Feshbach Projection Formalism

The Feshbach projection formalism derives an effective Hamiltonian in a chosen subspace by eliminating the complementary subspace with a resolvent. Unlike the Schrieffer–Wolff transformation, the resulting Hamiltonian is generally energy dependent. In scattering problems it can also be non-Hermitian after outgoing boundary conditions are imposed. Effective Hamiltonians and Scale Separation places this construction in the broader method map.

Projection Methods develops the foundational finite-dimensional block algebra, Schur complement, state reconstruction, and energy-derivative normalization. This page takes that construction into the regime where continuum boundary conditions, shifts, widths, and resonance poles are central. Folded Effective Hamiltonians instead treats the bound-state problem of converting Q-box energy dependence into one energy-independent model-space interaction.

The method is most useful when a small model space is coupled to a large or continuous set of eliminated states. It is a natural language for resonances, optical potentials, threshold effects, and effective interactions in model spaces. For the Hermitian second-order matrix used when several retained levels are close, see Quasi-Degenerate Perturbation Theory.

Projection Setup

Let $P$ project onto the retained model space and let

$$Q=I-P.$$

Assume

$$P^2=P, \qquad Q^2=Q, \qquad PQ=0.$$

Decompose a state as

$$|\psi\rangle = P|\psi\rangle + Q|\psi\rangle \equiv |\psi_P\rangle+|\psi_Q\rangle.$$

Write the Hamiltonian blocks as

$$H_{PP}=PHP, \qquad H_{PQ}=PHQ, \qquad H_{QP}=QHP, \qquad H_{QQ}=QHQ.$$

The stationary Schrodinger equation

$$H|\psi\rangle=E|\psi\rangle$$

becomes the pair of coupled equations

$$\begin{aligned} (E-H_{PP})|\psi_P\rangle &= H_{PQ}|\psi_Q\rangle, \\ (E-H_{QQ})|\psi_Q\rangle &= H_{QP}|\psi_P\rangle. \end{aligned}$$

Eliminating the Q Space

Formally solve the second equation:

$$|\psi_Q\rangle = \frac{1}{E-H_{QQ}} H_{QP}|\psi_P\rangle.$$

Substituting into the first equation gives

$$H_{\mathrm{eff}}(E)|\psi_P\rangle = E|\psi_P\rangle,$$

where

$$H_{\mathrm{eff}}(E) = H_{PP} + H_{PQ} \frac{1}{E-H_{QQ}} H_{QP}.$$

The second term is a self-energy. It accounts for excursions from $P$ into $Q$ and back.

In scattering, the resolvent needs a boundary prescription:

$$\frac{1}{E-H_{QQ}} \longrightarrow \frac{1}{E^+-H_{QQ}}, \qquad E^+=E+i0.$$

The $+i0$ selects outgoing-wave boundary conditions.

Energy Dependence

The Feshbach effective Hamiltonian depends on the same energy $E$ one is trying to find. This makes the eigenvalue problem nonlinear:

$$H_{\mathrm{eff}}(E)|\psi_P\rangle = E|\psi_P\rangle.$$

For bound states below all eliminated thresholds, the resolvent can be Hermitian and the equation may be solved iteratively. For scattering states, the effective Hamiltonian can acquire an imaginary part that represents loss of probability from the retained channel into eliminated open channels.

This energy dependence is not a defect. It is the exact price of eliminating degrees of freedom without expanding in a fixed small parameter.

Two-Level Check

For

$$H = \begin{pmatrix} E_P & v\\ v^* & E_Q \end{pmatrix},$$

choose $P$ to retain the first state and $Q$ to eliminate the second. Then

$$H_{\mathrm{eff}}(E) = E_P + \frac{|v|^2}{E-E_Q}.$$

The effective equation is

$$E = E_P + \frac{|v|^2}{E-E_Q}.$$

Multiplying by $E-E_Q$ gives

$$(E-E_P)(E-E_Q) = |v|^2,$$

which is exactly the characteristic equation of the original two-level Hamiltonian. If $|v|\ll|E_P-E_Q|$, one may approximate $E$ in the denominator by $E_P$ and obtain the usual second-order shift:

$$E \approx E_P + \frac{|v|^2}{E_P-E_Q}.$$

Resonance Interpretation

In scattering, the eliminated space can contain channels that are closed, open, or continuum-like. Coupling to those channels shifts levels and can turn bound states into resonances.

The self-energy

$$\Sigma(E) = H_{PQ} \frac{1}{E^+-H_{QQ}} H_{QP}$$

can be decomposed schematically as

$$\Sigma(E) = \Delta(E) - \frac{i}{2}\Gamma(E),$$

where $\Delta(E)$ is an energy shift and $\Gamma(E)$ is a width when decay channels are open. A resonance is associated with a pole of the continued resolvent or scattering matrix at a complex energy.

This is the operator language behind many Feshbach resonance descriptions: a state associated with one sector is shifted and broadened by coupling to another sector.

Relation to Schrieffer–Wolff

The Schrieffer–Wolff transformation produces an energy-independent effective Hamiltonian by perturbatively block diagonalizing the Hamiltonian with a unitary transformation. The Feshbach formalism instead gives an exact energy-dependent expression after formal elimination.

The two descriptions agree order by order when a well-separated subspace permits expansion of the resolvent. For example,

$$\frac{1}{E-H_{QQ}} \approx - \frac{1}{H_{QQ}-E_P} \cdots$$

near a retained energy $E_P$. This expansion yields familiar virtual-transition denominators.

The choice between the methods is practical:

• use Schrieffer–Wolff when a low-energy block can be perturbatively decoupled;
• use Feshbach when energy dependence, thresholds, continua, or scattering boundary conditions are central.

Applications

Feshbach projection ideas appear in:

• nuclear optical potentials,
• resonance scattering,
• Feshbach resonances in cold atoms,
• model-space effective interactions,
• open-channel and closed-channel decompositions,
• continuum-induced shifts and widths.

In each case, the retained space is chosen for the physical question. A different question may require a different $P$ space.

Common Mistakes

• Treating $H_{\mathrm{eff}}(E)$ as an ordinary energy-independent Hamiltonian.
• Dropping the boundary prescription in scattering problems.
• Ignoring imaginary parts above open-channel thresholds.
• Assuming the projected Hamiltonian $H_{PP}$ already contains virtual excursions into $Q$.
• Expanding the resolvent near a threshold where the expansion is not controlled.

Exercises

1. Starting from the coupled $P/Q$ equations, derive the Feshbach effective Hamiltonian.

Solution

The coupled equations are

$$\begin{aligned} (E-H_{PP})|\psi_P\rangle &= H_{PQ}|\psi_Q\rangle, \\ (E-H_{QQ})|\psi_Q\rangle &= H_{QP}|\psi_P\rangle. \end{aligned}$$

Solve the second equation:

$$|\psi_Q\rangle = (E-H_{QQ})^{-1}H_{QP}|\psi_P\rangle.$$

Substitute into the first:

$$(E-H_{PP})|\psi_P\rangle = H_{PQ}(E-H_{QQ})^{-1}H_{QP}|\psi_P\rangle.$$

Rearranging gives

$$\left[ H_{PP} + H_{PQ}(E-H_{QQ})^{-1}H_{QP} \right] |\psi_P\rangle = E|\psi_P\rangle.$$

2. For the two-level example, show that the Feshbach equation reproduces the exact eigenvalue equation.

Solution

The effective equation is

$$E = E_P + \frac{|v|^2}{E-E_Q}.$$

Multiplying by $E-E_Q$ gives

$$E(E-E_Q) = E_P(E-E_Q)+|v|^2.$$

Equivalently,

$$(E-E_P)(E-E_Q)-|v|^2=0,$$

which is the determinant condition

$$\det \begin{pmatrix} E_P-E & v\\ v^* & E_Q-E \end{pmatrix} =0.$$

3. Why can the effective Hamiltonian become non-Hermitian in scattering?

Solution

For scattering states, the eliminated space may include open continuum channels. The resolvent is defined with an outgoing prescription,

$$(E^+-H_{QQ})^{-1}.$$

When $E$ lies in the continuum, this resolvent has an imaginary part. The resulting self-energy can be written as a shift minus $i\Gamma/2$, representing probability flowing from the retained channel into eliminated outgoing channels.

References

• H. Feshbach, “Unified theory of nuclear reactions,” Annals of Physics 5, 357-390, 1958.
• H. Feshbach, “A unified theory of nuclear reactions. II,” Annals of Physics 19, 287-313, 1962.
• C. Mahaux and H. A. Weidenmuller, Shell-Model Approach to Nuclear Reactions, North-Holland, 1969.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826, 2011.