A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials

Abstract: In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schrödinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, using the ultracontractivity of the semigroup we obtain the log-concavity of the ground state which is strong if $Ω$ has positive Gauss curvature and if $V$ is strongly convex and regular enough.

• The paper establishes a convexity inequality for the first Dirichlet eigenvalue using semigroup and trace-class techniques.
• It leverages convex, Kato-decomposable potentials and advanced operator theory to extend classical geometric inequalities.
• The results imply strong log-concavity of the ground state, linking spectral theory with convex geometric analysis.

Brunn-Minkowski Inequality for Schrödinger Operators with Kato Class Potentials

Introduction and Context

The paper "A Brunn-Minkowski inequality for Schrödinger operators with Kato class potentials" (2603.29989) presents a detailed analysis of convexity properties of eigenvalues for Dirichlet realizations of Schrödinger-type operators with potentials belonging to the Kato class. Extending classical geometric inequalities, such as Brunn-Minkowski and Prékopa–Leindler, to spectral functionals associated with elliptic and parabolic PDEs has provided deep connections between convex geometry, analysis of PDEs, and spectral theory.

The Brunn-Minkowski inequality in its geometric form quantifies the concavity of volume under Minkowski addition. Analogous principles for spectral quantities—such as the first Dirichlet eigenvalue of elliptic operators—encode critical geometric and analytic behaviors. The analysis in this paper focuses on operators of the form $\mathcal{H}_V = -\operatorname{div}(A\nabla) + V$ under mild assumptions on $A$ (constant, symmetric, positive definite) and convex, Kato-decomposable potentials $V$.

Main Results

The central result establishes a Brunn-Minkowski-type convexity inequality for the first Dirichlet eigenvalue $\lambda_{1,V}(\Omega)$ of $\mathcal{H}_V$ over convex domains:

$$\lambda_{1,V}(\Omega_r) \le (1-r)\lambda_{1,V}(\Omega_0) + r\lambda_{1,V}(\Omega_1), \quad \Omega_r = (1-r)\Omega_0 + r\Omega_1.$$

This holds for all $r\in [0,1]$ and for arbitrary convex sets $\Omega_0, \Omega_1$, provided $V$ is convex and Kato-decomposable, and certain trace-class and integrability conditions are satisfied. The proof leverages the spectral structure of the semigroup generated by $\mathcal{H}_V$, employing trace class and ultracontractivity properties to pass convexity from the heat kernel to spectral data.

A direct consequence is the log-concavity of the Dirichlet ground state $A$0 under these hypotheses, with a strong log-concavity statement obtainable when both the domain and the potential have additional regularity (specifically, positive Gauss curvature and strong convexity of $A$1).

Technical Framework

Assumptions and Operator Classes

The analysis assumes:

• $A$2 is constant, symmetric, and positive definite.
• $A$3 is convex, Kato-decomposable ($A$4, $A$5), and $A$6 for all $A$7.

This framework admits both regular and singular potentials; the latter is exemplified by inverse-square singularities ($A$8), provided the Kato and integrability conditions are met.

Semigroup Approach and Spectral Analysis

The spectral convexity property is derived via analysis of the heat semigroup $A$9. The key steps are:

• Establishing the semigroup's trace class property (using Gaussian upper bounds, ultracontractivity, and the Golden-Thompson-Symanzik estimate).
• Relating the trace of the semigroup to the sum of exponentials of eigenvalues, allowing convexity arguments from the semigroup trace to be inherited by $V$0.
• For convex $V$1, the Trotter product formula and log-concavity of the heat kernel yield the critical log-concavity property for the partition function $V$2, with explicit control afforded by operator-theoretic spectral mapping results.

Log-Concavity and Strong Log-Concavity of Eigenfunctions

Leveraging the Brascamp-Lieb framework and isospectral transformations (notably for Ornstein-Uhlenbeck and Kolmogorov-type operators), the log-concavity of the ground state extends naturally from convex $V$3. For potentials with higher regularity and strongly convex domains, the ground state satisfies a nonlinear PDE whose Hessian can be analyzed via maximum principle and constant rank theorems to yield strict log-concavity.

Explicit Examples

The applicability of the framework is illustrated with:

• Kolmogorov operators: revealing the spectral equivalence with shifted harmonic oscillators and connecting Brunn-Minkowski convexity to classical Ornstein-Uhlenbeck settings.
• Schrödinger operators with singular potentials in bounded convex domains: demonstrating the persistence of the Brunn-Minkowski inequality even in the presence of singularities, as long as the Kato and trace-class criteria are satisfied.

Implications and Future Directions

The extension of Brunn-Minkowski inequalities to operators with Kato-class potentials not only reinforces the deep interplay between convexity, geometric analysis, and spectral theory but also enlarges the class of manageable PDEs, including those relevant for mathematical physics and probability (e.g., ground state dominance, Feynman-Kac representations, and isoperimetric inequalities in weighted spaces).

Potential directions for further research include:

• Quantitative stability in these inequalities under perturbations of the potential or domain.
• Extension to nonlocal and degenerate operators or those with merely measurable coefficients.
• Applications to concentration of measure phenomena, spectral gap estimates, and refined capacities in infinite-dimensional or stochastic settings.

Conclusion

This paper establishes a robust Brunn-Minkowski-type convexity inequality for the principal Dirichlet eigenvalue of Schrödinger operators with convex, Kato-decomposable potentials, using advanced operator semigroup techniques. It solidifies the link between geometric inequalities and spectral analysis in the broad context of both regular and singular PDEs, and generalizes log-concavity properties of ground states under optimal assumptions. These results provide a foundation for further advances in the geometric analysis of elliptic and parabolic operators and invite continued exploration of convexity phenomena in high-dimensional and irregular regimes.