Gaussian Fundamental Gap Conjecture

• The paper introduces a Gaussian analogue of the fundamental gap conjecture, establishing a sharp lower bound for the eigenvalue gap of the Ornstein–Uhlenbeck operator in convex domains.
• It leverages gauge transformation, log-concavity of eigenfunctions, and the maximum principle to reduce complex spectral problems to a one-dimensional model.
• The analysis demonstrates optimality through asymptotic methods, drawing parallels between Euclidean, spherical, and Gaussian spectral gap estimates.

The Gaussian Analogue of the Fundamental Gap Conjecture refers to the extension of the fundamental gap problem—originally formulated and resolved for Schrödinger and Laplacian operators in convex Euclidean domains—to the spectral theory of the Ornstein–Uhlenbeck operator on convex domains with respect to the standard Gaussian measure. The central assertion is that the gap between the first two Dirichlet eigenvalues in this weighted (Gaussian) setting admits a sharp lower bound determined by a one-dimensional model operator, analogous to the role played by intervals in the Euclidean case.

1. Formulation of the Gaussian Fundamental Gap Conjecture

In the Gaussian setting, the operator of interest is the Ornstein–Uhlenbeck operator

$L_{\mu} u = \Delta u - \langle x, \nabla u \rangle$

with respect to the Gaussian measure $$d\mu = (2\pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}} dx$$ on $\mathbb{R}^n$. For a bounded, strictly convex domain $\Omega \subset \mathbb{R}^n$ of Euclidean diameter $D$, let $\lambda_{1,\mu} < \lambda_{2,\mu}$ denote the first two Dirichlet eigenvalues of $L_{\mu}$. The Gaussian fundamental gap conjecture asserts the existence of a sharp lower bound: $$\lambda_{2,\mu} - \lambda_{1,\mu} \geq \bar{\lambda}_2(D) - \bar{\lambda}_1(D)$$ where $\bar{\lambda}_1(D)$ and $\bar{\lambda}_2(D)$ are the first two Dirichlet eigenvalues of the one-dimensional Schrödinger operator

$$- \frac{d^2}{ds^2} + \frac{1}{4} s^2, \quad s \in (-D/2, D/2),$$

with zero Dirichlet boundary conditions. It is further established that

$$\bar{\lambda}_2(D) - \bar{\lambda}_1(D) > \frac{3\pi^2}{D^2} \quad \forall D > 0,$$

thus providing a direct analogue to the Euclidean lower bound.

2. Proof Strategy and Key Techniques

The central proof approach, as developed by Sun and Wang (Sun et al., 18 Sep 2025), extends foundational methods from the Euclidean and spherical settings (e.g., Andrews–Clutterbuck (Andrews et al., 2010), Seto–Wang–Wei (Seto et al., 2016)) to the Gaussian space by leveraging a gauge transformation, variational principles, and maximum principle arguments.

• Gauge Transformation: For $u(x) = v(x)e^{-|x|^2/4}$, the Ornstein–Uhlenbeck eigenproblem reduces to

$$-\Delta v + \left( \frac{1}{4}|x|^2 + V(x) \right)v = (\lambda + n/2) v,$$

which is a standard Schrödinger operator with a strictly convex potential.

• Log-Concavity of Eigenfunctions: The ground state $\varphi_1$ is shown to satisfy strong log-concavity estimates compared to the one-dimensional model ground state $\bar{\varphi}_1$:

$$-\langle \nabla\log \varphi_1(y) - \nabla\log \varphi_1(x), \frac{y-x}{|y-x|} \rangle \geq -2\, \frac{d}{ds}\log \bar{\varphi}_1\left(\frac{|y-x|}{2}\right).$$

• Maximum Principle for Quotient Functions: Introducing the function $Q(x, y) = [w(x) - w(y)] / \bar{w}(|x-y|/2)$ with $w(x) = \varphi_2(x)/\varphi_1(x)$, the elliptic maximum principle coupled with second-variation analyses permit comparison to the one-dimensional model and yield the sharp gap bound.
• Dimension Reduction and Sharpness: Thin domain asymptotics demonstrate that for sequences of shrinking rectangles $$\Omega_\varepsilon = (-D/2, D/2) \times (-\varepsilon, \varepsilon)$$,

$$\lim_{\varepsilon \to 0^+} (\lambda_2^{\varepsilon} - \lambda_1^{\varepsilon}) = \bar{\lambda}_2(D) - \bar{\lambda}_1(D),$$

and

$$\lim_{D \to 0^+} [\bar{\lambda}_2(D) - \bar{\lambda}_1(D)] D^2 = 3\pi^2,$$

establishing both monotonicity and optimality.

3. Connections to Classical and Spherical Gap Theory

The Gaussian case exhibits close parallels to the classical Euclidean scenario as resolved by Andrews–Clutterbuck, along with extensions to spaces of constant curvature (spherical and hyperbolic), such as those studied by Seto, Wang, and Wei (Seto et al., 2016) and Bourni et al. (Khan et al., 2022). In all instances, the strategy involves reduction to a one-dimensional spectral model and comparison via concavity moduli or maximum principle techniques.

Unlike in negatively curved Riemannian manifolds—where the gap lower bound may fail catastrophically for domains with necks or flaring regions (Khan et al., 2022)—the Gaussian setting maintains a strict universal lower bound due to the strong convexity of the underlying potential $|x|^2/4$.

4. Weighted Spectral Geometry and Implications

The establishment of the Gaussian analogue is significant within weighted spectral geometry and for applications in probability, statistical mechanics, and diffusion processes. The extension of Euclidean gap estimates to the context of the Ornstein–Uhlenbeck operator confirms that the interplay between geometric domain properties (diameter, convexity) and measure-weighted operator structure governs spectral gaps in a manner robust under Gaussian weights.

Related results on isoperimetric minimizers (e.g., round $k$-cylinders) in symmetric Ehrhard-type inequalities (Livshyts, 2021) and fractional Schrödinger operators with harmonic (Gaussian) potentials (Bao et al., 2018) further underscore the role of convexity and symmetry in controlling spectral gaps in weighted or nonlocal settings.

5. Model Monotonicity and Sharpness Properties

The normalized one-dimensional gap,

$$\frac{[\bar{\lambda}_2(D) - \bar{\lambda}_1(D)] D^2}{3\pi^2},$$

is shown to be a strictly increasing function of diameter $D$, a property proved via variational arguments and scaling of the eigenfunctions. This monotonicity yields particularly sharp bounds for domains approaching degenerate (thin) configurations and reinforces the correspondence with intervals in the classical theory.

6. Extensions, Open Problems, and Broader Context

This rigorous approach provides a template for further developments:

• Extension to domains in Riemannian manifolds with Gaussian-type weights, while alerting to failures in presence of negative curvature (Khan et al., 2022).
• Possible adaptations to fractional Laplacians and weighted Schrödinger operators as in (Bao et al., 2018).
• Investigation of gap statistics in random matrix ensembles, where Painlevé-type differential equations and Hankel determinants govern gap probabilities (Lyu et al., 2018), potentially suggesting new analogues for spectral gaps in stochastic/ensemble settings.

A plausible implication is that the sharp log-concavity techniques and gauge-transform-driven reduction to one-dimensional models will continue to underpin future generalizations to other weighted and non-Euclidean spectral problems, provided the underlying potential preserves strong convexity.

7. Summary Table: Gap Estimates across Geometries

Geometry/Measure	Operator	Lower Bound Formula
Euclidean, Laplacian	$\Delta$	$\lambda_2 - \lambda_1 \geq \frac{3\pi^2}{D^2}$
Gaussian, O-U operator	$L_{\mu}$	$$\lambda_{2,\mu} - \lambda_{1,\mu} \geq \bar{\lambda}_2(D) - \bar{\lambda}_1(D) > \frac{3\pi^2}{D^2}$$
Sphere	Laplace-Beltrami	Gap bounded below by 1D model
Negative curvature	Laplace-Beltrami	No uniform lower bound

This unifying perspective centralizes dimension reduction (via one-dimensional models), log-concavity, and maximum principle methodology in the modern theory of spectral gaps, now rigorously extended to Gaussian spaces.