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.

Required background. Projection methods derives the Schur complement and reconstruction; Brillouin–Wigner theory explains energy self-consistency; resonances distinguishes poles from ordinary bound-state eigenvalues.

Outgoing elimination of continuum channels

Let $P=P^\dagger=P^2$ and $Q=I-P$ be orthogonal projectors. Write $H_{AB}=AHB$ for $A,B\in\{P,Q\}$, with compatible operator domains and self-adjoint diagonal blocks. For complex $z$ where the inverse exists,

$$H_{\mathrm{eff}}(z)=H_{PP}+\Sigma(z), \qquad \Sigma(z)=H_{PQ}(z-H_{QQ})^{-1}H_{QP}.$$

The exact projected resolvent is

$$P(z-H)^{-1}P=[z-H_{\mathrm{eff}}(z)]^{-1}.$$

At a continuum energy, the outgoing limit $z=E+i0$ replaces the bounded inverse. This boundary value is normally interpreted between suitable states or weighted spaces, not as a bounded operator on the entire Hilbert space.

If the incoming wave lies in $P$, there is no independent incident $Q$ amplitude and

$$|\psi_Q\rangle=(E+i0-H_{QQ})^{-1}H_{QP}|\psi_P\rangle.$$

More generally an incoming $Q$-channel solution $|\phi_Q^{\mathrm{in}}\rangle$ adds a homogeneous term. The retained equation then has a source:

$$[E-H_{\mathrm{eff}}(E+i0)]|\psi_P\rangle =H_{PQ}|\phi_Q^{\mathrm{in}}\rangle.$$

Omitting this source would describe the wrong scattering preparation.

Dispersive shifts and absorptive widths

The spectral boundary identity is

$$(E+i0-H_{QQ})^{-1} =\mathcal P\frac{1}{E-H_{QQ}}-i\pi\delta(E-H_{QQ}).$$

Therefore

$$\Sigma(E+i0)=\Delta(E)-\frac{i}{2}\Gamma(E), \qquad \Gamma(E)=2\pi H_{PQ}\delta(E-H_{QQ})H_{QP}.$$

The spectral density is positive, so $\Gamma$ is positive semidefinite wherever its density is defined. More generally this positivity holds after integrating against a nonnegative energy test function. The sign describes loss into outgoing eliminated channels; the full self-adjoint system remains unitary. $\Gamma$ is an energy width, and a corresponding rate carries a factor $1/\hbar$.

An energy-dependent optical Hamiltonian is not automatically an exact time-local generator. Fourier transformation generally produces memory, and threshold structure can make a constant-rate description inaccurate.

Example: a discrete state coupled to a finite band

Retain $|d\rangle$ with bare energy $E_d$, and eliminate a continuum whose product of density of states and squared coupling is a constant $J>0$ on $0<\epsilon<\Lambda$ and zero elsewhere. With this normalization, $J$ has units of energy. Then

$$\Sigma(z)=J\int_0^\Lambda\frac{d\epsilon}{z-\epsilon} =J[\log z-\log(z-\Lambda)],$$

using the branch analytic in the upper half-plane. For $0<E<\Lambda$,

$$\Delta(E)=J\log\!\frac{E}{\Lambda-E}, \qquad \Gamma(E)=2\pi J.$$

The discontinuous band edges cause logarithmic shifts in this model. Far from both edges and at weak coupling, evaluating the self-energy near the retained level gives the usual nearly constant-width resonance. Close to an edge, the strong energy dependence invalidates that approximation, even though $J$ itself is small. Outside the band the self-energy is real; bound poles can also occur.

Resonance poles and the choice of retained sector

A resonance pole solves

$$\det[z-H_{PP}-\Sigma^{\mathrm{II}}(z)]=0$$

on the appropriate analytically continued sheet. It is not obtained by simply evaluating the physical upper-half-plane resolvent at a lower-half-plane energy without specifying the continuation. For a narrow isolated pole, write $z_*=E_R-i\Gamma_{\mathrm{pole}}/2$; the probability rate is $\Gamma_{\mathrm{pole}}/\hbar$.

The matrix $\Gamma(E)$ on the real axis need not equal the pole width after energy dependence and channel mixing are included. Nor can the positive bound-state derivative norm from projection methods be interpreted as a probability for a nonnormalizable resonance.

Different choices of $P$ describe the same physics differently. One may retain a closed-channel molecular level and eliminate an open continuum; its self-energy then has an absorptive part. Or one may retain the open scattering channel and eliminate a closed level. Below that eliminated channel’s threshold its contribution can be real, while the resonance pole still emerges from propagation in the retained open channel. A Feshbach resonance does not require the eliminated sector to be open in every representation.

Bound-state reduction and unitary block diagonalization

For real $E$ outside the eliminated spectrum, $H_{\mathrm{eff}}(E)$ is self-adjoint under the stated block assumptions. The nonlinear eigenvalue problem remains exact. Replacing $E$ by a reference energy becomes a controlled approximation only when the relevant energy window is separated from eliminated poles.

Schrieffer–Wolff instead constructs an energy-independent Hermitian operator in an orthonormal transformed model space. Agreement with Feshbach reduction requires consistent state normalization and observable transformations, not just copying an energy-dependent matrix element.

Exercises

1. Show that the finite-band self-energy has the expected large-$z$ limit, and explain why an infinitely wide constant-density band needs a prescription for its real shift.

Solution

Expanding the integral gives $\Sigma(z)=J\Lambda/z+O(z^{-2})$, consistent with the total coupling weight $J\Lambda$. At a fixed energy inside the band, $\Delta(E)=J\log[E/(\Lambda-E)]$ diverges as $\Lambda\to\infty$ with fixed $J$. A wide-band idealization must specify how this cutoff-dependent real part is subtracted or absorbed into $E_d$; retaining only $-i\pi J$ silently makes that choice.

2. Suppose a scalar narrow resonance obeys $E_*=E_d+\Delta(E_*)$. Linearize the pole equation assuming $\Gamma(E)$ varies negligibly across its width and $1-\Delta'(E_*)>0$.

Solution

Writing $z=E_*+\delta z$ gives

$$[1-\Delta'(E_*)]\delta z+\frac{i}{2}\Gamma(E_*)\simeq0.$$

Thus $\Gamma_{\mathrm{pole}}\simeq\Gamma(E_*)/[1-\Delta'(E_*)]$. This linearization needs an isolated pole and a slowly varying self-energy over the resulting width. Near a threshold or when the derivative denominator becomes small, solve the continued equation without this approximation.

References

• S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many-body systems,” Annals of Physics 326, 2793–2826, 2011.
• 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. Weidenmüller, Shell-Model Approach to Nuclear Reactions, North-Holland, 1969.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.