An improved bound for the ground state of a Schrödinger operator on a loop

TL;DR

This work advances the Ovals Conjecture for the ground state of the Schrödinger operator with curvature potential by proving a new universal lower bound. The authors develop two complementary analytical routes—the $\phi^{-1}$ and the $x$/$y$ coordinates views—to bound the lowest eigenvalue $\lambda_{\Gamma}$ from below in terms of two simple functions $B_1$ and $B_2$, depending on a gap parameter $\Delta$ and a variation parameter $\tilde{\nu}$. By combining a total-variation analysis of the anti-periodic component $f$ of $\phi^{-1}$ with a Three Angles decomposition, they obtain a rigorous bound $\lambda_{\Gamma} > \inf_{\tilde{\nu},\Delta}\max\{B_1,B_2\}$, which they evaluate numerically to exceed $0.8246$, and they also derive a robust analytical bound of $0.81$. A refined argument focusing on the level set where $B_1=0.81$ shows that on this curve the function $B_2$ remains above $0.81$, establishing a sharper, explicit threshold. The key technical contribution lies in carefully tracking curvature fluctuations via Fourier-analytic decompositions and a constructive, variational minimization of total variation over admissible test functions.

Abstract

Consider a closed curve of length $2π$ with curvature $κ(s)$ and the Schrödinger operator $H$ with $κ^2$ as the potential term. Let $λ_Γ$ be the lowest eigenvalue of $H$. The Ovals Conjecture proposed by Benguria and Loss states that $λ_Γ\ge 1$. While the conjecture remains open, the present work establishes a new lower bound of $0.81$ on $λ_Γ$, improving on the previously best known estimate of $0.6085$.

Figures (8)

• Figure 1: Superposition of contour plots of $B_1$ (solid lines) and $B_2$ (dashed lines). The X marks the point where $\max(B_1,B_2)$ is minimal.
• Figure 2: By assumption, the $\pi$-periodic energy projection $I(t)$ (orange line) is equal to one at $\iota_1 > 0$ and $\iota_2 < \frac{\pi}{2}$ and it is larger than one in between. Four critical angles of $\Gamma$ are marked with diamonds in the chart. They occur in pairs which are separated by a distance of $\pi$, respectively. All critical angles below $\pi$ are contained in $[\iota_1,\iota_2]$ and $\tau_1$ and $\tau_2$ are the smallest and the largest of those, respectively. The critical angles are zeros of $f(t)$ (blue dotted line).
• Figure 3: To prove Theorem \ref{['theorem:main_result2']}, we have to show that at every point in $[0,1]\times(0,\frac{\pi}{2})$ at least one of the two functions $B_1$ and $B_2$ is greater than $0.81$. The monotonicity of $B_1$ guarantees this for $B_1$ below and to the left of the level set $\mathcal{L}$. The monotonicity of $B_2$ implies that no value of $B_2$ in the hatched area of the chart is larger than the infimum of $B_2$ restricted to $\mathcal{L}$.
• Figure 4: The function $f_1$(blue dotted line) is constructed from the function $\hat{f}$ (orange line). Here it is shown for $\nu_1 = 0.5$ and $\nu_2 = 0.9$.
• Figure 5: Estimating the values of $s_1(\nu_1,\nu_2)$ and $c_1(\nu_1,\nu_2)$ at the four corners of the square $\Sigma$ shows that the zero lines of $s_1$ and $c_1$ cross inside the square. The intersection corresponds to a choice of $\nu_1$ and $\nu_2$ which 'balances' $f_1$ with respect to the Fourier condition.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1: Upper bound on total variation of $f$
• proof
• Lemma 3.2: Lower bound on total variation of $f$
• Corollary 3.3: Relaxed lower bound on $V(f)$
• Corollary 3.4: Dual Use Corollary
• proof
• Lemma 3.5
• Lemma 3.6