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Award ID contains: 1954995

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  1. ; (, Inventiones mathematicae)
    Abstract We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems. 
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    Free, publicly-accessible full text available September 1, 2026
  2. ; ; (, Letters in Mathematical Physics)
    Abstract We prove a matrix inequality for convex functions of a Hermitian matrix on a bipartite space. As an application, we reprove and extend some theorems about eigenvalue asymptotics of Schrödinger operators with homogeneous potentials. The case of main interest is where the Weyl expression is infinite and a partially semiclassical limit occurs. 
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  3. ; ; (, Partial Differential Equations and Applications)
    Abstract In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains. 
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  4. (, Advanced Nonlinear Studies)
    Abstract We consider the problem of minimizing the lowest eigenvalue of the Schrödinger operator −Δ +Vin L 2 ( R d ) $${L}^{2}({\mathbb{R}}^{d})$$when the integral ∫e−tV dxis given for somet> 0. We show that the eigenvalue is minimal for the harmonic oscillator and derive a quantitative version of the corresponding inequality. 
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    Free, publicly-accessible full text available May 5, 2026
  5. ; (, Archive for Rational Mechanics and Analysis)
    Abstract We solve explicitly a certain minimization problem for probability measures involving an interaction energy that is repulsive at short distances and attractive at large distances. We complement earlier works by showing that in an optimal part of the remaining parameter regime all minimizers are uniform distributions on a surface of a sphere, thus showing concentration on a lower dimensional set. Our method of proof uses convexity estimates on hypergeometric functions. 
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  6. ; (, Communications in Mathematical Physics)
    Abstract By the Aharonov–Casher theorem, the Pauli operatorPhas no zero eigenvalue when the normalized magnetic flux$$\alpha $$ α satisfies$$|\alpha |<1$$ | α | < 1 , but it does have a zero energy resonance. We prove that in this case a Lieb–Thirring inequality for the$$\gamma $$ γ -th moment of the eigenvalues of$$P+V$$ P + V is valid under the optimal restrictions$$\gamma \ge |\alpha |$$ γ ≥ | α | and$$\gamma >0$$ γ > 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$$\gamma \ge 1$$ γ ≥ 1 . 
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  7. ; (, Calculus of Variations and Partial Differential Equations)
    Abstract We show that the Caffarelli–Kohn–Nirenberg (CKN) inequality holds with a remainder term that is quartic in the distance to the set of optimizers for the full parameter range of the Felli–Schneider (FS) curve. The fourth power is best possible. This is due to the presence of non-trivial zero modes of the Hessian of the deficit functional along the FS-curve. Following an iterated Bianchi–Egnell strategy, the heart of our proof is verifying a ‘secondary non-degeneracy condition’. Our result completes the stability analysis for the CKN-inequality to leading order started by Wei and Wu. Moreover, it is the first instance of degenerate stability for non-constant optimizers and for a non-compact domain. 
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  8. ; ; (, Letters in Mathematical Physics)
    Abstract We review some older and more recent results concerning the energy and particle distribution in ground states of heavy Coulomb systems. The reviewed results are asymptotic in nature: they describe properties of many-particle systems in the limit of a large number of particles. Particular emphasis is put on models that take relativistic kinematics into account. While non-relativistic models are typically rather well understood, this is generally not the case for relativistic ones and leads to a variety of open questions. 
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  9. ; (, Calculus of Variations and Partial Differential Equations)
    Abstract We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei. 
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  10. ; ; (, Journal of the European Mathematical Society)
    Lieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided. 
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    Free, publicly-accessible full text available July 8, 2026