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Abstract By the Aharonov–Casher theorem, the Pauli operatorPhas no zero eigenvalue when the normalized magnetic flux$$\alpha $$ satisfies$$|\alpha |<1$$ , but it does have a zero energy resonance. We prove that in this case a Lieb–Thirring inequality for the$$\gamma $$ -th moment of the eigenvalues of$$P+V$$ is valid under the optimal restrictions$$\gamma \ge |\alpha |$$ and$$\gamma >0$$ . Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order$$\gamma \ge 1$$ .
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This content will become publicly available on April 1, 2026
Optimizers For The Finite-Rank Lieb-Thirring Inequality
abstract: The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $$N$$ lowest eigenvalues of a Schr\odinger operator $$-\Delta-V(x)$ in terms of an $$L^p(\mathbb{R}^d)$$ norm of the potential $$V$$. We prove here the existence of an optimizing potential for each $$N$$, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $$\gamma>\max\{0,2-d/2\}$ on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in $$N$$, which sheds a new light on a conjecture of Lieb-Thirring. In dimension $d=1$ at $$\gamma=3/2$$, we show that the optimizers with $$N$$ negative eigenvalues are exactly the Korteweg-de Vries $$N$$-solitons and that optimizing sequences must approach the corresponding manifold. Our work also covers the critical case $$\gamma=0$$ in dimension $$d\geq3$$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.
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- Award ID(s):
- 1954995
- PAR ID:
- 10628071
- Publisher / Repository:
- Hopkins Press
- Date Published:
- Journal Name:
- American Journal of Mathematics
- Volume:
- 147
- Issue:
- 2
- ISSN:
- 1080-6377
- Page Range / eLocation ID:
- 503 to 560
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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