Our main result is that if a generic convex domain in $R^n$ collapses to a domain in $R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the do
main need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions.
Consider a quantum particle trapped between a curved layer of constant width built over a complete, non-compact, $mathcal C^2$ smooth surface embedded in $mathbb{R}^3$. We assume that the surface is asymptotically flat in the sense that the second fu
ndamental form vanishes at infinity, and that the surface is not totally geodesic. This geometric setting is known as a quantum layer. We consider the quantum particle to be governed by the Dirichlet Laplacian as Hamiltonian. Our work concerns the existence of bound states with energy beneath the essential spectrum, which implies the existence of discrete spectrum. We first prove that if the Gauss curvature is integrable, and the surface is weakly $kappa$-parabolic, then the discrete spectrum is non-empty. This result implies that if the total Gauss curvature is non-positive, then the discrete spectrum is non-empty. We next prove that if the Gauss curvature is non-negative, then the discrete spectrum is non-empty. Finally, we prove that if the surface is parabolic, then the discrete spectrum is non-empty if the layer is sufficiently thin.
The fundamental gap conjecture was recently proven by Andrews and Clutterbuck: for any convex domain in $R^n$ normalized to have unit diameter, the difference between the first two Dirichlet eigenvalues of the Laplacian is bounded below by that of th
e interval. In this work, we focus on the moduli spaces of simplices in all dimensions, and later specialize to the moduli space of Euclidean triangles. Our first theorem is a compactness result for the gap function on the moduli space of simplices in any dimension. Our second main result verifies a recent conjecture of Antunes-Freitas: for any Euclidean triangle normalized to have unit diameter, the fundamental gap is uniquely minimized by the equilateral triangle.
By introducing a weight function to the Laplace operator, Bakry and Emery defined the drift Laplacian to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose ei
genvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $R^n$ and Neumann eigenvalues of domains in $R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize emph{all the results in Riemannian geometry based on gradient estimates to Bakry-Emery manifolds}.
