Abstract
In this chapter, we look at some examples of applications of the results of Chap. 2. These include in particular self-adjoint operators with non-degenerate spectral band edges, operators with Dirac points in dispersion relation, as well as some non-self-adjoint cases.
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Notes
- 1.
Note that for each k ∈ V r ∖{k r}, λ r,j(k) ≠ 0 since \(V_{r} \cap F_{A, \mathbb {R}}=\{k_{r}\}\).
- 2.
In general, the converse of this statement is not true: e.g., consider A ∗ in this case then the zeroth-order coefficient of the transpose A ∗ is not necessarily nonnegative while \(\varLambda _{A^*}(0)=\varLambda _A(0) \geq 0\).
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Kha, M., Kuchment, P. (2021). Specific Examples of Liouville-Riemann-Roch Theorems. In: Liouville-Riemann-Roch Theorems on Abelian Coverings. Lecture Notes in Mathematics, vol 2245. Springer, Cham. https://doi.org/10.1007/978-3-030-67428-1_4
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