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Simon Larson

Publications and source records attributed to Simon Larson.

At least 19 recordsLinked to original sources
arXiv2026

Robin counting functions on cuboids in the critical scaling regime

We consider eigenvalue counting functions of Robin Laplace operators on cuboids where the Robin parameter and the spectral cut-off $\lambda$ are coupled. Our main focus is the critical regime, in which the Robin parameter is proportional to $\sqrt{\lambda}$. We obtain a two-term asymptotic expansion for the counting functions in this coupled setting. In the critical regime, the second term in the asymptotic expansion depends non-trivially on the proportionality constant and interpolates continuously between the corresponding second terms for the Dirichlet and Neumann Laplacians. Outside the critical regime, one recovers the corresponding Dirichlet or Neumann asymptotics. We also establish a P\'olya-type inequality for the counting function whenever the ratio of the Robin parameter and $\sqrt{\lambda}$ exceeds a dimension-dependent threshold.

math.SP↗
arXiv2026

An asymptotic shape optimization problem for Riesz means of Laplacian eigenvalues

We review our recent results on the problem of optimizing Riesz means of Laplace eigenvalues among convex sets of given measure in the regime where the cut-off parameter in the definition of the Riesz means tends to infinity. We show that for a certain range of Riesz exponents, the optimizing sets converge to a ball. We also present some new results where we optimize over disjoint unions of convex sets.

math.SP↗
arXiv2026

Optimizing Riesz means of Robin Laplace operators on cuboids in a semiclassical limit

We study asymptotic shape optimization for Riesz means of Robin Laplacian eigenvalues among cuboids of fixed measure. Our focus is the regime where the Robin parameter is proportional to the square root of the spectral parameter defining the Riesz means. Here, a transition emerges based on the precise ratio between the two parameters: as the spectral parameter tends to infinity, sequences of maximizers shift from converging to the unit cube to lacking convergent subsequences entirely. Key tools include two-term spectral asymptotics and uniform inequalities for the Riesz means. Notably, the transition point governing the behavior of optimizers may differ from the point at which the second asymptotic term changes sign. This shows that heuristics based solely on asymptotics for a fixed domain fail to accurately predict the asymptotic behavior of maximizers.

math.SP↗
arXiv2026

Uniform bounds for Neumann heat kernels and their traces in convex sets

We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends on the underlying domain only through very simple geometric characteristics. This is proved via a precise and uniform expansion of the on-diagonal heat kernel close to the boundary. Most of our results are valid without the convexity assumption and we also consider two-term asymptotics for the heat trace for Lipschitz domains.

math.AP↗
arXiv2025

Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrt\Lambda |\Omega|^{1/d}$, where $\Lambda$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|\Omega|^{1/d}$ is replaced by a generalized inradius of $\Omega$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.

math.SP↗
arXiv2024

Spectral asymptotics for Robin Laplacians on Lipschitz sets

We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.

math.SP↗
arXiv2024

Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains

We are interested in inequalities that bound the Riesz means of the eigenvalues of the Dirichlet and Neumann Laplacians in terms of their semiclassical counterpart. We show that the classical inequalities of Berezin-Li-Yau and Kr\"oger, valid for Riesz exponents $\gamma\geq 1$, extend to certain values $\gamma<1$, provided the underlying domain is convex. We also study the corresponding optimization problems and describe the implications of a possible failure of P\'olya's conjecture for convex sets in terms of Riesz means. These findings allow us to describe the asymptotic behavior of solutions of a spectral shape optimization problem for convex sets.

math.SP↗
arXiv2024

Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains

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.

math.SP↗
arXiv2024

On a Hardy-Morrey inequality

Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le \int_\Omega |Du|^p \,dx $$ for any open set $\Omega\subsetneq \mathbb{R}^n$. This inequality is valid for functions supported in $\Omega$ and with $\lambda$ a positive constant independent of $u$. The crucial hypothesis is that the exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of $\Omega$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality.

math.AP↗
arXiv2023

Decay of extremals of Morrey's inequality

We study the decay (at infinity) of extremals of Morrey's inequality in $\mathbb{R}^n$. These are functions satisfying $$ \displaystyle \sup_{x\neq y}\frac{|u(x)-u(y)|}{|x-y|^{1-\frac{n}{p}}}= C(p,n)\|\nabla u\|_{L^p(\mathbb{R}^n)} , $$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey's inequality. We prove that if $n \geq 2$ then any extremal has a power decay of order $\beta$ for any $$ \beta<-\frac13+\frac{2}{3(p-1)}+\sqrt{\left(-\frac13+\frac{2}{3(p-1)}\right)^2+\frac13}. $$

math.AP↗
arXiv2022

An inequality for the normal derivative of the Lane-Emden ground state

We consider Lane-Emden ground states with polytropic index $0\leq q-1\leq 1$, that is, minimizers of the Dirichlet integral among $L^q$-normalized functions. Our main result is a sharp lower bound on the $L^2$-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets $\Omega\subset\mathbb{R}^d$, without assuming convexity.

math.AP↗
arXiv2021

Discrete Schr\"odinger operators with decaying and oscillating potentials

We study a family of discrete one-dimensional Schr\"odinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential $V(n)=\lambda n^{-\alpha}\cos(\pi \omega n^\beta)$, with $1<\beta<2\alpha$, we prove that the spectrum is purely absolutely continuous on the spectrum of the Laplacian.

math.SP↗
arXiv2021

On the spectrum of the Kronig-Penney model in a constant electric field

We are interested in the nature of the spectrum of the one-dimensional Schr\"odinger operator $$ - \frac{d^2}{dx^2}-Fx + \sum_{n \in \mathbb{Z}}g_n \delta(x-n) \qquad\text{in } L^2(\mathbb{R}) $$ with $F>0$ and two different choices of the coupling constants $\{g_n\}_{n\in \mathbb{Z}}$. In the first model $g_n \equiv \lambda$ and we prove that if $F\in \pi^2 \mathbb{Q}$ then the spectrum is $\mathbb{R}$ and is furthermore absolutely continuous away from an explicit discrete set of points. In the second model the $g_n$ are independent random variables with mean zero and variance $\lambda^2$. Under certain assumptions on the distribution of these random variables we prove that almost surely the spectrum is $\mathbb{R}$ and it is dense pure point if $F < \lambda^2/2$ and purely singular continuous if $F> \lambda^2/2$.

math-ph↗
arXiv2020

Semiclassical asymptotics for a class of singular Schr\"odinger operators

Let $\Omega \subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schr\"odinger operators $-\Delta+ W$ on $\Omega$ with $W(x)\approx\mathrm{dist}(x, \partial\Omega)^{-2}$ as $\mathrm{dist}(x, \partial\Omega)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.

math.SP↗
arXiv2020

A sharp multidimensional Hermite-Hadamard inequality

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\,d\sigma(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al.

math.CA↗