In this chapter we introduce the spectral approach to delocalization in infinite disordered systems and provide a physical interpretation in context of the classical models discussed in Chap. 2. Here we argue that the spectral method offers an important contribution to transport theory since it avoids issues related to the use of periodic boundary conditions and finite-size scaling. Section 3.1 provides a simplified mathematical formulation of the spectral approach, which utilizes the concepts of cyclicity and measure decomposition. The relevant mathematical terminology is summarized in Appendix A. In Sect. 3.2, the proposed method is applied to 2D and 3D disordered lattices using proof-of-concept numerical simulations. The preliminary results from these simulations clearly indicate the existence of a metal-to-insulator transition in the critical two-dimensional case. Section 3.3 provides a physical interpretation of the spectral approach, which aims to connect methods in spectral theory to concepts in quantum mechanics. Finally, the limitations of the spectral approach are discussed in Sect. 3.4.
KeywordsSpectral approach Spectral theorem Measure decomposition Cyclicity Self-adjoint operators
- 1.T. Brandes, S. Kettenmann, The Anderson Transition and Its Ramifications—Localization, Quantum Interference, and Interactions (Springer, 2003)Google Scholar