2003/12/19 by Evans M. Harrell, Evans M. Harrell II, Harrell, Evans M.
Mathematics · #35J10 #47A75 #53Z05 #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.AP #math.SP #msc:35J10 #msc:47A75 #msc:53Z05
paper · pdf · doi:10.48550/arxiv.math/0312372
arxiv created 2003/12/19 · openalex publication_date 2003/12/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = -Δ+ V(x), acting on functions of a smooth, compact d-dimensional manifold M immersed in \bbrν, ν≥ d+1. Here Δ denotes the Laplace-Beltrami operator, and the real-valued potential--energy function V(x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature: Hg := -Δ+ g h2, where ν= d+1, g is a real parameter, and h := ∑j = 1d κj, with \κj\, j = 1, ..., d denoting the principal curvatures of M. For instance, by Theorem \refthm3.1 and Corollary \refcor4.5, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using H1/4. The "isoperimetric" parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.