[Paper Review] The fundamental gap for a one-dimensional Schrödinger operator with Robin boundary conditions

Ben Andrews, Julie Clutterbuck+1 more|arXiv (Cornell University)|Feb 17, 2020

Spectral Theory in Mathematical Physics14 references4 citations

TL;DR

This paper establishes that for one-dimensional Schrödinger operators with Robin or Neumann boundary conditions, the fundamental gap—the difference between the first and second eigenvalues—is minimized when the potential is constant, under convex or symmetric single-well conditions. The authors extend this result to the p-Laplacian operator and use a refined version of Lavine's method to avoid reliance on log-concavity of eigenfunctions.

ABSTRACT

For Schrödinger operators on an interval with either convex or symmetric single-well potentials, and Robin or Neumann boundary conditions, the gap between the two lowest eigenvalues is minimised when the potential is constant. We also have results for the $p$-Laplacian.

Motivation & Objective

• To determine the minimizer of the fundamental gap for one-dimensional Schrödinger operators under Robin or Neumann boundary conditions.
• To extend known results on the fundamental gap—previously established for Dirichlet conditions—to Robin and Neumann settings.
• To provide sharp lower bounds on the fundamental gap without relying on the log-concavity of the first eigenfunction, which does not hold for Robin problems.
• To generalize the results to the nonlinear p-Laplacian operator for 1 < p < ∞.
• To demonstrate that linear potentials do not minimize the gap, confirming that only constant potentials yield the minimal gap under the given conditions.

Proposed method

• Adapts Lavine's method for the fundamental gap in one dimension to Robin and Neumann boundary conditions, avoiding assumptions on eigenfunction log-concavity.
• Uses variational characterization via the Rayleigh quotient for both linear and nonlinear (p-Laplacian) eigenvalue problems.
• Applies a scaling transformation to map the interval (−1,1) to (0,2a^{1/3}) to analyze asymptotic behavior as a → ∞.
• Employs asymptotic analysis of Airy functions to derive eigenvalue expansions for large a, showing the gap tends to infinity.
• Derives differential identities for eigenfunctions under Robin conditions to analyze the first variation of the gap under linear potentials.
• Uses sign analysis and integral estimates involving eigenfunction differences to derive a contradiction when assuming a non-zero linear potential minimizes the gap.

Experimental results

Research questions

• RQ1 Does the fundamental gap for a one-dimensional Schrödinger operator with Robin or Neumann boundary conditions achieve its minimum when the potential is constant?
• RQ2 Can sharp lower bounds on the fundamental gap be established without relying on the log-concavity of the first eigenfunction?
• RQ3 How does the fundamental gap behave for linear potentials under Robin boundary conditions?
• RQ4 What is the behavior of the fundamental gap for the p-Laplacian operator under Robin conditions?
• RQ5 Is the constant potential the unique minimizer of the fundamental gap among convex or symmetric single-well potentials under Robin/Neumann conditions?

Key findings

• The fundamental gap for Schrödinger operators with Robin or Neumann boundary conditions is minimized when the potential is constant, among convex or symmetric single-well potentials.
• For linear potentials V(x) = ax, the gap Γ(ax) tends to infinity as a → ∞, and the minimum gap occurs only at a = 0, i.e., for the constant potential.
• The first variation of the gap under linear potentials vanishes only if ∫x(u₁² − u₀²)dx = 0, which leads to a contradiction under Robin conditions with α ≥ −1/2.
• The asymptotic expansion of eigenvalues for large a shows λᵢᵃ ∼ a²/³μᵢ − a + O(a²/³), implying the fundamental gap Γ₂(ax) ∼ a²/³(μ₂ − μ₁) → ∞ as a → ∞.
• The method successfully avoids the use of eigenfunction log-concavity, which fails for Robin problems, by using integral identities and sign analysis.
• The results extend to the p-Laplacian case, where similar minimization of the fundamental gap by constant potentials holds under the same potential classes.