Refined semiclassical asymptotics for fractional powers of the Laplace operator (Q256939)

The goal of the article is to provide asymptotic development of \(\frac{1}{N}\sum_{l=1}^N\lambda^s_l\) as \(N\) goes to infinity (this development is denoted by \(\text{Tr}(H_\Omega)_-)\), the family \((\lambda_l^s)_{1\leq l\leq N}\) stands for the eigenvalues associated to the perturbed fractional Laplacian operator \(H_\Omega=(-h^2\Delta)^s-1\) defined in the set of square-integrable measurable functions on a bounded domain \(\Omega\subset\mathbb R^n\) such that the boundary \(\partial \Omega\) belongs to \(C^{1,\alpha}\) with \(\alpha\in(0,1]\) and \(h>0\). Precisely, the authors state that NEWLINENEWLINE\[NEWLINE\text{Tr}(H_\Omega)_-=L^{(1)}_{s,n}|\Omega|h^{-n}-L^{(2)}_{s,n}|\partial \Omega|h^{1-n}+R_h,NEWLINE\]NEWLINE such that \(|\cdot|\) represents the Lebesgue measure, \(L_{s,n}^{(1)}=\frac{1}{(2\pi)^n}\int_{\mathbb R^n}\big(|p|^{2s}-1\big)_-dp\), \(L^{(2)}_{s,n}\) is an explicit positive constant and it is given in terms of the \(L_1([0,\infty))\)-norm of an explicit integral function, where it is discussed comprehensively in the sixth section, and \(R_h=o(h^{1-n})\) when \(h\) is close to zero, besides \(R_h\) satisfies the property of \(h\)-continuous frame, i.e., a lower and an upper bound of \(R_h\) is given in terms of a function of \(h\) (Theorem 1.1). The proof is essentially based on showing that the expression NEWLINENEWLINE\[NEWLINE\text{Tr}\big(\psi H_\Omega\psi\big)_--L_{s,n}^{(1)}\int_\Omega[\psi(x)]^2h^{-n}dx+L_{s,n}^{(2)}\int_{\partial\Omega}[\psi(x)]^2h^{1-n}d\sigma(x)NEWLINE\]NEWLINE satisfies the property of continuous frame such that \(\psi\) is a suitable bump function with a compact support in a Euclidean ball (Proposition 2.5). The case when \(\Omega\) is substituted by the upper half-space is also treated, see the third section. In Appendix A, the authors provide an equivalent form of \(\text{Tr}(H_\Omega)_-\).