Quantum / CONCEPT

Middle-vertex spectral weights and integer moments of odd paths with end transfer at time pi

Let H be the Hamiltonian of a weighted path on the 2m + 1 vertices 0, ..., 2m, with positive edge weights and real potentials, whose propagator exp(i pi H) has the entry 1 at the last and first vertices. Then H is invariant under the reversal of the vertices, its eigenvalues are distinct integers that are even on a class A of m + 1 indices and odd on the other m, the eigenvectors of the odd class vanish at the middle vertex c = m, and the eigenvector of index k in A has squared modulus prod_{l not in A} (z k - z l) / prod_{l in A, l != k} (z k - z l) at c. The diagonal entries (H^p)(c, c) are integers congruent to the binomial coefficients C(m, p) modulo 2.