Polynomial splines and eigenvalue approximations on quantum graphs

A notion of splines is introduced on a quantum graph Γ. It is shown that eigen values of a Hamiltonian on a finite graph Γ can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polynomial splines on Γ. In particular, a bounded set of eigenvalues can be determined using a space of such polynomial splines with a fixed set of singularities. It is also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polynomial splines with appropriate fixed set of singularities.