Quantum / CONCEPT

Middle-edge spectral weights, integer moments and Gram identity of even paths with end transfer at time pi

Let H be the Hamiltonian of a weighted path on the 2k + 2 vertices 0, ..., 2k + 1, with positive edge weights and real potentials, whose propagator exp(i pi H) has the entry 1 at the last and first vertices. Then the eigenvalues of H are distinct integers that are even on a class A of k + 1 indices and odd on the other k + 1, the eigenvectors of the odd class have opposite entries at the two middle vertices a = k and b = k + 1, and the eigenvector of index j in A satisfies |W(a, j) + W(b, j)|^2 = prod_{l not in A} (z j - z l) / (rho prod_{l in A, l != j} (z j - z l)), where rho is the weight of the middle edge. With S p the sum of the four entries of H^p at a and b, the number rho S p is the integer coefficient of X^k in the remainder of X^p P_B modulo P_A, for the nodal polynomials P_A and P_B of the two classes, and S 0 S 2 - (S 1)^2 = 4 r'^2 for the weight r' of the edge following the middle edge.