Eigenvalues with respect to a weight for general boundary value problems on networks

In this work we analyze self-adjoint boundary value problems on networks for Schrödinger operators, in which a part of the with Neumann condition is always considered. We first characterize when energy positive semi-definite space functions satisfying null conditions. To do this, fundamental tools are Doob transform and discrete version trace function. Then, raise eigenvalue respect to weight general prove Mercer Theorem. Finally, apply obtained results Dirichlet-Robin problem star-like network.