Weyl's law for Neumann Schrödinger operators on Hölder domains (Q6566027)

In this survey paper, based on the author's paper [``Semiclassical estimates for Schrödinger operators with Neumann boundary conditions on Hölder domains'', Preprint, \url{arXiv:2304.01587}], the main points of her own results on the Weyl asymptotic expansion formula for the number of eigenvalues of a Schrödinger operator defined on a bounded \(\gamma\)-Hölder domain under Neumann boundary conditions are given. As a byproduct, it also explains that another proof of the Weyl asymptotic expansion formula for the Neumann Laplacian on a bounded \(\gamma\)-Hölder domain obtained by \textit{Y. Netrusov} and \textit{Y. Safarov} [Commun. Math. Phys. 253, No. 2, 481--509 (2005; Zbl 1076.35085)] can be obtained.\N\NFirst, the results of Rayleigh, Sommerfeld, Lorentz, Weyl, Rozenblum, Kac and Ivrii, etc. about the famous Weyl asymptotic expansion formula for the Dirichlet Laplacian are mentioned. Next, it is explained that the situation for the Neumann Laplacian is quite different from that for the Dirichlet Laplacian, citing the Hempel-Seco-Simon result [\textit{R. Hempel} er al., J. Funct. Anal. 102, No. 2, 448--483 (1991; Zbl 0741.35043)] that zero belongs to the essential spectrum in case of the Neumann Laplacian.\N\NThen, Netrusov-Safarov's Neumann-Laplacian results are introduced. That is, under the assumption that a bounded open set \(\Omega\subset\mathbb{R}^d\) (\(d\geqq2\)) is a \(\gamma\)-Hölder domain, two results are shown: \((i)\) the Weyl asymptotic expansion formula holds for all \(\gamma\) satisfying \(\frac{d-1}{d}<\gamma<1\), and \((ii)\) if \(\gamma\) satisfies \(0<\gamma\leqq \frac{d-1}{d}\), then there exists a \(\gamma\)-Hölder domain \(\Omega\) that does not satisfy the Weyl asymptotic expansion formula. In this theorem, \(\Omega\) is called a \(\gamma\)-Hölder domain if it is a domain and \(\partial\Omega\), the boundary of \(\Omega\), is locally the graph of a \(\gamma\)-Hölder continuous function \(f\) satisfying \(\left|f(x)-f(y)\right|\leqq c\left|x-y\right|^{\gamma}\) for all \(x\) and \(y\) in the domain of \(f\) and for some \(c>0\).\N\NIn connection with the above result, the author provides two interpretations of the threshold \(\frac{d-1}{d}\) of \(\gamma\), the Hausdorff dimension of the boundary \(\partial\Omega\) and the semi-classical theory.\N\NAfter mentioning the key points of the proof of their results, the main results obtained by the author herself are finally stated.\N\NThe author's own theorem can be stated using some of the necessary notation as follows. The Neumann Laplacian defined on a bounded domain \(\Omega\left(\subset\mathbb{R}^d\right)\) is denoted by \(-\Delta^N_{\Omega}\), the volume of the unit ball in \(\mathbb{R}^d\) is denoted by \(\left|B^d_1(0)\right|\), and the number of eigenvalues of operator \(A\) is denoted by \(N(A)\).\N\NThe required condition on the potential function \(V(x)\) is then as follows:\N\NCondition \((V)\) \(\colon\) Let \(V(x)\) be a non-positive-valued measurable function defined on \(\Omega\).\N\NFirst, the results for the so-called Cwikel-Lieb-Rozenblum-type bound are as follows. Let \(\Omega\) be a \(\gamma\)-Hölder domain. Under the assumptions that the space dimension \(d\) satisfies \(d\geqq 2\) and \(\gamma\in\left[\frac{2(d-1)}{2d-1} , 1\right)\), there exists a positive constant \(C=C\left(d,\gamma,\Omega\right)\) and \(p_{d,\gamma}\left(>\frac{d}{2}\right)\) such that for every \(V\) satisfying \((V)\) and \(V\in L^{p_{d,\gamma}}\), it holds that \N\[\NN\left(-\Delta^N_{\Omega}+ V\right)\leqq C_{\Omega}\left(1 + \left|\left|V\right|\right|^{\frac{d}{2}}_{p_{d,\gamma}}\right).\N\]\NMoreover, \(\lim_{\gamma\to1}p_{d,\gamma}=\frac{d}{2}\).\N\NThen, by using this result, Weyl law for Schrödinger operators on \(\gamma\)-Hölder domain is proved in the following form:\N\NAssume \(d\geqq 2\), and \(\gamma\in\left(\frac{2(d-1)}{2d-1},1\right)\). Also assume that \(\Omega\) is bounded, \(\gamma\)-Hölder domain, and \(V\) satisfies \((V)\) and \(V\in L^{\frac{d}{2}}\). Then the following asymptotic formula holds: \N\[\NN\left(\Delta^N_{\Omega}+ \lambda V\right)=\frac {|B^d_1(0)|}{(2\pi)^d}{\lambda}^{\frac{d}{2}}\int_{\Omega}|V(x)|^{\frac{d}{2}}dx+o\left({\lambda}^{\frac{d}{2}}\right) \qquad \text{as} \qquad \lambda \to \infty.\N\]\NHere, the range of \(\gamma\) in this result is narrower than in the result by Netrusov-Safarov. However, it is explained that it is possible to recover the range of \(\gamma\) in their result by assuming \(V\) to be a constant potential and replacing the \(L^{\frac{d}{2}}\)-norm of the integral on the right-hand-side with some weighted \(L^{\tilde{p}}\)-norm in the author's own theorem above, thus their result itself is obtained again. Here, \(\tilde{p}\) is given by \N\[\N\tilde{p}=\frac{1}{2d}\left(\frac{d-1}{\gamma}+1\right)^2\left(>\frac{d}{2}\right).\N\]\NAccording to Netrusov-Safarov result, the proof involves covering the domain \(\Omega\) with a smaller domain called an oscillatory domain, so that the corresponding Schrödinger operator has only one negative eigenvalue on each oscillatory domain. Furthermore, it is mentioned that the proof of the theorem requires a new covering theorem (which is needed for handling near the boundary \(\partial\Omega\)), Sobolev inequality for oscillatory domains, Poincarê inequality for oscillatory domains with Neumann boundary conditions, Poincarê-Sobolev inequality for oscillatory domains, etc. (which were newly derived by the author himself), in addition to the Besicovitch covering lemma used in Netrusov-Safarov result.