Partitioning technique, perturbation theory, and rational approximations

Instead of the Schródinger equation ℋΨ = E Ψ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (ℋ − ℰ)Ψ ℰ = a φ, where ℰ is a complex variable, with the same boundary condition, which gives a = 〈φ|ℋ|Ψ ℰ 〉 − ℰ = ℰ 1 − ℰ. Introducing the projector P = 1 − |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψ ℰ = (1 − P ℋ/ℰ) −1 φ = (1 + T ℰ ℋ)φ, where T ℰ = (ℰ − P ℋ) −1 P = P (ℰ − P ℋ P ) −1 P is the reduced resolvent associated with the auxiliary Hamiltonian H̄ = P ℋ P . The existence of these operators is discussed. It is shown that, if the parameter ℰ is real in the “discrete part” of the spectrum to ℋ, then ℰ and ℰ 1 = 〈φ|ℋ|Ψ ℰ 〉 = 〈φ|ℋ + ℋ T ℰ ℋ|Φ〉 ≡ f (ℰ) bracket a true eigenvalue E satisfying the relation E = f ( E ). The Newton‐Raphson solution to the equation F (ℰ) = ℰ − f (ℰ) = 0 is related to the variation principle. Putting ℋ = ℋ 0 + V and expanding the inverse (ℰ − P ℋ 0 − PV ) −1 in terms of powers of V or ( V − α), one gets various expansions relating to finite‐order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = { h 1 , h 2 ,…, h n } is a linearly independent set, the inner projection Á n = | h 〉 〈 h | A −1 | h 〉 −1 〈 h | is a “rational approximation” to the operator A which converges toward A when n →∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.