Rigorous estimation for the difference quotients of multiple eigenvalues (2305.14063v5)

Abstract: In spectral theory, the multiplicity of nearly degenerate eigenvalues presents significant challenges. This paper introduces a new difference quotient formula to capture the behavior of nearly degenerate Laplacian eigenvalues resulting from domain perturbations. Additionally, we propose a novel numerical algorithm for rigorously estimating the difference quotient of these multiple eigenvalues in response to domain deformation, using a recently developed guaranteed computation method for eigenvalue problems. As an application, we solve the open problem of the simplicity of the second Dirichlet eigenvalue for nearly equilateral triangles, offering a partial solution to Conjecture 6.47 in A. Henrot's book ``Shape Optimization and Spectral Theory."