# Eigenvalue bounds for the fractional Laplacian: A review

@article{Frank2016EigenvalueBF, title={Eigenvalue bounds for the fractional Laplacian: A review}, author={Rupert L. Frank}, journal={arXiv: Spectral Theory}, year={2016} }

We review some recent results on eigenvalues of fractional Laplacians and fractional Schr\"odinger operators. We discuss, in particular, Lieb-Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

## 29 Citations

Kroger's upper bound types for the Dirichlet eigenvalues of the fractional Laplacian.

- Mathematics
- 2020

We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific…

On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians

- MathematicsJournal of Differential Equations
- 2022

On the asymptotic behavior of $p$-fractional eigenvalues

- Mathematics
- 2021

In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the p−fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.

Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded Euclidean domain

- Mathematics, Philosophy
- 2019

This paper is devoted to interior, i.e. away from the boundary, estimates for eigenfunctions of the fractional Laplacian in an Euclidean domain of $\mathbb R^d$.

Nonradiality of second eigenfunctions of the fractional Laplacian in a ball

- MathematicsProceedings of the American Mathematical Society
- 2022

Using symmetrization techniques, we show that, for every
N
≥
2
N \geq 2
, any second eigenfunction of the fractional Laplacian in the
N
N
-dimensional unit ball with…

Pólya's conjecture fails for the fractional Laplacian

- MathematicsJournal of Spectral Theory
- 2018

The analogue of P\'olya's conjecture is shown to fail for the fractional Laplacian (-Delta)^{alpha/2} on an interval in 1-dimension, whenever 0 < alpha < 2. The failure is total: every eigenvalue…

Interior estimates for the eigenfunctions of the fractional Laplacian on a bounded domain

- MathematicsAdvances in Mathematics
- 2021

Minimizers for the fractional Sobolev inequality on domains

- Mathematics
- 2017

We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we…

Static and dynamical, fractional uncertainty principles

- Mathematics
- 2022

We study the process of dispersion of low-regularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional…

Variational methods for nonpositive mixed local-nonlocal operators

- Mathematics
- 2022

. We prove the existence of a weak solution for boundary value problems driven by a mixed local–nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive deﬁnite.

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