We consider how distribution of eigenvalues depends on boundary conditions of a discrete Laplacian operator on lattices. We study the Laplacian with boundary conditions given by a linear combination of Dirichlet and Neumann conditions. In particular, we derive a secular equation and investigate the Laplacian operator's eigenvalues with different boundary conditions, including the interlacing property, the first eigenvalue gaps, and the monotonicity property.
"Boundary Effects on Eigen-problems of Discrete Laplacian in Lattices." Taiwanese J. Math. Advance Publication 1 - 24, 2022. https://doi.org/10.11650/tjm/220202