessary background information will be introduced in the talk.

Event Date: 
September 30, 2021 - 3:30pm to 4:30pm
Online (See URL)
Calendar Category: