The Fundamental Gap Conjecture for Polygonal Domains

In [22], M. van de Berg made a “fundamental gap conjecture” in the context of a free boson gas confined in a container in Rn with hard walls, and in [24], Yau generalized the conjecture to what is now known as the “fundamental gap conjecture.” For a convex domain Ω ⊂ Rn, (0.1) ξ(Ω) := d (λ2(Ω)− λ1(Ω)) ≥ 3π where d is the diameter of the domain, and 0 < λ1(Ω) < λ2(Ω) are the first two eigenvalues of the Euclidean Laplacian on Ω with Dirichlet boundary condition. The scalar invariant ξ is the gap function. We restrict attention to planar domains. Our main result is a compactness theorem for the gap function when the domain is a triangle in R2. This result shows that for any triangles which collapse to the unit interval, the gap function is unbounded. Due to numerical methods, we expect that the fundamental gap conjecture holds for all triangular domains in R2. We show with examples that the behavior of the gap for collapsing polygonal domains is quite delicate. These examples motivate a technical result for collapsing polygonal domains giving conditions under which the gap function either remains bounded or becomes infinite. Our work initiates a general program to prove the fundamental gap conjecture using convex polygonal domains. 1. Motivation and results For a Schrödinger operator on a compact convex domain, after the first eigenvalue the next natural object to study is the gap between the first two eigenvalues, known as the fundamental gap. This includes the work of [2], [20], [22], [25], and many other authors. While it is always interesting to understand the interaction between the eigenvalues of a differential operator and the geometry of the domain, beyond purely mathematical implications the fundamental gap has physical implications. For the heat equation, the gap controls the rate of collapse of any initial state toward a state dominated by the first eigenvalue and is of central interest in statistical mechanics and quantum field theory. In analysis, the gap is important to refinements of the Poincaré inequality and à priori estimates. Numerically, the gap can be used to control the rate of convergence of numerical computation methods such as discretization or finite element method by which one uses matrices to approximate a differential operator. The ability to solve for the first eigenvalue and eigenvector of these matrices is controlled by the size of the gap between the first eigenvalue and the rest of the spectrum. Understanding the behavior of the gap for collapsing convex polygonal domains is also relevant to computer graphics image