Essential spectrum of the Laplacian for the Neumann problem in a model region of complicated structure
The essential spectrum \(\sigma_{\text{ess}}(-\Delta^N_\Omega)\) is studied for the Neumann problem for the Laplace operator in a domain \(\Omega\) consisting of rectangles \(\Omega_i\) with the length \(a_i\) and width \(b_i\) connected by segments \(P_i\) with length \(l_i\) and width \(\alpha_i\). The domain \(\Omega\) is supposed to be bounded, i.e. \(\sum^\infty_{i=1}(a_i+b_i)<\infty\) and the sequence \(\{b_i\}^\infty_{i=1}\) is bounded. Under these conditions the following statements are valid: 1) if \(\alpha_i=o(a^2_il^{-1/2}_i,a^{1/2}_il^{3/2}_i)\) then \(\sigma_{\text{ess}}(-\Delta^N_\Omega)= (\bigcup^\infty_{i=1}\sigma(-\Delta^N_{\Omega_i}))^{cl}= (\bigcup^\infty_{n=1}\bigcup^\infty_{m=1} (\pi_m/b_m)^2)^{cl}\); 2) if \(\sum^\infty_{i=1}\alpha_i(a^{-2}_il^{1/2}_i+a^{1/2}_il^{-3/2}_i)<\infty\) then the absolutely continuous spectrum is \(\sigma_{ac}(-\Delta^N_\Sigma)=0\).
scientific article; zbMATH DE number 907285