Equivalence of operator-ideal and Weyl-coefficient criteria in the overlapping order range

Show, by direct computation, whether the conditions obtained via the operator-ideal method for convergence class and type (expressed in terms of integrals of directional entries for diagonal Hamiltonians) are equivalent to the conditions obtained via the Weyl-coefficient approach (expressed in terms of the integral of K_H(t;r)) in the overlapping range of orders between 1 and 2.

Background

Two complementary approaches yield criteria for dense and sparse spectrum: the operator-ideal method (Part I) provides integral and sequential conditions on the Hamiltonian’s directional entries, while the Weyl-coefficient method (Part I) expresses the Stieltjes transform of the eigenvalue-counting function in terms of an explicit kernel K_H. In the intermediate order regime (between 1 and 2), both sets of criteria apply.

Establishing equivalence would unify the theories and possibly lead to streamlined criteria that are more directly computable from Hamiltonian data.