Abstract
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.
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Notes
- 1.
By our conventions, we have implicitly agreed to use the same symbols V , V −1 to denote the left and right shifts of scalar (d = 1) and vector (d > 1) sequences. As a consequence, e.g., V ( Φ|ψ〉) = (V Φ)|ψ〉, illustrating how V may appear in multiples places of an equation with meanings determined by its use.
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Alase, A. (2019). Mathematical Foundations to the Generalized Bloch Theorem. In: Boundary Physics and Bulk-Boundary Correspondence in Topological Phases of Matter. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-31960-1_5
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