Optimal bounds on the fundamental spectral gap with single-well\n potentials

2018/07/22 by Evans M. Harrell, Zakaria El Allali, Harrell, Evans M. +1 · 1 citation
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.1807.08328

Abstract

We characterize the potential-energy functions V(x) that minimize the gap\n\Γ between the two lowest Sturm-Liouville eigenvalues for \H(p,V) u :=\n-
fracddx
left(p(x)
fracdudx
right)+V(x) u =
lambda u,
quad
quad\nx
in [0,
pi ], where separated self-adjoint boundary conditions are imposed\nat end points, and V is subject to various assumptions, especially convexity\nor having a "single-well" form. In the classic case where p=1 we recover with\ndifferent arguments the result of Lavine that \Γ is uniquely minimized\namong convex V by the constant, and in the case of single-well potentials,\nwith no restrictions on the position of the minimum, we obtain a new, sharp\nbound, that \Γ > 2.04575\….\n

Cited by

• Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space

Related

• On the Fundamental Eigenvalues gap of Sturm-Liouville Operators
• Proof of the fundamental gap conjecture
• The fundamental gap for a one-dimensional Schrödinger operator with Robin boundary conditions
• The dual eigenvalue problems for p-Laplacian
• Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic space