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The fundamental gap conjecture

·2 mins

Daniel Hauer (The University of Sydney, School of Mathematics and Statistics) #

Note that this seminar will be online!!! #

At the end of his study [J. Statist. Phys., 83] on thermodynamic functions of a free boson gas, van den Berg conjectured that the difference between the two smallest eigenvalues \[ \Gamma^{V}(\Omega):=\lambda_{2}^{V}(\Omega)-\lambda_{1}^{V}(\Omega), \] of the Schr"odinger operator \(-\Delta + V\) on a convex domain \(\Omega\) in \(\mathbb{R}^d\), \(d\ge 1\), equipped with homogeneous Dirichlet boundary conditions satisfies

\[ \Gamma^{V}(\Omega)\ge \Gamma\left(I_D \right)=\frac{3\pi^{2}}{D^2}, \tag{1} \] where \(I_D\) is the interval \((-D/2,D/2)\) of length \(D=\text{diameter}(\Omega)\). The term \(\Gamma^{V}(\Omega)\) is called the fundamental gap and describes an important physical quantity: for example, in statistical mechanics, \(\Gamma^{V}(\Omega)\) measures the energy needed to jump from the ground state to the next excited eigenstate, or computationally, it can control the rate of convergence of numerical methods to compute large eigenvalue problems [SIAM, 2011]. Thus, one is interested in (optimal) lower bounds on \(\Gamma^{V}(\Omega)\). Since the late 80s, the fundamental conjecture (1) attracts consistently the attention of many researcher including M. S. Ashbaugh & R. Benguria [Proc. Amer. Math. Soc., 89], R. Schoen and S.-T. Yau [Camb. Press, 94.] (see also [Geneva, 86]), B. Andrews and J. Clutterbuck [J. Amer. Math. Soc., 11].

In this talk, I present new results on the fundamental gap conjecture (1) for the Schr"odinger operator \(-\Delta + V\) on a convex domain \(\Omega\) equipped with Robin boundary conditions. In particular, we present a proof of this conjecture in dimension one, and mention results for the \(p\)-Laplacian.

The talk is based on the joint works [1,2] with B. Andrews and J. Clutterbuck.

Bibliography #

  1. Ben Andrews, Julie Clutterbuck and Daniel Hauer, Non-concavity of the Robin ground state. Camb. J. Math., 8(2):243–310, 2020.
  2. Ben Andrews, Julie Clutterbuck and Daniel Hauer, The fundamental gap for a one-dimensional Schr"{o}dinger operator with Robin boundary conditions. Proc. Amer. Math. Soc., 149(4): 1481–1493, 2021.
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