Invariants for Laplacians on periodic graphs

Mathematische Annalen. 2020. Vol. 337. P. 723–758.

Korotyaev Evgeny, Saburova N.

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite
number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of
the periodic graph, we introduce a subclass of minimal forms with a minimal number
I of edges in their supports on the period. We obtain a specific decomposition of the
Laplacian into a direct integral in terms of minimal forms, where fiber Laplacians
(matrices) have the minimal number 2I of coefficients depending on the quasimomentum
and show that the number I is an invariant of the periodic graph. Using this
decomposition, we estimate the position of each band, the Lebesgue measure of the
Laplacian spectrum and the effective masses at the bottom of the spectrum in terms
of the invariant I and the minimal forms. In addition, we consider an inverse problem:
we determine necessary and sufficient conditions for matrices depending on the
quasimomentum on a finite graph to be fiber Laplacians.Moreover, similar results for
Schrödinger operators with periodic potentials are obtained.

Ключевые слова: графы, graphs, инварианты, Invariants, Laplacian, Лапласиан