Lieb-Thirring inequalities for the Dirac operator on spheres

In this paper, we obtain bounds for the best constants in two inequalities which can be seen as analogues of the Lieb-Thirring inequality, but with the Dirac operator, on the $n-$sphere. We then apply these results in order to improve the known upper bounds on the classical Lieb-Thirring constant on the $n$-sphere for $n\geq 5$.

💡 Research Summary

The paper “Lieb‑Thirring inequalities for the Dirac operator on spheres” by Uwe Kaehler, André Pedroso Kowacs, and Michael Ruzhansky develops Lieb‑Thirring‑type spectral estimates for the Dirac operator on the n‑dimensional sphere (S^{n}). Classical Lieb‑Thirring inequalities bound the (\gamma)-moments of negative eigenvalues of the Schrödinger operator (-\Delta-V) and involve a universal constant (L_{\gamma,n}) (or its equivalent form (k_{n})). Extending such bounds to manifolds requires handling the zero eigenvalue of the Laplace–Beltrami operator; this is usually done by projecting away the constant functions.

The authors focus on the spherical Dirac operator (/!!D = \Gamma - \frac{n-1}{2}), where (\Gamma = -\sum_{j<k} e_{j}e_{k}L_{jk}) is built from the angular momentum operators (L_{jk}=x_{j}\partial_{x_{k}}-x_{k}\partial_{x_{j}}). Its square satisfies the Weitzenböck formula (/!!D^{2}= (\frac{n-1}{2})^{2} - \Delta_{LB}), so the spectrum of (/!!D) is explicitly known: eigenvalues (\pm(k+\frac{n}{2})) with multiplicities (2^{\lfloor n/2\rfloor}\binom{k+n-1}{k}). This explicit knowledge enables the authors to formulate two Dirac‑type Lieb‑Thirring inequalities.

For an orthonormal family ({\psi_{j}}{j=1}^{N}\subset H^{1}(S^{n},\mathbb C^{d})) with zero mean, define the density (\rho(x)=\sum{j=1}^{N}|\psi_{j}(x)|^{2}). The paper proves:

1. Positive‑spectrum inequality
   \