Refined semiclassical asymptotics for fractional powers of the Laplace operator

Abstract

Abstract. 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 the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.

Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain

“…This completes the proof of (35). Next, we discuss the localization error coming from (12). We claim that…”

Regularity of spectral fractional Dirichlet and Neumann problems

“…The properties of the restricted Dirichlet fractional Laplacian defined in the introduction were studied e.g. in Blumenthal and Getoor , Landkof , Hoh and Jacob , Kulczycki , Chen and Song , Jakubowski , Silvestre , Caffarelli and Silvestre , Frank and Geisinger , Ros‐Oton and Serra , , Felsinger, Kassmann and Voigt , Grubb , , Bonforte, Sire and Vazquez , Servadei and Valdinoci , Binlin, Molica Bisci and Servadei , and many more papers referred to in these works (see in particular the list in ).…”

Refined eigenvalue bounds on the Dirichlet fractional Laplacian

“…We know that 0 ≤ −ϕ k (t) ≤ β for t ≥ 0 where β > 0 is given by (19). Since ϕ k (0) > 0 due to (14) let us first define…”

“…Note that we use the same notation for eigenvalues and eigenfunctions to illuminate the striking similarities though they might be different for each E f . Defining Φ k as in (14), we obtain (15) immediately. However, (16) needs to be re-written as the following inequality…”

“…One may consult the articles [2,3,4,14,17,20,24,25,30] and references therein for a through literature review. It is worth pausing here for a moment to consider the Dirichlet Laplacian results relevant to our main result.…”