Mathematical Foundations to the Generalized Bloch Theorem
This chapter presents the mathematical results that were used in establishing the generalized Bloch theorem in Chap. 2. We work directly with possibly non-Hermitian block-Toeplitz matrices, so as to keep the formalism as general as possible. There are two main takeaways as part of the proof of the generalized Bloch theorem: First, a simple yet effective separation of the time-independent Schrödinger equation into bulk and boundary equations is what really allows us to capture the exact interplay between the bulk and the BCs; second, the Smith normal form of matrix polynomials emerges as a natural tool in the treatment of systems with boundary.
KeywordsNon-Hermitian banded block-Toeplitz matrices Smith normal form Bulk-boundary correspondence Lattice models
- 4.M. Fardad, The operator algebra of almost Toeplitz matrices and the optimal control of large-scale systems, in 2009 American Control Conference (IEEE, Piscataway, 2009), pp. 854–859Google Scholar