Continuum Green’s Function Method

• Continuum Green’s Function Method is a set of analytical techniques that represent spectra and impulse responses directly through Green’s functions without resorting to box discretization.
• It is applied in RMF and Skyrme-HFB theories to precisely extract resonance energies and widths by identifying poles and analyzing the density of states in nuclear systems.
• Beyond nuclear structure, the method extends to dispersive PDEs and stochastic models, using contour integration and spectral curves to achieve self-similar modal expansions and wave propagation analysis.

In the supplied literature, the expression continuum Green’s function method denotes a family of techniques in which Green’s functions are used to represent spectra or impulse responses without reducing the continuum to a purely box-discretized problem. In nuclear structure, the method is formulated in coordinate space within relativistic mean-field (RMF) and Skyrme Hartree–Fock–Bogoliubov (HFB) theories, where bound states, weakly bound states, resonances, and continuum states are treated on the same footing, and resonance energies and widths are extracted from poles, extrema, or densities of states (Wang et al., 2021, Chen et al., 2020, Huo et al., 2022). In other domains, the same label is attached to analytical reconstruction of finite- and semi-infinite-domain kernels from infinite-domain Green’s functions in the complex Fourier transform domain, to jointly continuous random kernels for stochastic PDEs, and to contour-integral constructions on spectral curves (Zhu, 2022, Alberts et al., 2022, Boris, 2014). This suggests a methodological umbrella rather than a single standardized formalism.

1. Formal structure and recurring definitions

A central nuclear realization begins from the Dirac equation in RMF theory,

$$\left[\bm{\alpha}\cdot\bm{p}+V(\bm{r})+\beta\big(M+S(\bm{r})\big)\right]\psi_n(\bm{r})=\varepsilon_n\psi_n(\bm{r}),$$

and defines the single-particle Green’s function through

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$

Its spectral representation,

$$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$

makes explicit that the analytic structure is controlled by poles at the eigenenergies $\varepsilon_n$. Because the Dirac spinor has upper and lower components, $\mathcal G$ is a $2\times 2$ matrix, and under spherical symmetry it is decomposed into $\kappa$-resolved radial Green’s functions built from a solution regular at $r\to 0$, an outgoing or decaying solution at $r\to\infty$, and an $r$-independent Wronskian (Wang et al., 2021).

In continuum HFB theory, the same logic is lifted to Nambu space. The Green’s function is defined as the resolvent of the coordinate-space HFB matrix operator,

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$0

and the generalized density matrix is recovered by contour integration in the complex quasiparticle-energy plane. For even-even nuclei this contour encloses all negative quasiparticle energies; for odd-$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$1 nuclei additional contours isolate blocked poles (Huo et al., 2022, Sun et al., 2019).

Outside nuclear structure, the same continuum orientation appears in analytically and probabilistically distinct forms. For the parabolic Anderson model, the kernel $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$2 is constructed as a jointly continuous fundamental solution of

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$3

with the singularity at $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$4 isolated by factoring out the deterministic heat kernel (Alberts et al., 2022). In the discrete finite-gap setting, the Green’s function is written as a contour integral on the spectral curve involving a discrete Baker–Akhiezer wave function, its involution-related dual, and a meromorphic differential (Boris, 2014). Across these examples, a plausible implication is that the defining feature is not a specific equation class, but the decision to encode continuum behavior directly in the kernel and its analytic continuation.

2. RMF continuum spectroscopy and direct pole searching

The most explicit use of the term in nuclear spectroscopy is the RMF continuum Green’s function method for single-particle resonances. In this setting, resonant states are identified as poles in the fourth quadrant of the complex energy plane,

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$5

so that $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$6 is the resonance energy and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$7 is the resonance width. Because $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$8, the Green’s function becomes large near a genuine eigenvalue. One practical implementation is therefore to scan $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$9 and search for extrema of $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$0, or of the integrated diagnostic quantity

$$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$1

whose peak position gives $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$2 and whose imaginary part gives $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$3 (Wang et al., 2021).

For $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$4Sn with the RMF functional PK1, the coordinate-space solution uses $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$5 fm, $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$6 fm, and a complex-energy scan step $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$7 for both real and imaginary parts. The paper states that this scan gives resonance energies and widths accurate to about $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$8 keV, and even better if the scan step is reduced. For the $$\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\sum_n \frac{\psi_n(\bm{r})\psi_n^\dagger(\bm{r}')}{\varepsilon-\varepsilon_n},$$9 neutron resonance, the extremum occurs at

$\varepsilon_n$0

hence

$\varepsilon_n$1

For the broader $\varepsilon_n$2 resonance, the extremum is found at

$\varepsilon_n$3

corresponding to

$\varepsilon_n$4

The same work reports that the $\varepsilon_n$5 component is much larger than $\varepsilon_n$6, but both yield the same resonance location, and that results depend only very slightly on the coordinate-space box size when $\varepsilon_n$7 fm are compared (Wang et al., 2021).

A closely related RMF study formulates the practical criterion as searching for poles of the Green’s function or, equivalently in practice, the extremes of the density of states. That work emphasizes that the strategy is effective for both narrow and broad resonances, identifies four new broad resonant states in $\varepsilon_n$8Sn—$\varepsilon_n$9, $\mathcal G$0, $\mathcal G$1, and $\mathcal G$2—and improves the width of the very narrow $\mathcal G$3 state to about $\mathcal G$4. It also reports that the RMF-GF results are very close to those by the complex momentum representation method and the complex scaling method (Chen et al., 2020).

3. Density of states, partial-wave analysis, and symmetry studies

A second major formulation uses the Green’s function as the source of the density of states. For spherical systems, the partial-wave density is

$\mathcal G$5

In this representation, bound states appear as sharp $\mathcal G$6-function peaks below threshold, continuum states form a smooth background above threshold, and resonant states appear as pronounced peaks in $\mathcal G$7 on top of the free-particle continuum background. A small imaginary part $\mathcal G$8 turns ideal $\mathcal G$9-peaks into Lorentzians with full width at half maximum $2\times 2$0, making the resonance peaks numerically visible (Sun et al., 2019).

Applied to the Dirac equation with a Woods–Saxon potential in $2\times 2$1Pb, the method uses

$2\times 2$2

with $2\times 2$3, $2\times 2$4, $2\times 2$5, and $2\times 2$6. The coordinate-space calculation employs $2\times 2$7 fm, $2\times 2$8 fm, $2\times 2$9 MeV, and $\kappa$0 MeV. The paper lists resonances such as $\kappa$1 at $\kappa$2 MeV with $\kappa$3 MeV, $\kappa$4 at $\kappa$5 MeV with $\kappa$6 MeV, and $\kappa$7 at $\kappa$8 MeV with $\kappa$9 MeV (Sun et al., 2019).

The same framework is used there to investigate spin and pseudospin symmetry in resonant states. The resonant spectrum contains spin doublets such as $r\to 0$0, $r\to 0$1, and $r\to 0$2, and pseudospin doublets such as $r\to 0$3, $r\to 0$4, $r\to 0$5, and $r\to 0$6. The study reports that the threshold effect plays an important role in resonant pseudospin doublets, that some pseudospin splittings become negative so that the level ordering is reversed in the continuum, that width splittings are systematically positive for both spin and pseudospin doublets and are typically small, usually below $r\to 0$7 MeV, and that the splittings of the energies and widths are independent. It also shows that $r\to 0$8 remains very similar between spin partners, whereas $r\to 0$9 remains very similar between pseudospin partners (Sun et al., 2019).

4. Continuum Skyrme-HFB Green’s functions for even-even and odd-$r\to\infty$0 nuclei

In continuum Skyrme-HFB theory, the Green’s function method is used to build local densities and pairing fields self-consistently from the analytic structure of the quasiparticle spectrum. The formal motivation is the weak-binding regime near the drip line, where the Fermi energy lies close to zero and pairing can scatter nucleons into the continuum. The method is attractive because it imposes the correct asymptotic behavior of wave functions in coordinate space, treats bound states, weakly bound states, resonances, and continuum states on the same footing, can provide resonant energies and widths directly, avoids the nonphysical density cutoff at the box boundary, and is well suited to calculating local densities and pairing fields self-consistently (Huo et al., 2022).

For neutron-rich Ca, Ni, Zr, and Sn isotopes, the implementation uses the SLy4 Skyrme parameter set in the particle-hole channel and a density-dependent delta interaction in the pairing channel,

$r\to\infty$1

with $r\to\infty$2 MeV·fm$r\to\infty$3, $r\to\infty$4 fm$r\to\infty$5, $r\to\infty$6, and $r\to\infty$7. The radial mesh uses $r\to\infty$8 fm and $r\to\infty$9 fm, with $r$0, $r$1 MeV, and a rectangular contour $r$2 of height $r$3 MeV, length $r$4 MeV, and energy step $r$5 MeV (Huo et al., 2022).

The same paper ties the continuum formalism directly to observables. Separation energies $r$6 and $r$7, pairing energies,

$r$8

rms radii,

$r$9

and the occupations $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$00 and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$01 are all computed from the Green’s-function reconstruction of densities. The paper attributes halo-like density extensions to orbitals around the Fermi surface, especially low-angular-momentum states such as $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$02, $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$03, and some $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$04-waves, and emphasizes that the resulting densities are essentially independent of box size when the boundary conditions are chosen properly (Huo et al., 2022).

For odd-$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$05 nuclei, the method is extended through blocking with the equal filling approximation. The generalized density matrix acquires the blocked-state correction

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$06

and the contour formula correspondingly includes $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$07 and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$08, which must include the blocked quasiparticle pole but cannot intrude into the continuum area. The required condition is

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$09

When $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$10, the paper reports an unphysical peak in $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$11 and an abnormal increasing tail in $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$12. In the example of neutron-rich $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$13Sn with blocking of the quasiparticle state $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$14, the large-$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$15 density tail is dominated by weakly bound orbitals near the Fermi surface, with the $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$16 orbit identified as the dominant contributor at very large radius (Sun et al., 2019).

5. Analytical continuum kernels in dispersive and stochastic equations

In dispersive linear PDEs, the continuum Green’s function method is formulated as a reflection and transmission analysis in the complex Fourier transform domain. The core procedure is to solve the infinite-domain impulse response exactly in transform space, interpret it as a self-similar propagating wave packet, use reflection and transmission matrices to generate boundary-reflected copies of the infinite-domain Green’s function, and re-sum the reflected-wave series with a matrix geometric series. For the Euler–Bernoulli beam,

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$17

the infinite-domain Green’s function satisfies

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$18

and inverse transformation yields a self-similar Green’s function expressed through a complex-valued function $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$19. The same machinery is then applied to finite domains, semi-infinite domains, and coupled multi-domain systems, including classical boundary conditions such as pinned, clamped, free, and sliding (Zhu, 2022).

A notable analytical outcome is a new modal expansion derived from the matrix geometric series

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$20

which the paper interprets as analytically proving wave-mode duality. The new expansion yields the same modes and frequencies as the traditional modal expansion but does not require calculation of each mode’s inner product. The paper also states a time-complementary convergence behavior: the self-similar/reflection expansion is excellent for short response times, whereas the traditional modal expansion is better for long times. For the simply supported beam, the reported crossover is around a dimensionless time $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$21; for the heat equation, the threshold is

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$22

The same framework is used to argue that non-propagating waves also possess wave speed, and that heat conduction can be treated as propagating waves (Zhu, 2022).

In the stochastic setting of the parabolic Anderson model, the continuum Green’s function is a random kernel $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$23 constructed simultaneously for all $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$24, $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$25, and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$26. After normalizing by the deterministic heat kernel, the random factor extends continuously to the diagonal with value $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$27. Theorem $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$28 in that work gives an event $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$29 of probability one on which there exists a jointly continuous regular version $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$30 that is strictly positive and finite everywhere, with $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$31 agreeing, for fixed $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$32, with the unique continuous adapted mild solution. The same kernel generates all solutions with admissible initial data through the superposition principle

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$33

satisfies the Chapman–Kolmogorov property, and is proved to be strictly totally positive: $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$34 for strictly ordered $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$35-tuples $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$36 and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$37 (Alberts et al., 2022).

6. Continuum spectra, spectral-curve analogues, and methodological limits

A further use of continuum Green’s functions appears in a five-dimensional warped model with a UV brane, an IR brane, and a linear-dilaton-like extension beyond the IR brane. There the Green’s functions are fully analytic and exhibit a mass gap

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$38

a continuum for $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$39, and poles in the second Riemann sheet of the form

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$40

For gauge bosons with Neumann or Dirichlet UV boundary conditions, the resonance masses and widths satisfy the approximate equation

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$41

with $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$42 the $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$43-th branch of the Lambert function. The paper’s stated phenomenological point is that the spectrum above the gap is continuous and the resonant poles are broad, so there are no sharp isolated resonance peaks of the Randall–Sundrum type (Megias et al., 2021).

An algebro-geometric analogue is provided by the five-point discretization of a two-dimensional finite-gap Schrödinger operator. There the Green’s function is constructed as a contour integral over a spectral curve, using a discrete wave function $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$44, its dual $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$45, a meromorphic differential $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$46, and a special contour $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$47. The final formula,

$$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$48

satisfies $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$49 and an exponential growth estimate controlled by the quasi-momenta $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$50 and $$[\varepsilon-\hat{h}(\bm{r})]\mathcal{G}(\bm{r},\bm{r}';\varepsilon)=\delta(\bm{r}-\bm{r}').$$51. The paper presents this construction explicitly as the discrete analog of the continuum finite-gap Green’s function method (Boris, 2014).

Several limitations recur across the literature. In RMF resonance searches, the Green’s-function-extremum method is described as efficient and less time-consuming, but it does not provide as intuitive a picture of the single-particle spectrum as the density-of-states method (Wang et al., 2021). In continuum HFB, box discretization is criticized because it imposes artificial boundary conditions, discretizes continuum states into box states, can produce nonphysical density drops at the box edge, and can depend strongly on box size when densities are extended, whereas the Green’s-function method avoids those artifacts only when the asymptotic boundary conditions and contour prescriptions are chosen properly (Huo et al., 2022, Sun et al., 2019). In the complex-Fourier PDE construction, the self-similar expansion is more accurate at short times but becomes less efficient at long times because many reflections accumulate (Zhu, 2022). A common misconception is therefore that “continuum Green’s function method” names one uniform algorithm; the supplied literature instead shows several exact or asymptotically exact kernel constructions linked by a shared commitment to continuum boundary conditions, analytic structure, and non-discretized spectral information.