The spectral structure of the electronic black box Hamiltonian (Q372943)

Let \(\mathcal{H}=\mathcal{H}_l \oplus \mathcal{H}_S \oplus \mathcal{H}_r\) be a finite dimensional Hilbert space \(\mathcal{H_S}\) coupled to a left and a right reservoir with Hilbert spaces \(\mathcal{H}_l\) and \(\mathcal{H}_r\). Given \(\chi_l \in \mathcal{H}_l\), \(\chi_r \in \mathcal{H}_r\), and \(\delta_l, \delta_r \in \mathcal{H}_S\), this paper considers the Hamiltonian \(H_{\lambda,\nu}\) defined as \(H_{\lambda,\nu}=H_0+\lambda[(\chi_l,\cdot)\delta_l+(\delta_l,\cdot)\chi_l]+\nu[(\chi_r,\cdot)\delta_r+(\delta_r,\cdot)\chi_r]\) , where \(H_0\) is a non-interacting Hamiltonian on \(\mathcal{H}\) and \(\lambda,\nu \in \mathbb{R}\) are control parameters. Particularly, it addresses the absence of singular continuous spectrum of \(H_{\lambda,\nu}\) as it is crucial for the rigorous derivation of the Landauer-Büttiker formula in transport theory.