A Spectral Gap Estimate and Applications

Original · EN

We consider the Schrödinger operator -d²/d x² + V on an interval [a,b] with Dirichlet boundary conditions, where V is bounded from below and prove a lower bound on the first eigenvalue λ₁ in terms of sublevel estimates: if wᵥ(y) = |Iy|, where Iy:= { x ∈ [a,b]: V(x) ≤ y }, then λ₁ ≥ 1/250 > V(1/wᵥ(y)² + y). The result is sharp up to a universal constant if { x ∈ [a,b]: V(x) ≤ y } is an interval for the value of y solving the minimization problem. An immediate application is as follows: let Ω⊂ R² be a convex domain with inradius ρ and diameter D and let u:Ω→ R be the first eigenfunction of the Laplacian -Δ on Ω with Dirichlet boundary conditions on ∂ Ω. We prove u ₗ∞ 1ρ (ρD)¹/⁶ uₗ₂, which answers a question of van den Berg in the special case of two dimensions.