Eigenvalue bounds for the fractional Laplacian: A review

  title={Eigenvalue bounds for the fractional Laplacian: A review},
  author={Rupert L. Frank},
  journal={arXiv: Spectral Theory},
  • 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. 
Kroger's upper bound types for the Dirichlet eigenvalues of the fractional Laplacian.
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
On the asymptotic behavior of $p$-fractional eigenvalues
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
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
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
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
Minimizers for the fractional Sobolev inequality on domains
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
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
. 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.


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
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
We prove analyticity of solutions in $\mathbb{R}^{n}$, $n\ge1$, to certain nonlocal linear Schr\"odinger equations with analytic potentials.
On Fractional Laplacians
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
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
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
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