Perturbation Theory of Product Hamiltonians through Fourth Order

Abstract

Expressions are derived through fourth order in a $V$ representation for the eigenvalues and off‐diagonal elements of Hamiltonians expressible as sums of products of operators from two Hilbert spaces ($V$ and $R$ ), $$\mathbf{H}{=}^0{\mathbf{H}}_V{+}^0{\mathbf{H}}_R+{\displaystyle \sum _p}({\mathbf{H}}_V{\mathbf{H}}_R{)}_p$$ . The zeroth order Hamiltonian is assumed separable, and the eigenvalue differences in the $V$ space are assumed to be an order of magnitude larger than the eigenvalue differences in the $R$ space. The method involves successive contact transformations chosen to yield results in terms of matrix elements in the $V$ space and operators in the $R$ space. The technique allows for the exclusion of interactions between resonant states in the $V$ space for subsequent numerical diagonalization.