###### Abstract  
We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number \(\mu\), approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a \(\tau\) preconditioning when the variable coefficient wave number \(\mu\) is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented. C pacs: 03.67.-a, 03  
for every \(i,j\in\mathbb{Z}\). The discrete version of the fractional Laplacian in such a setting is given by  
\[(-\Delta_{h})^{\alpha/2}(u(x,y)):=\frac{1}{h^{\alpha}}\sum_{k_{1},k_{2}\in \mathbb{Z}}b^{(\alpha)}_{k_{1},k_{2}}u(x+k_{1}h,y+k_{2}h), \tag{2}\]  
where \(b^{(\alpha)}_{k_{1},k_{2}}\) are the Fourier coefficients of the function  
\[t_{\alpha}(\eta,\Psi)=\left[4\sin^{2}\Bigl{(}\frac{\eta}{2}\Bigr{)}+4\sin^{2} \biggl{(}\frac{\Psi}{2}\biggr{)}\right]^{\frac{6}{2}}, \tag{3}\]  
that is  
\[b^{(\alpha)}_{k_{1},k_{2}}=\frac{1}{4\pi^{2}}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi }t_{\alpha}(\eta,\Psi)e^{-i(k_{1}\eta+k_{2}\Psi)}d\eta d\Psi,\]  
where i is the imaginary unit. Proceeding as in [1] we trace back the original problem to solving the following linear system  
\[A_{\mathbf{n}}u:=(B_{\mathbf{n}}+D_{\mathbf{n}}(\mu))u=f,\quad\mathbf{n}=(n,n), \tag{4}\]  
where \(B_{\mathbf{n}}=\frac{1}{\mu! }\,\hat{B}_{\mathbf{n}}\), and \(\hat{B}_{\mathbf{n}}\) is the two-level symmetric Toeplitz matrix generated by \(t_{\alpha}(\eta,\Psi)\), i.e., \(B_{\mathbf{n}}=T_{\mathbf{n}}(t_{\alpha})\) with  
\[B_{\mathbf{n}}=\left[ \begin{array}{cccccc}B_{0}&B_{1}&B_{2}&\cdots&B_{n-2}& B_{n-1}\\ B_{1}&B_{0}&B_{1}&\cdots&B_{n-3}&B_{n-2}\\ B_{2}&B_{1}&B_{0}&\cdots&B_{n-4}&B_{n-3}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ B_{n-2}&B_{n-3}&B_{n-4}&\cdots&B_{0}&B_{1}\\ B_{n-1}&B_{n-2}&B_{n-3}&\cdots&B_{1}&B_{0}\end{array} \right],\]  
\[B_{j}=\left[ \begin{array}{cccccc}b^{(\alpha)}_{0,j}&b^{(\alpha)}_{1,j}&b^{( \alpha)}_{2,j}&\cdots&b^{(\alpha)}_{n-2,j}&b^{(\alpha)}_{n-1,j}\\ b^{(\alpha)}_{1,j}&b^{(\alpha)}_{0,j}&b^{(\alpha)}_{1,j}&\cdots&b^{(\alpha)} _{n-3,j}&b^{(\alpha)}_{n-2,j}\\ b^{(\alpha)}_{2,j}&b^{(\alpha)}_{1,j}&b^{(\alpha)}_{0,j}&\cdots&b^{(\alpha)} _{n-4,j}&b^{(\alpha)}_{n-3,j}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ b^{(\alpha)}_{n-2,j}&b^{(\alpha)}_{n-3,j}&b^{(\alpha)}_{n-4,j}&\cdots&b^{( \alpha)}_{0,j}&b^{(\alpha)}_{1,j}\\ b^{(\alpha)}_{n-1,j}&b^{(\alpha)}_{n-2,j}&b^{(\alpha)}_{n-3,j}&\cdots&b^{( \alpha)}_{1,j}&b^{(\alpha)}_{0,j}\end{array} \right].\]  
For the sake of simplicity, the previous equation is rewritten in the following scaled form  
\[\hat{A}_{\mathbf{n}}u:=(\hat{B}_{\mathbf{n}}+l^{\alpha}D_{\mathbf{n}}(\mu))u =v,\quad\mathbf{n}=(n,n). \tag{5}\]  
For the two-level notations and the theory regarding Toeplitz structures, refer to [3]. In the case where \(\mu(x,y)=1/(x+\mathrm{i}y)^{\gamma}\) we can give sufficient conditions on the coefficient \(\gamma\), depending on \(\alpha\), in order to guarantee that \(\{l^{\alpha}D_{\mathbf{n}}(\mu)\}_{\mathbf{n}}\) is zero distributed in the eigenvalue/singular value sense, thus obtaining the spectral distribution of the sequence \(\{\hat{A}_{\mathbf{n}}\}_{\mathbf{n}}\) which, under mild conditions, has to coincide with that of \(\{\hat{B}_{\mathbf{n}}\}_{\mathbf{n}}\). In the next section, we first introduce the necessary tools and then present theoretical results completing those in [1; 2] and related numerical experiments. The numerical experiments concern the visualization of the distribution/clustering results and the optimal performances of the related preconditioning when the preconditioned GMRES is used. We highlight that the spectral analysis for the considered preconditioned and nonpre-conditioned matrix-sequences for unbounded \(\mu(x,y)\) is completely new.