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The Laplacian matrix is the representation of a graph in matrix form.
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Regularity of the eigenfunctions associated to perturbed laplacian on a compact manifold
Let $M$ be a closed manifold, I consider first order laplacian perturbation associated to a density $\rho \in \mathcal{C}^\infty(M)$ with $\rho > 0$ of the form :
$$
\Delta_{\rho} f = \Delta f + \langle … rangle.
$$
It is well known that if $\rho =1$ we have an hilbert basis of $L^2(dx)$ given by the eigenfunctions $(\phi_i)_i$ such that
$$
\Delta \phi_i = - \lambda_i \phi_i.
$$
If we take the perturbed laplacian …