Optimal Lieb-Thirring Inequalities

• Optimal Lieb-Thirring inequalities are precise eigenvalue bounds for Schrödinger operators, addressing both real and complex potentials across varied geometric settings.
• They utilize advanced methods such as convex duality, Fourier analysis, and variational techniques to determine sharp constants and optimal parameters.
• These inequalities play a crucial role in stability of matter, semiclassical analysis, and quantum mechanics, while opening new research directions in non-selfadjoint and fractional operator theory.

Optimal Lieb-Thirring type inequalities constitute a central branch of modern spectral analysis, providing precise control over eigenvalue distributions of Schrödinger operators in both self-adjoint and non-selfadjoint regimes, on Euclidean and non-Euclidean manifolds, and in both deterministic and random settings. These inequalities are fundamental to the analysis of stability of matter, semiclassical analysis, quantum mechanics, and partial differential equations such as those arising in fluid dynamics and statistical mechanics. In the optimal form, they seek not only the sharp exponent but also the best (typically semiclassical) constant, and address the impact of geometry, complex potentials, and interactions.

1. Fundamentals and Formulation of Optimal Lieb-Thirring Inequalities

Classical Lieb-Thirring inequalities provide upper bounds for sums of powers of negative eigenvalues $\{E_j\}$ of Schrödinger operators $H = -\Delta + V$, typically in the form

$$\sum_{j} |E_j|^\gamma \leq L_{\gamma,d} \int_{\mathbb{R}^d} [V(x)]_-^{\gamma + d/2}\,dx,$$

where $[V(x)]_-$ denotes the negative part of the potential and $L_{\gamma,d}$ is an optimal constant. The semiclassical (Weyl) constant $L_{\gamma,d}^{cl}$ arises from phase-space considerations and plays a pivotal role in the conjectured optimality for large $\gamma$.

Recent research has extended this framework:

• To non-selfadjoint (complex) Schrödinger and Jacobi operators, requiring additional weight functions in the eigenvalue sums to control eigenvalues that may accumulate at the essential spectrum via non-real routes (Bögli et al., 2 Oct 2025).
• To operators on manifolds with curvature, such as the sphere, torus, or hyperbolic space, where geometry imposes new spectral subtleties and modified constants or conditions (Ilyin, 2011, Ilyin et al., 2023).
• To systems with positive density backgrounds, fractional Laplacians, or with interactions replacing antisymmetry as the mechanism enforcing exclusion (Frank et al., 2011, Dubuisson, 2014, Kögler et al., 2020, Duong et al., 1 Jan 2025).

The core challenge in optimal Lieb-Thirring theory is to determine the exact constant $L_{\gamma,d}$ and clarify the necessity, role, and sharpness of all corrective factors or weights in general settings.

2. Extension to Complex Potentials and Weighted Power Sums

In the case of complex-valued potentials, the nature of the discrete spectrum changes dramatically. Discrete eigenvalues can appear in the complex plane and accumulate near the essential spectrum [0, ∞) in intricate ways, invalidating classical inequalities unless appropriately modified. The optimal inequalities in this setting take the form

$$\sum_{\lambda \in \sigma(H_V)} \frac{[\operatorname{dist}(\lambda, [0, \infty))]^p}{|\lambda|^{d/2}}\, f\left( -\log \left[\frac{\operatorname{dist}(\lambda, [0,\infty))}{|\lambda|}\right] \right) \leq C_{p,d,f} \int_{\mathbb{R}^d} |V(x)|^p dx,$$

with a positive, typically monotone weight function $f$ that must satisfy an integrability condition $\int_0^\infty f(t)\,t\,dt < \infty$. This structure is mirrored in the case of Jacobi (discrete) and functional-difference operators, with corresponding adjustments for spectral endpoints and geometry (Bögli et al., 2 Oct 2025, Bögli, 2021, Laptev et al., 2021).

Optimality is rigorously characterized: the integrability condition on $f$ is shown both necessary and sufficient, with divergence estimates quantifying the rate of blowup when it is not satisfied. Explicit models demonstrate that, in the absence of integrability, eigenvalue sums can exhibit logarithmic or polynomial divergence superior to semiclassical predictions for real potentials (Bögli et al., 2 Oct 2025).

3. Sharp Constants, Variational Problems, and Numerical Insights

Determining the optimal constant is central to the theory, and several methodologies have been developed:

• For certain operators and parameter regimes, the sharp constant matches the semiclassical value $L_{\gamma,d}^{cl}$, particularly for $\gamma$ large and in the presence of symmetry or strong repulsive effects (Nam, 2017, Corso et al., 2 Sep 2024).
• Numerical and variational methods have produced new lower bounds and insights into the structure and multiplicity of optimizers, clarifying when the sharp constant is attained by one-bound-state potentials, and when more complex or infinite-rank configurations are needed (Levitt, 2012).
• Recent works formulated the optimization of $L_{\gamma,d}$ as a convex-analytic (or dual) variational problem, explicitly solved via Fourier analysis, convolution formulations, Fenchel–Rockafellar duality, and complex-analytic techniques involving Hardy-like spaces and Hadamard-type interpolation (Corso et al., 7 Mar 2024). This approach yields the best constants achievable by convolution and real-variable methods, with explicit construction of optimizers via Blaschke products and associated integral representations.

A summary of these methodologies is presented in the table below:

Method	Scope/Setting	Key Results/Tools
Monotonic fixed-point, FEM	Schrödinger in $\mathbb{R}^d$	Lower bounds for $L_{\gamma,d}$, optimizer shape
Convex duality, Fourier	General inequalities (CLR/LT)	Optimal constants, analytic minimizer formula
Randomization, Schatten-norm	Complex/random potentials in $\mathbb{R}^d$	Improved bounds for sums of eigenvalues
Complex analysis, zeros	Non-selfadjoint, fractional Laplacian	Weighted sum inequalities, optimal exponents

4. Geometry, Manifolds, and the Role of Curvature

Lieb-Thirring inequalities on manifolds require careful adaptation of techniques to accommodate curvature, topology, and boundary conditions:

• On compact manifolds (sphere S², torus T²), optimal constants can be achieved for functions orthogonal to the constants, even matching the flat-space value (Ilyin, 2011).
• For hyperbolic spaces $\mathbb{H}^d$, the inequality retains the phase-space form (with the Laplace–Beltrami spectrum beginning at $(d-1)^2/4$), but extra multiplicative constants emerge unless further geometric improvement is achieved (Ilyin et al., 2023). In specific cases, such as carefully structured domains, numerical evidence indicates that the classical constant may be sufficient, supporting a Polya-type conjecture for these spaces.
• Formulations for the counting function and Riesz means reflect essential spectral features such as the structure of the threshold, resonance effects, and multi-parameter dependence on curvature.

5. Extensions: Fractional Cases, Interactions, and Many-Body Systems

Recent contributions have greatly generalized the class of physical systems and operators for which optimal Lieb–Thirring-type inequalities are available:

• For fractional Laplacians $(–Δ)^s$, optimal inequalities involve new exponents and require precise Schatten–class and complex-analytic control; extensions to complex potentials have been established, integrating results on Blaschke-type zero distributions and determinant bounds (Dubuisson, 2014).
• In many-body systems, anti-symmetry can be replaced by strong repulsive interactions (homogeneous, nearest-neighbor, or Hardy-like) without loss in optimal scaling, showing that sufficiently singular interaction potentials force exclusion and lead to the optimal (Gagliardo–Nirenberg) constants in the strong-coupling limit (Kögler et al., 2020, Duong et al., 1 Jan 2025). This reformulates the classical stability mechanism from a statistical to an energetic exclusion principle.
• Analogous results for Hardy–Lieb–Thirring inequalities demonstrate that further singular behavior can be accommodated (Duong et al., 1 Jan 2025).

6. Randomized Potentials and Non-Selfadjoint Extensions

The challenge of establishing eigenvalue and power-sum bounds for random or generic non-selfadjoint potentials is addressed by combining:

• The Birman–Schwinger principle with resolvent operator estimates in Schatten classes.
• Probabilistic tools such as Anderson-type randomization and square-root cancellation to overcome known counterexamples that obstruct boundedness in deterministic settings.
• Carefully calibrated weights and entropy estimates to extend classical bounds to broader ranges of $L^p$ indices, even when decay conditions are nearly twice as weak as the threshold for deterministic models (Cuenin et al., 2023). These techniques yield nearly optimal inequalities for sums of eigenvalues, up to logarithmic corrections, and clarify the robustness of spectral localization under randomness or complex-valued perturbations.

7. Open Problems and Future Directions

A number of open questions and current research directions remain central:

• Identifying the exact parameter regimes (in dimension $d$, exponent $\gamma$, and geometric/analytic settings) where the semiclassical constant is optimal and where excess constants (beyond phase-space) are unavoidable (Schimmer, 2022, Corso et al., 2 Sep 2024).
• Completing the classification of optimizers and phase transitions between different types of maximizers (single-bound-state, multi-bound-state, or delocalized) in variational problems for LT and CLR inequalities (Levitt, 2012, Frank et al., 2020).
• Achieving sharper analytic understanding of the Polya-type inequalities on curved manifolds and closing the gap between general and numerically observed constants (Ilyin et al., 2023).
• Refining the treatment of random, non-selfadjoint, or operator-valued potentials, especially sharpening divergence rate estimates and the dependence on weight functions in spectral sums (Bögli et al., 2 Oct 2025, Bögli, 2021, Cuenin et al., 2023).

Plausible developments include further integration of convex-analytic, complex-analytic, and probabilistic methods to broaden the reach of optimal inequalities to even more general operators and to connect the theory to a wider spectrum of applications in mathematical physics, partial differential equations, and quantum information theory.