Effective Schrödinger equations for nonlocal and/or dissipative systems

Arnold Neumaier

Institut für Mathematik, Universität Wien Strudlhofgasse 4, A-1090 Wien, Austria email: Arnold.Neumaier@univie.ac.at WWW: http://www.mat.univie.ac.at/∼neum/

January 14, 2002

Abstract.

The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential.

It is also shown how to recover all solutions of the time-independent Schrödinger equation in terms of solutions of the effective Schrödinger equation in the reduced state space and a Schrödinger equation in a reference state space.

For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.

Keywords: Feshbach projection, effective Hamiltonian, nonlocal Schrödinger equation, Bethe-Salpeter equation, coupled reaction channels, resonating group method, dissipative quantum system, optical potential, form factor, doorway operator, time-independent perturbation theory, backward error analysis, multichannel scattering

E-print Archive No.: hep-th/0201085

1998 PACS Classification: primary: 03.65.-w secondary: 31.15.Dv

1 Introduction

In many applications, a quantum system of interest is part of a much bigger system, and the latter’s state influences the system state. If the big system is represented by solutions of a Schrödinger equation in a big state space, it is desirable to find an effective Schrödinger equation in a small state space that describes how the small system of interest is affected by the embedding into the big system. Under certain conditions, a Schrödinger equation in the big state space can indeed be reduced to an effective Schrödinger equation in the small state space, in such a way that the interesting part of any solution of the full Schrödinger equation satisfies exactly the effective Schrödinger equation.

Much effort has gone into solving this reduction problem, and in a sense it is well understood [1, 6, 13, 17, 28]. Exact expressions for the effective Hamiltonian can be given. In its exact form, the effective Hamiltonian is energy dependent and usually acquires a nonhermitian part; its eigenvalues describe the bound states and resonances of the reduced system. There are approximation schemes that compute the effective Hamiltonian (at least in principle) to arbitrary accuracy.

Less known (but proved here) is that certain solutions of the full Schrödinger equation can be reconstructed from solutions of the effective Schrödinger equation, using little more than what is already available from the reduction process.

The reduction is usually done for bound state calculations by the variational principle discussed in every textbook on quantum mchanics. For resonance calculations, the reduction may be done within the Feshbach [7, 9] projection formalism (for an exposition see, e.g., Kukulin et al. [17, Chapter 4]). In both cases, the reduced state space is finite-dimensional. For the calculation of scattering states, the reduction may be done by means of coupled reaction channel equations, also called the resonating group method; a nice exposition is given in Wildermuth & Tang [28]. In this case, the reduced state space is a direct sum of finitely many function spaces, one for each energetically admissible arrangement of particles into clusters, with states parameterized by coordinates or momenta of cluster centers only.

The coupled reaction channel equations are numerically easy to handle, but the approximations involved in their derivation make an assessment of their accuracy difficult. On the other hand, the projection formalism gives in principle exact results, limited in accuracy only by the approximations made in the calculation of the effective Hamiltonian. However, this involves projection operators, which are clumsy to use if an orthogonal basis is not easily available.

Moreover, if the full Hamiltonian is already nonhermitian, or if the Hamiltonian is energy-dependent (such as in relativistic calculations; cf. [8, 14, 15, 16, 12, 18, 19, 22, 26]), the standard derivation of the effective Hamiltonian is no longer valid. In the following, we shall remedy both defects.

The paper is organized as follows. We first extend the projection formalism such that it applies to the reduction of nonlocal systems and of dissipative systems modeled by an optical potential. Thus we work throughout with complex symmetric Hamiltonians, introduced in Section 2, and generalize in Sections 3 and 4 the traditional Feshbach projection approach to this more general situation. We then show in Section 5 that the formalism in fact allows to recover all solutions of the time-independent Schrödinger equation in terms of solutions of two Schrödinger equations, one in the reduced state space and the other in a reference state space. The resulting formulas are closely related to those of time-independent perturbation theory.

For practical calculations, the resulting formalism is revised in Section 6 so that it can be used without computing any projection operators. Section 7 shows how the use of doorway operators gives flexible approximation schemes for the exact formulas derived in Section 6. A backward error analysis discussed in Section 8 allows the estimation of the reliability of approximate solutions of Schrödinger equations obtained by this or any other method.

Specific choices of the embedding map lead in Section 9 to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information, in which solutions of the full Schrödinger equation can be obtained from solutions of coupled reaction channel equations for the effective Hamiltonian. In principle it is capable of arbitrarily accurate approximations to the full dynamics, and shares this feature with the two Hilbert space method of Chandler & Gibson [3, 4], which partly inspired the present investigations.

The theory is presented in a fully rigorous manner, allowing for unbounded operators by using in place of Hilbert spaces a pair of dual topological vector spaces. The most useful results are in Sections 6, 7, and 9.

The reader interested in the results but not in mathematical rigor may omit all references to spaces and topology, may think of all spaces as finitedimensional and of operators as matrices, and may skip all proofs. In particular, of Section 2 introducing the basic terminology, only (1)–(3) are essential, and most of Sections 3–5 can be skimmed.

2 Symmetric operators

Lin(𝕍, 𝕎) denotes the space of continuous linear mappings between two topological vector spaces 𝕍 and 𝕎, 𝕍^{∗} = Lin(𝕍, ℂ) denotes the dual space of continuous, complex-valued linear functionals on 𝕍, and Lin 𝕍 = Lin(𝕍, 𝕍) denotes the algebra of continuous linear transformations of 𝕍.

In the following, ℍ is a complex topological vector space with a definite, continuous symmetric bilinear form, providing a natural embedding of ℍ into the dual space ℍ^{∗}. We refer to ℍ^{∗} as a state space, and to the ψ ∈ ℍ^{∗} as states. We write the pairing and the bilinear inner product as

φ^{T} ψ = ψ^{T} φ for φ ∈ ℍ, ψ ∈ ℍ^{∗}.

The notation is chosen such that it looks as closely as possible like standard finite-dimensional linear algebra.

We say that a sequence (or net) ψ_{l} ∈ ℍ^{∗} converges weakly to ψ ∈ ℍ^{∗}, and write ψ_{l} ⇀ ψ, if

lim_{l→∞} φ^{T} ψ_{l} = φ^{T} ψ∀φ ∈ ℍ.

We extend the bilinear inner product to arbitrary pairs (φ, ψ) ∈ ℍ^{∗} × ℍ^{∗} for which

φ^{T} ψ = lim_{l→∞} φ_{l}^{T} ψ_{l}

is defined and independent of the weakly converging sequences (or nets) φ_{l} ⇀ φ, ψ_{l} ⇀ ψ of φ_{l}, ψ_{l} ∈ ℍ. (In the applications, this allows to form the inner product of state vectors corresponding to bound states and resonances but not that of scattering states.)

Complex conjugation is denoted by a bar, and, with the notation ψ^{∗} = ψ̄^{T}, the Hermitian inner product on ℍ is

⟨φ|ψ⟩ = φ^{∗}ψ for φ, ψ ∈ ℍ.

The associated Euclidean norm is

∥ψ∥ = \sqrt{ψ^{∗}ψ},

and ℍ̄ is the closure of ℍ in ℍ^{∗} with respect to the Euclidean norm. Thus ℍ ⊆ ℍ̄ ⊆ ℍ^{∗}, and ℍ̄ is a Hilbert space.

In the applications, ℍ is a space of sufficiently nice functions (namely arbitrarily often differentiable, with compact support) on some finite- or infinitedimensional manifold, the inner product of two functions is some integral of their pointwise product induced by a nonnegative measure on the manifold, and ℍ ⊆ ℍ̄ ⊆ ℍ^{∗} is a Gelfand triple (or rigged Hilbert space). (For these concepts, see Gelfand & Vilenkin [10], Maurin [21]. For an exposition in physicists’ terms see Kukulin [17, Appendix A]; cf. also Böhm [2].)

The transpose of a linear operator A ∈ Lin(ℍ, ℍ^{∗}) is the linear operator A^{T} ∈ Lin(ℍ, ℍ^{∗}) defined by

(A^{T} φ)^{T} ψ = φ^{T} Aψ for φ, ψ ∈ ℍ,

A^{∗} = Ā^{T} defines the adjoint of A, and

Re A = \frac{1}{2}(A + A^{∗}), Im A = \frac{1}{2i}(A − A^{∗})

define the real and imaginary part of A. Clearly, (AB)^{T} = B^{T} A^{T} and (AB)^{∗} = B^{∗}A^{∗}. The operator A ∈ Lin ℍ is called symmetric if A^{T} = A on ℍ, Hermitian if A^{∗} = A on ℍ, and positive semidefinite if

ψ^{∗}Aψ ≥ 0 for all ψ ∈ ℍ.

In particular, A is Hermitian if and only if Im A = 0. We extend symmetric operators A ∈ Lin ℍ to Lin ℍ^{∗} by defining Aψ ∈ ℍ^{∗} for ψ ∈ ℍ^{∗} by

φ^{T} Aψ = (Aφ)^{T} ψ∀φ ∈ ℍ.

In the following, L ∈ Lin ℍ is always a symmetric operator such that

Im L is positive semidefinite.  (1)

(In particular, this includes the case where L is Hermitian since then Im L = 0.)

Since ψ^{∗}Lψ = ψ^{∗}(Re L)ψ + iψ^{∗}(Im L)ψ and both ψ^{∗}(Re L)ψ and ψ^{∗}(Im L)ψ are real, (1) is equivalent to

Im ψ^{∗}Lψ ≥ 0 for all ψ ∈ ℍ.  (2)

In particular, the spectrum of L is in the complex upper half plane. If L has a spectral resolution then, since |λ + iε| ≥ ε for Im λ ≥ 0, we conclude that (L+iε)^{−1} exists for all ε > 0 as a bounded operator on ℍ̄ with spectral norm

∥(L + iε)^{−1}∥ ≤ ε^{−1} for all ε > 0.

The traditional situation is the one where L = E − H with a complex energy E satisfying Im E ≥ 0 and a Hamiltonian H = H_{s}− \frac{i}{2}Γ. Here H_{s}, Γ ∈ Lin ℍ are symmetric and Hermitian, and Γ is positive semidefinite. In the most important case of a conservative system, Γ = 0 and H is Hermitian. However, care is taken that all our results hold in the nonhermitian case, corresponding to dissipative systems with an optical potential that contributes to Γ.

The use of L helps to avoid a multitude of expressions involving E − H or E −H +iε. Since the (time-independent) Schrödinger equation Hψ = Eψ takes the simple form

Lψ = 0,  (3)

this makes the formal manipulations independent of energy and free of references to the Hamiltonian H, and thus much more readable. (Of course, in actual calculations, E and H reappear.)

More generally, (3) also covers nonlocal problems with a nonlinear dependence of L on E, and therefore can be used for the Bethe-Salpeter equations arising in bound state and resonance calculations for relativistic systems [14, 15, 18, 19, 26].

3 Reduction of the state space

Let ℍ and ℍ_{eff} be topological vector spaces with a definite, continuous bilinear inner product, related by the embedding map P, an injective, closed linear operator from ℍ_{eff}^{∗} to ℍ^{∗} satisfying P̄ = P and

P ψ_{eff} ∈ ℍ for all ψ_{eff} ∈ ℍ_{eff}.

We want to relate a Schrödinger equation Lψ = 0 in the full state space ℍ^{∗} to an effective Schrödinger equation L_{eff}ψ_{eff} = 0 in the reduced state space ℍ_{eff}^{∗}.

P^{∗} = P^{T} maps ℍ^{∗} to ℍ_{eff}^{∗} and ℍ to ℍ_{eff}. Moreover, P^{∗}P : ℍ_{eff}^{∗} → ℍ_{eff}^{∗} is invertible since P is closed and injective. The pseudo inverse

P^{I} := (P^{∗}P)^{−1}P^{∗}  (4)

maps ℍ^{∗} to ℍ_{eff}^{∗} and possesses the properties

(P P^{I})^{T} = P P^{I}, P^{T} P P^{I} = P^{T} , P^{I}P = 1.  (5)

This implies that

Q := 1 − P P^{I} = 1 − (P^{I})^{T} P^{T} ∈ Lin ℍ^{∗}  (6)

satisfies

P^{I}Q = P^{T} Q = 0, QP = 0, Q^{2} = Q^{T} = Q.  (7)

and hence is the orthogonal projection to the orthogonal complement of the range of P.

A Q-resolvent of a symmetric operator L ∈ Lin ℍ is a symmetric operator G ∈ Lin ℍ satisfying

GP = 0, GLQ = Q.  (8)

Since GQ = G(1 − P P^{I}) = G, the symmetry of G implies

QG = G = GQ, QLG = Q = GLQ.  (9)

Formally, G = Q(QLQ)^{−1}Q, but (QLQ)^{−1} is only defined on the range Qℍ^{∗} of Q.

The following discussion generalizes the Feshbach projection formalism which is obtained in the special case where L = E −H and H is Hermitian and G is the ordinary resolvent of QHQ in Qℍ^{∗} (cf. the development in Kukulin et al. [17, Chapter 4]; in their notation, G = G_{Q}(z) is called the ‘orthogonalized resolvent’).

3.1 Proposition.

(i) If L has a Q-resolvent G, it is uniquely determined, the operator

L_{eff} := P^{T} LP − P^{T} LGLP = P^{T} (L − LGL)P = P^{T} L(P − GLP) (10)

is symmetric, and Im L_{eff} is positive semidefinite.

(ii) For arbitrary ψ_{eff} ∈ ℍ_{eff}^{∗}, the vector

ψ = (P − GLP)ψ_{eff} ∈ ℍ^{∗}  (11)

satisfies

P^{T} Lψ = L_{eff}ψ_{eff}, QLψ = 0, ψ_{eff} = P^{I}ψ.  (12)

Proof. If (9) holds with G′ in place of G then

G′ = G′Q = G′QLG = G′LG = G′LQG = QG = G,

giving uniqueness. The vector (11) satisfies

QLψ = QL(P − GLP)ψ_{eff} = (Q − QLG)LP ψ_{eff} = 0,
P^{T} Lψ = P^{T} L(P − GLP)ψ_{eff} = L_{eff}ψ_{eff},

and, since P^{I}G = P^{I}QG = 0,

P^{I}ψ = P^{I}(P − GLP)ψ_{eff} = P^{I}P ψ_{eff} = ψ_{eff}.

This proves (ii). Since P̄ = P, we have

ψ^{∗}Lψ = ψ^{∗}((P^{I})^{T} P^{T} + Q)Lψ = ψ^{∗}(P^{I})^{T} L_{eff}ψ_{eff}
= ψ^{∗}(P^{I})^{∗}L_{eff}ψ_{eff} = ψ_{eff}^{∗}L_{eff}ψ_{eff}.

Thus, for arbitrary ψ_{eff} ∈ ℍ_{eff}^{∗}, we have Im ψ_{eff}^{T}L_{eff}ψ_{eff} = Im ψ^{∗}Lψ ≥ 0. This

proves (i).  ⊓ ⊔

The second term

Δ := P^{T} LGLP  (13)

in the definition of L_{eff} is called the optical potential induced by the reduction process. (The name is explained in Taylor [27, p.385].)

In the special case where P^{T} P = 1 and L = E −H, G = G(E) and hence the optical potential Δ(E) = P^{T} (E−H)G(E)(E−H)P is energy-dependent (and nonlocal) and we have L_{eff} = E − H_{eff}(E) with the effective Hamiltonian

H_{eff}(E) = P^{T} HP + Δ(E).

Thus the optical potential causes energy shifts in the eigenvalues of the projected Hamiltonian P^{T} HP. We also note that the reduced Schrödinger equation is generally a nonlinear eigenvalue problem

H_{eff}(E)ψ = Eψ.

Frequently, the energy-dependence is ignored; however, nonlinear eigenvalue problems for nonlocal Schrödinger equations arising from Bethe-Salpeter equations for relativistic problems were actually solved, e.g., [14, 15, 18, 19, 26].

Note that the resonating group method (Wildermuth & Tang [28]) works with coupled reaction channel equations derived from a projected Hamiltonian and hence misses the optical potential; an energy-independent term is added instead on a phenomenological basis [28, Chapter 8.2]. An alternative exact method is derived below.

We return to the general case.

3.2 Proposition. The following identities hold:

L_{eff}P^{I} = P^{T} (1 − LG)L, (P^{I})^{T} L_{eff} = L(P − GLP),  (14)

and, if L and L_{eff} are invertible,

P^{T} L^{−1}P = P^{T} P L_{eff}^{−1}P^{T} P.  (15)

Proof. Since (L − LGL)Q = L(Q − GLQ) = 0, we have

L_{eff}P^{I} = P^{T} (L − LGL)P P^{I} = P^{T} (L − LGL)(1 − Q) = P^{T} (L − LGL).

This gives the first equation in (14), and the transpose gives the second equation. For invertible L, L_{eff}, we conclude from (14) and GP = GQP = 0 that

L_{eff}P^{I}L^{−1}P = P^{T} (1 − LG)P = P^{T} P,
hence P^{T} P L_{eff}^{−1}P^{T} P = P^{T} P P^{I}L^{−1}P = P^{T} L^{−1}P.  ⊓ ⊔

(15) implies that for invertible L and L_{eff}, the part of L^{−1} accessible from ℍ_{eff}^{∗} can be computed from the knowledge of L_{eff} alone. Similarly, our next result says that, if L has a Q-resolvent, all solutions of the Schrödinger equation Lψ = 0 can be computed from solutions of the reduced Schrödinger equation L_{eff}ψ_{eff} = 0 and a knowledge of the correction operator

R = GLP  (16)

occurring in (10) and (11).

3.3 Theorem. Suppose that L has a Q-resolvent G.

(i) For any ψ_{eff} ∈ ℍ_{eff}^{∗} with L_{eff}ψ_{eff} = 0, the vector ψ ∈ ℍ^{∗} defined by (11) satisfies Lψ = 0 and

ψ_{eff} = P^{I}ψ ∈ ℍ_{eff}^{∗}.  (17)

(ii) For any ψ ∈ ℍ^{∗} with Lψ = 0, the vector (17) satisfies L_{eff}ψ_{eff} = 0, and we can reconstruct ψ from (11).

Proof. (i) Multiplication of the second equation of (14) with ψ_{eff} gives

0 = (P^{I})^{T} L_{eff}ψ_{eff} = L(P − GLP)ψ_{eff} = Lψ,

and (17) follows from Proposition 3.1(ii).

(ii) Multiplication of the first equation of (14) with ψ gives

L_{eff}ψ_{eff} = L_{eff}P^{I}ψ = P^{T} (1 − LG)Lψ = 0.

Since P ψ_{eff} = P P^{I}ψ = (1 − Q)ψ = ψ − Qψ, (11) follows from

\begin{aligned}(1 − GL)ψ & = & (1 − GL)ψ − (Q − GLQ)ψ = (1 − GL)(1 − Q)ψ \\ & = & (1 − GL)P ψ_{eff} = (P − GLP)ψ_{eff} \end{aligned}

since

GLψ = 0.  (18)
⊓ ⊔

Note that, while L_{eff}ψ_{eff} = 0 looks like a Schrödinger equation, this is generally a nonlinear eigenvalue problem. Indeed, if L = L(E) = E − H then L_{eff}(E) is generally a nonlinear analytic function of E, and the effective Hamiltonian

H_{eff}(E) := E − L_{eff}(E)

has a nonlinear dependence on the energy.

In many cases of interest, the Q-resolvent does not exist for the Hamiltonian H of interest. But frequently the Q-resolvents G_{ε} for the perturbed Hamiltonians H − iε corresponding to L_{ε} = L + iε exist for all ε > 0, and are given by

G_{ε} = Q(QLQ + iε)^{−1}Q.

If the limit

R := lim_{ε↓0} G_{ε}L_{ε}P  (19)

exists, most of the preceding proof still goes through, with R in place of GLP and R^{T} in place of P^{T} LG, and the effective Hamiltonian (usually) acquires a nonhermitian part.

The only exception is the second half of statement (ii), which must be modified. (This is most conspicuously seen when P = 0, where L_{eff} = 0 and the reduced Schrödinger equation provides no information at all.) Inspection of the proof shows that (18) fails. Thus, if ψ ∈ ℍ^{∗} satisfies Lψ = 0, the expression

ψ^{⊥} := ψ − (P − R)ψ_{eff}  (20)

need not vanish (as predicted by (11) under the stronger assumptions), but only the much weaker equation P^{T} ψ^{⊥} = 0 follows. Thus the reduction does no longer allow one to recover all solutions of Lψ = 0. By Theorem 3.3(i), ψ^{⊥} is also a solution of the Schrödinger equation in ℍ^{∗}, and since P^{T} ψ = 0 implies ψ_{eff} = 0 and hence ψ^{⊥} = ψ we have

{ψ^{⊥} | Lψ = 0} = {ψ | Lψ = 0, P^{T} ψ = 0}.

We discuss later (Theorem 5.2) how to access this missing part.

4 Perturbation theory

The computation of the Q-resolvent is traditionally done using perturbation theory. It is assumed that the inhomogeneous Schrödinger equation for a related reference problem is explicitly solvable, and the problem of interest is considered as a perturbation of the reference Schrödinger equation.

We therefore assume that we have a symmetric operator L_{ref} ∈ Lin ℍ satisfying

L_{ref}Q = QL_{ref};  (21)

this commutation relation can always be achieved by taking an arbitrary symmetric approximation L_{0} to L and putting

L_{ref} = L_{0} − (1 − Q)L_{0}Q − QL_{0}(1 − Q).  (22)

Let G be a Q-resolvent of L and put

V = L_{ref} − L, T = V + V GV, Ω := 1 + GV.  (23)

V is called the interaction, T the (Q-version of the) transition operator or T-matrix, and Ω the (Q-version of the) Möller operator. This terminology is justified by the close formal relations of the properties of T and Ω, derived below, with those of the T-matrix and the Möller operators of standard scattering theory. Indeed, for P = 0, Q = 1 (which is uninteresting from the point of view of effective Schrödinger equations), the results reduce to those of standard perturbation theory.

4.1 Proposition. If (21) holds then

ΩP = (1 − GL)P, Ω^{T} P = P,  (24)
T = V Ω = Ω^{T} V,  (25)
P^{T} LΩP = L_{eff} = P^{T} (L_{ref} − T)P.  (26)

Proof. Since

GL_{ref}P = GQL_{ref}P = GL_{ref}QP = 0

we have

GLP = GL_{ref}P − GV P = −GV P,

P^{T} LG = (GLP)^{T} = −(GV P)^{T} = −P^{T} V G,

hence

ΩP = P + GV P = (1 − GL)P.

Together with Ω^{T} P = (1 + V G)P = (1 + V GQ)P = P, this gives (24). (25) follows directly from (23). Since

P^{T} LΩP = P^{T} L(1 − GL)P = L_{eff}
= P^{T} LP − P^{T} LGLP = P^{T} LP + P^{T} LGV P
= P^{T} (L_{ref} − V)P − P^{T} V GV P = P^{T} (L_{ref} − T)P,
(26) follows.  ⊓ ⊔

4.2 Proposition. If Ω is invertible then

G_{ref} = Ω^{−1}G  (27)

is a Q-resolvent of L_{ref}, and

Ω = (1 − W)^{−1}, where W = G_{ref}V.  (28)

Proof.

Ω(1 − W) = Ω − ΩG_{ref}V = Ω − GV = 1,

gives (28). Since G_{ref}P = Ω^{−1}GP = 0 and

G_{ref}L_{ref}Q = G_{ref}LQ + G_{ref}V Q = Ω^{−1}GLQ + W Q = (1 − W)Q + W Q = Q,

G_{ref} is a Q-resolvent of L_{ref}.  ⊓ ⊔

4.3 Theorem. Suppose that G_{ref} is a Q-resolvent of a symmetric operator L_{ref} satisfying (21), and suppose that Ω with (28) exists. Then G = ΩG_{ref} is a Q-resolvent of L = L_{ref} − V, and with

T = V + V GV,  (29)

(23)–(27) hold. Moreover, we have

Ω = 1 + W Ω = 1 + ΩW,  (30)
T = V + T W = V + W^{T} T,  (31)
G = G_{ref} + W G = G_{ref} + GW^{T} .  (32)

Proof. (30) follows directly from Ω(1 − W) = (1 − W)Ω = 1 and implies Ω = 1 + ΩG_{ref}V = 1 + GV, hence (23). Since GP = ΩG_{ref}P = 0 and

GLQ = GL_{ref}Q − GV Q = ΩG_{ref}L_{ref}Q − GV Q = ΩQ − GV Q = Q,

G is a Q-resolvent of L. Thus Proposition 4.1 applies, and gives (24)–(26).

(27) is obvious. The first equality in (31) follows from (25) and (30), and the second follows by transposing the first equation. The first equation in (32) follows from

W G = G_{ref}V G = G_{ref}QV QG = G_{ref}(QL_{ref}Q − QLQ)G = G − G_{ref},
and the second follows again by transposing the first equation.  ⊓ ⊔

Inserting the definition (28) of W into (32) gives the (Q-version of the) Dyson equation

G = G_{ref} + G_{ref}V G = G_{ref} + GV^{T} G_{ref}.  (33)

If the spectral norm of W = G_{ref}V is smaller than one, (30)–(32) can be used to calculate iteratively the Möller operator Ω, the transition operator T and the Q-resolvent G. To lowest order, we get

Ω ≈ 1 + G_{ref}V, T ≈ V, G ≈ G_{ref} + G_{ref}V G_{ref}.

When ℍ_{eff} = {0}, this is the Born approximation, and when ℍ_{eff} is finitedimensional, this is a version of the distorted wave Born approximation (see, e.g., Newton [24, Section 9.1]). In the Born approximation, we simply get

L_{eff} ≈ P^{T} (L_{ref} − V)P = P^{T} LP;

in second order,

L_{eff} ≈ P^{T} LP − P^{T} V G_{ref}V P,  (34)

giving the approximation Δ ≈ P^{T} V G_{ref}V P for the optical potential (13). Further iteration gives the Born series

T = V + V G_{ref}V + V G_{ref}V G_{ref}V + …,

giving the exact optical potential

Δ = P^{T} V G_{ref}V P + P^{T} V G_{ref}V G_{ref}V P + ….

Thus we have recovered a generalized version of traditional perturbation theory. For ℍ_{eff} = {0}, P = 0, Q = 1, the equations obtained above reduce to those of standard perturbative scattering theory; the missing part – that one gets the scattering solutions of the Schrödinger equations – follows in the next section. Of course, from the point of view of effective Schrödinger equations, the case P = 0 is completely uninteresting; however, it is instructive in that it shows that the methods used to solve scattering problems apply with small modifications to the problem of finding effective Hamiltonians.

Similarly, if ℍ_{eff} is the eigenspace of H_{ref} corresponding to a bound state of the reference system and P the orthogonal projector to this space, we get the situation leading (in the Born approximation) to Fermi’s Golden Rule; cf. Kukulin et al. [17, Section 4.4]. For a nondegenerate bound state, ℍ_{eff} is one-dimensional, and again the point of view of effective Schrödinger equations is empty. For degenerate bound states, however, we recover a nontrivial low-dimensional eigenvalue problem as effective Schrödinger equation.

5 Solving the Schrödinger equation

We are now ready to express all solutions of the Schrödinger equation in terms of the solutions of a reduced Schrödinger equation and special solutions of a reference Schrödinger equation.

5.1 Proposition. If

ψ_{eff} = P^{I}ψ, ψ_{ref} = Q(1 − W)ψ  (35)

then

P^{T} ψ_{ref} = 0, ψ = Ω(P ψ_{eff} + Qψ_{ref}).  (36)

If (36) holds then

L_{eff}ψ_{eff} = P^{T} Ω^{T} Lψ,  (37)
L_{ref}ψ_{ref} = QLψ,  (38)
Lψ = (P^{I})^{T} L_{eff}ψ_{eff} + L_{ref}ψ_{ref} − (1 − Q)Ω^{T} L_{ref}ψ_{ref}.  (39)

Proof. Since

QW = QG_{ref}V = G_{ref}V = W,

(35) implies

\begin{aligned}Qψ_{ref} & = Q(1 − W)ψ = (Q − W)ψ \\ & = (1 − P P^{I} − W)ψ = (1 − W)ψ − P ψ_{eff}, \end{aligned}

hence

Ω(P ψ_{eff} + Qψ_{ref}) = Ω(1 − W)ψ = ψ,
P^{T} ψ_{ref} = P^{T} Q(1 − W)ψ = 0.

This gives (36). Now suppose that (36) holds. By (24),

P^{T} Ω^{T} L = (ΩP)^{T} L = P^{T} (1 − LG)L = P^{T} L(1 − GL),  (40)

and

P^{T} Ω^{T} LQ = P^{T} L(Q − GLQ) = 0.  (41)

Since (1 − QW)Ω = (1 − W)Ω = 1, this implies

P^{T} Ω^{T} LΩ = P^{T} Ω^{T} L(1 − QW)Ω = P^{T} Ω^{T} L.  (42)

Using (36), (42), (41), and (40), we find

P^{T} Ω^{T} Lψ = P^{T} Ω^{T} LΩ(P ψ_{eff} + Qψ_{ref}) = P^{T} Ω^{T} L(P ψ_{eff} + Qψ_{ref})
= P^{T} Ω^{T} LP ψ_{eff} = P^{T} L(1 − GL)P ψ_{eff} = L_{eff}ψ_{eff},

giving (37). Since QL_{ref}W = QL_{ref}G_{ref}V = QV, we have

QLΩ = (QL_{ref} − QV)Ω = QL_{ref}(1 − W)Ω = QL_{ref} = L_{ref}Q,

hence

QLψ = QLΩ(P ψ_{eff} + Qψ_{ref}) = L_{ref}Q(P ψ_{eff} + Qψ_{ref})
= L_{ref}Qψ_{ref} = L_{ref}ψ_{ref},

by (35), giving (38). Finally, X := 1 − (1 − Q)Ω^{T} satisfies

XP = P − (1 − Q)Ω^{T} P = P − (1 − Q)P = QP = 0

by (24), hence

XQL = X(1 − P P^{I})L = XL = (1 − (1 − Q)Ω^{T})L = L − (P^{I})^{T} P^{T} Ω^{T} L,

so that

Lψ = ((P^{I})^{T} P^{T} Ω^{T} + XQ)Lψ = (P^{I})^{T} L_{eff}ψ_{eff} + XL_{ref}ψ_{ref}
by (37) and (38).  ⊓ ⊔

Since the formulas in Proposition 5.1 and L_{eff} = P^{T} LΩP do not involve G, we can take limits and obtain:

5.2 Theorem. Let G_{ref,ε} be a Q-resolvent of L_{ref,ε} with lim_{ε↓0} L_{ref,ε} = L_{ref}.

Suppose that

W := lim_{ε↓0} G_{ref,ε}V and Ω = (1 − W)^{−1}

exist. If

ψ_{eff} = P^{I}ψ, ψ_{ref} = Q(1 − W)ψ

then

P^{T} ψ_{ref} = 0, ψ = Ω(P ψ_{eff} + Qψ_{ref}).

Moreover, with L_{eff} := P^{T} LΩP, we have

Lψ = 0 ⇔ L_{eff}ψ_{eff} = 0, L_{ref}ψ_{ref} = 0.

6 State space reduction without projections

Projection operators and the associated Q-resolvents are clumsy to use if an orthogonal basis is not easily available. We therefore revise the above formalism so that it can be used without computing any projection operators or Q-resolvents. Additional flexibility is gained by using in place of the reference operator L_{ref} ∈ Lin ℍ an appropriate operator P_{0} : ℍ_{0}^{∗} → ℍ^{∗} from the dual of a reference space ℍ_{0}. For exact solutions, this reference space must be at least as big as ℍ; however, in Section 7, we choose ℍ_{0} to be a smaller space in which numerical calculations are tractable, and obtain practical approximation schemes for the correction operator and the effective Hamiltonian. (This is related to the two Hilbert space method of Chandler & Gibson [3, 4], and indeed was inspired by their work.)

As before, L ∈ Lin ℍ is assumed to be symmetric, and P : ℍ_{eff}^{∗} → ℍ^{∗} is assumed to be a closed injective linear mapping satisfying P̄ = P and

P ψ_{eff} ∈ ℍ for all ψ_{eff} ∈ ℍ_{eff}.

Let P_{0} : ℍ_{0}^{∗} → ℍ^{∗} be a closed, surjective linear mapping satisfying

P_{0}ψ_{0} ∈ ℍ for all ψ_{0} ∈ ℍ_{0}.

Then P_{0}^{T} maps ℍ^{∗} to ℍ_{0}^{∗} and ℍ to ℍ_{0}. Moreover, P_{0}P_{0}^{T} : ℍ^{∗} → ℍ^{∗} is invertible since P_{0} is closed and surjective.

The pseudo inverse P^{I} of the injective P satisfies as before

P^{I} = (P^{T} P)^{−1}P^{T} , P^{I}P = 1, P^{T} P P^{I} = P^{T}

but the pseudo inverse P_{0}^{I} of the surjective P_{0} satisfies

P_{0}^{I} = P_{0}^{T} (P_{0}P_{0}^{T})^{−1}, P_{0}P_{0}^{I} = 1, P_{0}^{I}P_{0}P_{0}^{T} = P_{0}^{T} .

6.1 Proposition. If there are linear mappings R : ℍ_{eff}^{∗} → ℍ^{∗} and L′ : ℍ_{eff}^{∗} → ℍ_{eff}^{∗} such that

Rψ_{eff} ∈ ℍ for all ψ_{eff} ∈ ℍ_{eff}; L′ψ_{0} ∈ ℍ_{eff} for all ψ_{0} ∈ ℍ_{eff}

and

P_{0}^{T} LR + P_{0}^{T} P L′ = P_{0}^{T} LP, P^{T} R = 0,  (43)

then

L_{eff} := P^{T} LP − R^{T} LR ∈ Lin ℍ  (44)

is a symmetric operator satisfying

L_{eff} = P^{T} L(P − R), (P^{I})^{T} L_{eff} = L(P − R).  (45)

Moreover, if L and L_{eff} are invertible then

P^{T} L^{−1}P = P^{T} P L_{eff}^{−1}P^{T} P.  (46)

Proof. Symmetry is obvious. Multiplication of (43) with Q(P^{I})^{T} gives

QLP = QLR + QP L′ = QLR.

Now

\begin{aligned}R^{T} LR & = & R^{T} L(Q + (P^{I})^{T} P^{T})R = R^{T} LQR \\ & = & (QLR)^{T} R = (QLP)^{T} R = P^{T} LQR = P^{T} LR \end{aligned}

since QR = (1 − (P^{I})^{T} P^{T})R = R, hence

L_{eff} = P^{T} LP − R^{T} LR = P^{T} LP − P^{T} LR = P^{T} L(P − R).

This is the first half of (45). By multiplication with (P^{I})^{T}, we find

\begin{aligned}(P^{I})^{T} L_{eff} & = & (P^{I})^{T} P^{T} L(P − R) = (1 − Q)L(P − R) \\ & = & L(P − R) − QLP + QLR = L(P − R) \end{aligned}

giving the second half of (45). If L and L_{eff} are invertible then (45) implies

P^{T} L^{−1}(P^{I})^{T} L_{eff} = P^{T} (P − R) = P^{T} P,

hence

P^{T} P L_{eff}^{−1}P^{T} P = P^{T} L^{−1}(P^{I})^{T} P^{T} P = P^{T} L^{−1}P,
giving (46).  ⊓ ⊔

It is not difficult to see that R and L_{eff} from Section 3 are an instance of this construction, with ℍ_{0} = ℍ and P_{0} = 1.

6.2 Proposition.

(i) For arbitrary ψ ∈ ℍ^{∗}, the vectors

ψ_{eff} = P^{I}ψ ∈ ℍ_{eff}^{∗}, ψ_{0} = P_{0}^{I}(ψ − (P − R)ψ_{eff}) ∈ ℍ_{0}^{∗}  (47)

satisfy

P^{T} P_{0}ψ_{0} = 0, ψ = (P − R)ψ_{eff} + P_{0}ψ_{0}.  (48)

(ii) If (48) holds for some ψ_{eff} ∈ ℍ_{eff}^{∗}, ψ_{0} ∈ ℍ_{0}^{∗} then

L_{eff}ψ_{eff} = (P − R)^{T} Lψ,  (49)
P_{0}^{T} LP_{0}ψ_{0} = (P_{0} − (P − R)P^{I}P_{0})^{T} Lψ,  (50)
Lψ = (P^{I})^{T} L_{eff}ψ_{eff} + (P_{0}^{I})^{T} (P_{0}^{T} LP_{0}ψ_{0}).  (51)

Proof. (i) follows from

P_{0}ψ_{0} = P_{0}P_{0}^{I}(ψ − (P − R)ψ_{eff}) = ψ − (P − R)ψ_{eff},

\begin{aligned}P^{T} P_{0}ψ_{0} & = & P^{T} ψ − P^{T} (P − R)ψ_{eff} = P^{T} ψ − P^{T} P ψ_{eff} \\ & = & P^{T} ψ − P^{T} P P^{I}ψ = 0. \end{aligned}

(ii) Since by (43)

\begin{aligned}R^{T} Lψ & = & R^{T} LP_{0}ψ_{0} = (P_{0}^{T} LR)^{T} ψ_{0} = (P_{0}^{T} LP − P_{0}^{T} P L′)^{T} ψ_{0} \\ & = & P^{T} LP_{0}ψ_{0} − L^{′T} P^{T} P_{0}ψ_{0} = P^{T} LP_{0}ψ_{0}, \end{aligned}

(49) follows from (45) and (48):

\begin{aligned}L_{eff}ψ_{eff} & = & P^{T} L(P − R)ψ_{eff} = P^{T} L(ψ − P_{0}ψ_{0}) \\ & = & P^{T} Lψ − P^{T} LP_{0}ψ_{0} = P^{T} Lψ − R^{T} Lψ = (P − R)^{T} Lψ. \end{aligned}

(50) holds since by (48), (45) and (49),

P_{0}^{T} LP_{0}ψ_{0} = P_{0}^{T} Lψ − P_{0}^{T} L(P − R)ψ_{eff} = P_{0}^{T} Lψ − P_{0}^{T} (P^{I})^{T} L_{eff}ψ_{eff}
= P_{0}^{T} Lψ − P_{0}^{T} (P^{I})^{T} (P − R)^{T} Lψ
= (P_{0} − (P − R)P^{I}P_{0})^{T} Lψ.

(51) holds since by (47) and (45),

(P_{0}^{I})^{T} P_{0}^{T} LP_{0}ψ_{0} = LP_{0}ψ_{0} = LP_{0}P_{0}^{I}(ψ − (P − R)ψ_{eff})
= L(ψ − (P − R)ψ_{eff}) = Lψ − (P^{I})^{T} L_{eff}ψ_{eff}.
⊓ ⊔

We now have the following projector-free and Q-resolvent-free version of Theorem 5.2, constructing all solutions of the Schrödinger equation Lψ = 0 in terms of two simpler Schrödinger equations.

6.3 Theorem.

(i) If Im L is positive semidefinite then Im L_{eff} is positive semidefinite. If also P̄_{0} = P_{0} then also Im P_{0}^{T} LP_{0} is positive semidefinite.

(ii) For arbitrary ψ ∈ ℍ^{∗}, ψ_{eff} ∈ ℍ_{eff}^{∗}, ψ_{0} ∈ ℍ_{0}^{∗} satisfying (47) or (48),

Lψ = 0 ⇔ L_{eff}ψ_{eff} = 0, P_{0}^{T} LP_{0}ψ_{0} = 0.

In particular, to find all solutions of Lψ = 0, it suffices to solve the two problems

L_{eff}ψ_{eff} = 0,
P_{0}^{T} LP_{0}ψ_{0} = 0, P^{T} P_{0}ψ_{0} = 0.

Proof. (i) For arbitrary ψ_{eff} ∈ ℍ_{eff}^{∗}, we define ψ by (48) with ψ_{0} = 0. Then P^{T} ψ = P^{T} (P − R)ψ_{eff} = P^{T} P ψ_{eff}, hence ψ_{eff} = P^{I}ψ. Now (51) gives

Lψ = (P^{I})^{T} L_{eff}ψ_{eff},

and since (P^{I})^{∗} = (P^{I})^{T}, we get

ψ^{∗}Lψ = ψ^{∗}(P^{I})^{T} L_{eff}ψ_{eff} = (P^{I}ψ)^{∗}L_{eff}ψ_{eff} = ψ_{eff}^{∗}L_{eff}ψ_{eff}.

Hence ψ_{eff}^{∗}(Im L_{eff})ψ_{eff} = ψ^{∗}(Im L)ψ ≥ 0, and Im L_{eff} is positive semidefinite. If also P̄_{0} = P_{0} then P_{0}^{∗} = P_{0}^{T} and Im P_{0}^{T} LP_{0} is positive semidefinite since

ψ_{0}^{∗}(Im P_{0}^{T} LP_{0})ψ_{0} = (P_{0}ψ_{0})^{∗}(Im L)(P_{0}ψ_{0}) ≥ 0.

(ii) The forward implication follows directly from (49) and (50), the reverse

implication from (51).  ⊓ ⊔

6.4 Remark. If ψ_{0} = 0 then φ^{T} ψ = φ_{eff}^{T}(P^{T} P + R^{T} R)ψ_{eff}, so that

G_{eff} = P^{T} P + R^{T} R

is the effective metric induced on ℍ_{eff}. Note that it is generally not the original metric in ℍ_{eff}, not even when P is an orthogonal projection.

7 Form factors

In this section we show how to obtain efficiently approximations to the correction operator R. This leads in Section 9 to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.

We emphasize that the formulas derived in this section no longer involve a pseudo inverse. In particular, they can be used even when P is not injective and P_{0} is not surjective. (Hovever, since the assumptions under which the formulas are derived are then violated, they lose some information and hence give only approximate effective Hamiltonians.)

7.1 Theorem. Let

L_{0} := P_{0}^{T} LP_{0}, P_{1} := P^{T} P_{0}, U := P_{0}^{T} LP − P_{0}^{T} P L̃  (52)

with a symmetric L̃ ∈ Lin ℍ_{eff} (and hence in Lin ℍ_{eff}^{∗}), and write

L̂_{ε} := ( \begin{matrix}L_{ε} & P_{1}^{T} \\ P_{1} & −iε \end{matrix} ) ∈ Lin(ℍ_{0} ⊕ ℍ_{eff}) with L_{ε} = L_{0} + iε.

(i) If the strong limit

lim_{ε↓0} L̂_{ε}^{−1}\binom{U}{0} = \binom{F_{0}}{F_{1}}  (53)

exists then (43) is solved by

R = P_{0}F_{0}, L′ = L̃ + F_{1}.  (54)

(ii) Relation (53) holds with

F_{0} = lim_{ε↓0} F_{0ε}, F_{1} = lim_{ε↓0} F_{1ε},  (55)

where

F_{1ε} = (P_{1}L_{ε}^{−1}P_{1}^{T} + iε)^{−1}P_{1}L_{ε}^{−1}U, F_{0ε} = L_{ε}^{−1}(U − P_{1}^{T} F_{1ε}), (56)

if these limits exist.

Proof. (i) follows from

\binom{P_{0}^{T} LR + P_{0}^{T} P L′}{P^{T} R} = \binom{P_{0}^{T} LP_{0}F_{0} + P_{0}^{T} P F_{1} + P_{0}^{T} P L̃}{P^{T} P_{0}F_{0}}

= L̂_{0}\binom{F_{0}}{F_{1}} + \binom{P_{0}^{T} P L̃}{0}

= lim_{ε↓0} L̂_{0}L̂_{ε}^{−1}\binom{U}{0} + \binom{P_{0}^{T} P L̃}{0}

= \binom{U}{0} + \binom{P_{0}^{T} P L̃}{0} = \binom{P_{0}^{T} LP}{0}

by looking at the upper and the lower part separately. (ii) follows from the equation

L̂_{ε}\binom{F_{0ε}}{F_{1ε}} = \binom{U}{0},

which is easily verified by substitution.  ⊓ ⊔

In principle, L̃ in (52) may be chosen arbitrarily. However, for numerical calculations it may be advisable to choose L̃ in such a way that U (which replaces the interaction V in the projection approach) becomes small in some sense. This has the beneficial consequence that then the numerical approximation errors have a much smaller effect on the calculated solution.

In practice it is impossible to compute the exact correction operator and hence the exact effective Hamiltonian, since these tend to be exceedingly complicated. One therefore exploits physical intuition to select a space ℍ_{0} of manageable complexity whose dual contains the doorway states believed to mediate the interaction of the reduced system and the unmodelled environment. ℍ_{0}^{∗} is embedded into ℍ^{∗} by means of a doorway operator P_{0} that is now no longer surjective. Fortunately, the formulas (43) and (44) defining R and L_{eff} do not depend on pseudo inverses, and hence make also sense in this case. (43) and (44) now only yield approximate solutions for the correction operator and an approximate effective Hamiltonian. However, these approximations become better and better as the range of P_{0} covers a bigger and bigger part of ℍ.

The situation is fully analogous to numerical discretization schemes that are necessary to solve all but the simplest partial differential equations; the only difference is that in the present context it frequently makes sense to consider approximations in manageable function spaces, so that one does not discretize completely.

A proper choice of the doorway operator P_{0} makes the computations more tractable. At the same time, it limits the formal complexity of the optical potential

Δ = R^{T} LR = F_{0}^{T} (P_{0}^{T} LP_{0})F_{0},  (57)

the second term in L_{eff}. As one can see, P_{0} specifies the allowed form of the optical potential while the form factor F_{0} specifies the coefficients in the optical potential, and thus introduces energy-dependent running coupling constants.

The art in applying the reduction technique consists in finding embeddings P that ‘dress’ the subsystem of interest in a sufficiently accurate way, a doorway operator P_{0} embedding the relevant doorway states, and an operator L̃ such that U is small, and the limit (53) exists and can be approximated efficiently.

Setting P_{0} = 0 gives R = 0, Δ = 0, and hence the trivial approximation

L_{eff} ≈ P^{T} LP.  (58)

This simply amounts to discarding the interaction of the subsystem with the rest of the system. The choice ℍ_{0} = ℍ, P_{0} = P is not better since then P^{T} P F_{0} = P^{T} P_{0}F_{0} = P^{T} R = 0 by (43), and hence F_{0} = 0, R = 0. This is not surprising since we expect that the doorway operator P_{0} should incorporate additional information about doorway states not yet represented in the subsystem but significantly interacting with it.

If ℍ_{0} and ℍ_{eff} are finite-dimensional then (52) defines finite-dimensional matrices, and the computation of the form factor amounts to solving the matrix equation

( \begin{matrix}L_{0} & P_{1}^{T} \\ P_{1} & 0 \end{matrix} ) \binom{F_{0}}{F_{1}} = \binom{U}{0}  (59)

with a complex symmetric (and for L = L^{∗}, P̄_{0} = P_{0} Hermitian) coefficient matrix. (59) can be solved efficiently by sparse matrix methods (cf. Duff et al. [25, 5]) if suitable localized basis functions are used to construct P and P_{0}.

In practice, it may be useful to employ in combination with discretization methods a complex absorbing potential in place of the +iε. In particular, if one proceeds as in Neumaier & Mandelshtam [23] one gets a quadratic eigenvalue problem that can handle all energies in a certain range simultaneously using harmonic inversion (Mandelshtam & Taylor [20]).

If ℍ_{eff} is finite-dimensional but ℍ_{0} is a function space then L_{0} is a differential or integral operator on ℍ_{0}. By solving suitable differential or integral equations we can find the vector-valued functions

B_{0} := lim_{ε↓0} L_{ε}^{−1}P_{0}^{T} LP, B_{1} := lim_{ε↓0} L_{ε}^{−1}P_{0}^{T} P.

and the complex symmetric matrix

G_{1} := P_{1}B_{1} = lim_{ε↓0}(P_{1}L_{ε}^{−1}P_{1}^{T} + iε).

Noting that L′ = F_{1} if L̃ = 0, the formula (55) for the form factor becomes

F_{0} = B_{0} − B_{1}L′, where L′ = G_{1}^{−1}P_{1}B_{0}.

8 The quality of approximate state vectors

In practice, it is usually impossible to find exact solutions of a Schrödinger equation Lψ = 0. On the other hand, in real applications, L is never precisely known either. Hence it makes sense to assess the quality of an approximate state vector ψ by trying to modify L a little to an operator L̃ that satisfies L̃ψ = 0 exactly. If the modification L̃ − L is within the accuracy to which L is known, we are confident that ψ is a good approximation to the true but unknown L.

This way of assessing the quality of an approximate solution of a problem is widely used in numerical analysis (see, e.g., Wilkinson [29]) and is known under the name of backward error analysis. Here we give a backward error analysis for the equation Lψ = 0, and deduce guidelines for quality assessment of approximate state vectors.

8.1 Theorem. Let L ∈ Lin ℍ be symmetric. Then, for arbitrary ψ ∈ ℍ^{∗} with finite ψ^{T} ψ ≠ 0, the modified operator

L̂ = (1 − \frac{ψψ^{T}}{ψ^{T} ψ} )L(1 − \frac{ψψ^{T}}{ψ^{T} ψ} ).  (60)

satisfies L̂ψ = 0 and

tr(L̂ − L)^{2} = τ_{L}(ψ),  (61)

where

τ_{L}(ψ) := 2\frac{(Lψ)^{T} (Lψ)}{ψ^{T} ψ} − (\frac{ψ^{T} Lψ}{ψ^{T} ψ} )^{2}.  (62)

Proof. Since the formulas are invariant under scaling ψ we may assume that ψ is normalized to norm 1. Then

ψ^{T} ψ = 1, ψ^{T} Lψ =: λ, ψ^{T} L^{2}ψ =: μ,

with real λ, μ. The operator

Δ := L − L̂ = ψψ^{T} L + Lψψ^{T} − λψψ^{T}

satisfies Δψ = Lψ, hence L̂ψ = 0. Since

ΔLψ = λLψ + (μ − λ^{2})ψ,

we find

Δ^{2} = Lψψ^{T} L + (μ − λ^{2})ψψ^{T} .

Δ^{2} maps ℍ to the two-dimensional space spanned by ψ and Lψ, hence is trace class. Using the formula tr φψ^{T} = ψ^{T} φ, we find

tr Δ^{2} = μ + (μ − λ^{2}) = 2μ − λ^{2} = τ_{L}(ψ),

giving (61).  ⊓ ⊔

In the conservative case where Im L = 0, and hence L is Hermitian, it is possible to show that the choice (60) is best possible. Note that a state vector ψ ∈ ℍ^{∗} with ψ̄ = ψ and finite ψ^{T} ψ is now in the Hilbert space ℍ̄.

8.2 Theorem. Let L ∈ Lin ℍ be Hermitian, and suppose that ψ ∈ ℍ̄\{0} satisfies ψ̄ = ψ and Lψ ∈ ℍ̄.

(i) Any symmetric and Hermitian L̃ ∈ Lin ℍ with L̃ψ = 0 satisfies

tr(L̃ − L)^{2} ≥ τ_{L}(ψ).  (63)

Equality in (63) is achieved precisely when L̃ = L̂.

(ii) We always have

0 ≤ \frac{(Lψ)^{T} Lψ}{ψ^{T} ψ} ≤ τ_{L}(ψ) ≤ 2\frac{(Lψ)^{T} Lψ}{ψ^{T} ψ} .  (64)

In particular, τ_{L}(ψ) = 0 if and only if Lψ = 0.

Proof. By the preceding theorem, L̃ = L̂ gives equality in (63), and is a good choice since L̂ψ = 0. Hence suppose that L̃ ≠ L̂. Without loss of generality, (L̃ − L)^{2} is trace class (otherwise the trace is infinity and (63) is trivially satisfied). Since (L̃ − L̂)ψ = L̃ψ − L̂ψ = 0 we have

\begin{aligned}tr(L − L̂)(L̃ − L̂) & = tr Δ(L̃ − L̂) \\ & = ψ^{T} L(L̃ − L̂)ψ + ψ^{T} (L̃ − L̂)Lψ − λψ^{T} (L̃ − L̂)ψ = 0. \end{aligned}

Therefore

tr(L̃ − L)^{2} − τ_{L}(ψ) = tr(L̃ − L)^{2} − tr(L̂ − L)^{2}
= tr(L̃ + L̂ − 2L)(L̃ − L̂)
= tr(L̃ − L̂)^{2} − 2 tr(L − L̂)(L̃ − L̂)
= tr(L̃ − L̂)^{2} > 0

since L̃ ≠ L̂. Therefore, (63) holds for L̃ ≠ L̂ with strict inequality. This proves (i). Since ψ̄ = ψ, the second term in (61) is nonnegative. This gives the upper bound in (64) and implies the final assertion. The lower bound follows from the Cauchy-Schwarz inequality (ψ^{T} Lψ)^{2} ≤ ψ^{T} ψ ·(Lψ)^{T} Lψ. ⊓ ⊔

(60) is the orthogonal projection of L to the orthogonal complement of ψ, and \sqrt{τ_{L}(ψ)} measures, in a sense, its deviation from L. Therefore, τ_{L}(ψ) (or its square root) serves as a useful measure for the quality of an approximate solution ψ of the Schrödinger equation with Hermitian L.

In the nonhermitian case, it seems possible that τ_{L}(ψ) = 0 even if Lψ ≠ 0. (A finite-dimensional example is L = \binom{α i}{i 0}, ψ = \binom{1}{0} which has L^{2} = 0 and τ_{L}(ψ) = 0 for α = \sqrt{2}, and for α = 1 even τ_{L}(ψ) < 0.) Thus, unless L^{∗} = L, the measure τ_{L}(ψ) might be sometimes too optimistic. However, in practice L is nearly Hermitian and the use of τ_{L}(ψ) should cause no problems.

9 Multichannel scattering

We now apply the above to the multichannel approach discussed below. A projection approach to multichannel scattering leading to effective Hamiltonians is discussed, e.g., in Newton [24, Section 16.6], but the equations derived there appear not to be suitable to numerical approximation. A more useful formulation is given by the present equations from Theorem 7.1, with P and P_{0} as given below.

An arrangement is a partition A of the system of particles into clusters i ∈ A, with correct assignment of distinguishability. At a fixed energy E, those arrangements are relevant that contain channels defined by cluster bound states with energies E_{i} such that

∑_{i∈A} E_{i} ≤ E + ΔE

where ΔE is zero or a small quantity. These open or nearly open channels are assumed to correspond approximately to states in a n_{A}-dimensional space

ℍ_{A0} ⊆ ⨂_{i∈A} ℍ_{i}

with basis functions

φ_{Ak}(x) = ∏_{i∈A} φ_{il_{ik}}(x_{i}),  (65)

where x_{i} is the vector of coordinates of particles in cluster i, and the φ_{il}(x_{i}) are translation invariant basis functions from the cluster Hilbert space ℍ_{i}, used in all possible combinations in (65). The motion of the clusters is described by a space ℍ_{A} of functions of a system of relative coordinates r_{A} between the cluster centers.

Consider, for example, a 3-particle reaction XY + Z ⇌ X + Y Z. Then ℍ_{XY +Z,0} consists of products of approximate bound states ψ_{XY} and the ground state ψ_{Z}; ℍ_{X+Y Z,0} consists of products of the ground state ψ_{X} and approximate bound states ψ_{Y Z}, and ℍ_{XY Z,0} consists of sufficiently many states localized in the transition region to resolve the resonances of interest. Usually, one would keep the arrangements (XY, Z) and (X, Y Z) in the reduced description, and use the states belonging to the arrangement (XY Z) as doorway states for the transition regime.

Let C be the set of arrangements considered relevant for the reduced description. The reduced multichannel state space is then the dual of the space

ℍ_{eff} = ⨁_{A∈Ceff} ℍ_{A}^{n_{A}}

or the properly symmetrized subspace in case of indistinguishable clusters. The inner product in ℍ_{eff} is given by

φ_{eff}^{T}ψ_{eff} = ∑_{A∈Ceff} ∫ dr_{A}φ_{A}(r_{A})^{T} ψ_{A}(r_{A}).

The embedding map P : ℍ_{eff}^{∗} → ℍ^{∗} is given by

P ψ_{eff} = ∑_{A∈Ceff} P_{A}ψ_{A},

where the kth component of P_{A} : (ℍ_{A}^{n_{A}})^{∗} → ℍ^{∗} maps a function of r_{A} to the kth basis vector of ℍ_{A} modified to have the corresponding dependence on the center of mass r_{A}(x) of the cluster coordinates x,

(P_{A}ψ_{A})_{k}(x) = ψ_{A}(r_{A}(x))φ_{Ak}(x).

The transpose P^{T} : ℍ^{∗} → ℍ_{eff}^{∗} is given by

(P^{T} ψ)_{A} = P_{A}^{T} ψ for all A ∈ C_{eff}.

By construction, P^{T} LP is a direct sum of contributions of the form

P_{A}^{T} LP_{A} = L_{A} − H_{A},

where H_{A} is the free Hamiltonian for the motion of the cluster centers and L_{A} is a symmetric n_{A} ×n_{A}-matrix. In a basis of cluster eigenstates with energies E_{i}, L_{A} is the diagonal matrix formed by the energy differences E − E_{i}.

The construction of (ℍ_{0}, P_{0}) is completely analogous, using a larger set of arrangements and/or channels that contain the doorway states.

Thus the calculations have the same complexity as those for the coupled reaction channel equations (or resonating group method), as described, e.g., by Wildermuth & Tang [28]). However, the present scheme is more flexible in that it can incorporate information from doorway states. In principle, by increasing the size of the doorway state space, it is capable of arbitrarily accurate approximations to the full dynamics, and shares this feature with the two Hilbert space method of Chandler & Gibson [3, 4] and with a technique by Goldflam & Kowalski [11].

To solve the reduced Schrödinger equation (and, if a similar construction is used for the doorway operator, the equations for the form factor), the whole arsenal of methods developed in the applications is available. Binary arrangements can be handled by Lippmann-Schwinger equations (see, e.g., Wildermuth & Tang [28], Adhikari & Kowalski [1, Chapter 3]), and 3-cluster arrangements by the Faddeev [6] connected kernel approach (see, e.g., Glöckle [13, Chapter 3], Kukulin et al. [17], Adhikari & Kowalski [1, Chapter 7]).

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