On discrete graphs with purely discrete spectrum, the summed eigenvalue difference between the Laplacian with and without a potential equals the weighted sum of the potential.
Spectral asymptotics for Robin Laplacians on Lipschitz sets
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abstract
We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.
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Spectral comparison results for Laplacians on discrete graphs
On discrete graphs with purely discrete spectrum, the summed eigenvalue difference between the Laplacian with and without a potential equals the weighted sum of the potential.