# 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}
}```
• R. Frank
• Published 31 March 2016
• Mathematics
• arXiv: Spectral Theory
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.
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• 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\$.
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• 2018
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Variational methods for nonpositive mixed local-nonlocal operators
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## References

SHOWING 1-10 OF 106 REFERENCES
Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces
Abstract We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L 2 ( Ω ) (or L 2 ( Ω ,  C m
A short proof of Weyl's law for fractional differential operators
We study spectral asymptotics for a large class of differential operators on an open subset of Rd with finite volume. This class includes the Dirichlet Laplacian, the fractional Laplacian, and also
ESTIMATES FOR THE SUMS OF EIGENVALUES OF THE FRACTIONAL LAPLACIAN ON A BOUNDED DOMAIN
• Mathematics
• 2013
The purpose of this paper is two-fold. Firstly, we state and prove a Berezin–Li–Yau-type estimate for the sums of eigenvalues of , the fractional Laplacian operators restricted to a bounded domain Ω
Real analyticity of solutions to Schr\"odinger equations involving fractional Laplacians
• Mathematics
• 2012
We prove analyticity of solutions in \$\mathbb{R}^{n}\$, \$n\ge1\$, to certain nonlocal linear Schr\"odinger equations with analytic potentials.
On Fractional Laplacians
• Mathematics, Philosophy
• 2014
We compare two natural types of fractional Laplacians (− Δ) s , namely, the “Navier” and the “Dirichlet” ones. We show that for 0 < s < 1 their difference is positive definite and positivity
Refined Semiclassical Asymptotics for Fractional Powers of the Laplace Operator
• Mathematics
• 2011
We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for
Equivalence of Sobolev inequalities and Lieb-Thirring inequalities
• Mathematics
• 2010
We show that, under very general definitions of a kinetic energy operator T, the Lieb–Thirring inequalities for sums of eigenvalues of T - V can be derived from the Sobolev inequality appropriate to
Critical Lieb-Thirring bounds for one-dimensional Schrodinger operators and Jacobi matrices with regular ground states
This paper has been withdrawn by the author in favor of a stronger result proven by the author with R. Frank and T. Weidl in arXiv:0707.0998
An Extension Problem Related to the Fractional Laplacian
• Mathematics
• 2007
The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the