Renormalization of a Finite Matrix Hamiltonian
Asher Peres · Journal of Mathematical Physics · 1969
We investigate the eigenvalues of a finite matrix Hamiltonian H = H0 + g0V, where H0 is diagonal with eigenvalues 1, 2, …, N, and where all the elements of V are equal to 1. We are interested in the case N → ∞. The radius of convergence of the perturbation series is (ln N)−1, but nevertheless the exact eigenvalues of H tend to well-defined limits when N → ∞. It is shown that if we define g=(g0−1+lnN)−1 and if we let g0 → 0 as N → ∞ in such a way that g is constant, then it is possible to obtain a perturbation series with the ``renormalized'' coupling constant g, provided that suitable counter terms are introduced. We also investigate a different model (where Vmn = mn) and show that no such renormalization is possible there.