Green Functions and Resolvents

A Green function is an inverse kernel together with a boundary prescription. In quantum mechanics, Green functions solve sourced equations, encode spectral information, select incoming or outgoing behavior, and organize linear response. The same language later expands into many-body correlators and relativistic field propagators.

For a time-independent self-adjoint Hamiltonian, the central operator is the resolvent

$$R(z) = (zI-H)^{-1},$$

defined when $z$ lies in the resolvent set of $H$. Its coordinate kernel is

$$G(x,x';z) = \langle x\rvert R(z)\lvert x'\rangle,$$

and it satisfies

$$(z-H_x)G(x,x';z) = \delta(x-x').$$

This equation alone does not determine a unique physical Green function when $z$ approaches the spectrum. The choice of domain, asymptotic behavior, temporal support, and limiting prescription is part of the definition.

What This Chapter Owns

This chapter is the canonical home for

• Green functions as inverse operators and source-response kernels;
• the resolvent set, spectrum, and operator-valued resolvent;
• coordinate-space energy Green functions;
• incoming, outgoing, retarded, and advanced boundary prescriptions;
• discrete, continuous, and mixed spectral representations;
• poles, residues, branch cuts, and spectral densities;
• total and local density of states from resolvent boundary values;
• introductory bridges to linear response and the Kubo formula;
• the resolvent structure behind the Lippmann–Schwinger equation;
• the translation from one-particle quantum mechanics to many-body and QFT Green functions.

The chapter does not own the time-evolution kernel itself; that belongs to Propagators and Kernels. It previews response without replacing the canonical open-systems and response treatment. It previews scattering-state construction without replacing the full Lippmann–Schwinger Equation. It also introduces QFT terminology without importing field-theory locality, vacuum structure, or renormalization into a one-particle setting.

Inverse Operators Need Boundary Data

For a linear operator $L$, a Green kernel formally satisfies

$$L_xG(x,x') = \delta(x-x').$$

If $G_1$ and $G_2$ solve the same source equation, their difference obeys the homogeneous equation:

$$L_x \left[ G_1(x,x')-G_2(x,x') \right] = 0.$$

Boundary or support conditions select among these possibilities. For the Schrödinger operator, common choices include

• square-integrable behavior for bound-state problems;
• outgoing or incoming waves in stationary scattering;
• retarded support after a source acts;
• advanced support before a designated source time;
• time ordering for Feynman correlators;
• periodicity in imaginary time for thermal problems.

Calling two kernels “solutions of the same equation” does not make them the same Green function. Their global prescriptions carry physical information.

Resolvent and Spectrum

The resolvent set $\rho(H)$ consists of complex $z$ for which $zI-H$ has a bounded inverse. The spectrum is its complement:

$$\sigma(H) = \mathbb C\setminus\rho(H).$$

For self-adjoint $H$, every nonreal $z$ lies in the resolvent set. The spectral theorem gives

$$R(z) = \int_{\mathbb R} \frac{1}{z-\lambda} \,dE_H(\lambda),$$

where $E_H$ is the projection-valued spectral measure. In a finite discrete system,

$$R(z) = \sum_n \frac{P_n}{z-E_n}.$$

An isolated eigenvalue produces a pole whose residue is the projector onto the corresponding eigenspace. Continuous spectrum appears through boundary values and discontinuities across the real axis rather than isolated normalizable-state poles.

The norm estimate

$$\lVert R(z)\rVert \le \frac{1}{\lvert\operatorname{Im}z\rvert}$$

for nonreal $z$ shows why the resolvent becomes sensitive near the real spectrum. A finite imaginary part is both a regularization and, in numerical work, an energy broadening that must be reported.

Boundary Values on the Spectrum

At a real energy in the continuous spectrum, use limiting boundary values

$$R^\pm(E) = \lim_{\eta\to0^+} (E-H\pm i\eta)^{-1}.$$

For standard scattering conventions,

$$R^+(E) \quad \text{selects outgoing behavior},$$

while

$$R^-(E) \quad \text{selects incoming behavior}.$$

The operator-valued distribution identity is

$$R^+(E)-R^-(E) = -2\pi i\,\delta(E-H).$$

Consequently,

$$-\frac{1}{\pi} \operatorname{Im}R^+(E) = \delta(E-H).$$

These identities connect analytic boundary values to spectral projectors and density of states. They are not ordinary pointwise operator equalities on the spectrum; interpret them through matrix elements, traces, or test-function smearing.

Time-Domain Green Operators

For a time-independent Hamiltonian, one common retarded normalization is

$$G^R(t,t') = -\frac{i}{\hbar} \theta(t-t') e^{-iH(t-t')/\hbar}.$$

It satisfies

$$\left( i\hbar\frac{\partial}{\partial t}-H \right) G^R(t,t') = \delta(t-t')I$$

and vanishes for $t\lt t'$. The advanced solution is

$$G^A(t,t') = \frac{i}{\hbar} \theta(t'-t) e^{-iH(t-t')/\hbar}.$$

With a consistent Fourier convention, the retarded and advanced transforms yield the upper- and lower-half-plane boundary values of the resolvent. Factors of $i$, $\hbar$, and $2\pi$ vary across conventions, so every comparison should begin by writing the defining source equation and transform pair.

Spectra and Density of States

For a finite system, the exact density of states is

$$\rho(E) = \sum_n\delta(E-E_n) = \operatorname{Tr}\delta(E-H).$$

The retarded resolvent gives

$$\rho(E) = -\frac{1}{\pi} \operatorname{Im} \operatorname{Tr}R^+(E).$$

In a position basis, the local density of states is

$$\rho(x,E) = -\frac{1}{\pi} \operatorname{Im} G^+(x,x;E).$$

A finite broadening $\eta$ replaces exact delta peaks by broadened lines. That can model resolution or aid visualization, but it is not the exact finite-system spectrum. State whether $\rho$ is total, local, per volume, per cell, or resolved by spin or another internal label.

Page Map

Question	Canonical page	Main object
What does “Green function” mean across quantum mechanics?	What Is a Green Function?	inverse kernel plus prescription
Where is the inverse defined as an operator?	Resolvent Operator	$R(z)=(zI-H)^{-1}$
How does the resolvent become a coordinate-space source solution?	Energy Green Function	$G(x,x';z)$
How do support and $i0$ signs select causal response?	Retarded and Advanced Green Functions	$G^R$ and $G^A$
How do poles, continua, and spectral densities enter?	Spectral Representation of Green Functions	spectral measures and discontinuities
How is state counting recovered from a resolvent?	Green Functions and Density of States	$-\pi^{-1}\operatorname{Im}\operatorname{Tr}R^+$
How do retarded commutators encode weak-source response?	Green Functions and Response Preview	susceptibility and Kubo bridge
How do free resolvents generate scattering states?	Lippmann–Schwinger Equation Preview	$\lvert\psi^\pm\rangle$ and $R_0^\pm V$
What changes for many-body systems and quantum fields?	Green Functions from QM to QFT	correlators, ordering, and fields

These nine articles form the planned chapter. The response and scattering pages are deliberately labeled previews because their detailed canonical homes require additional physical assumptions and machinery.

Suggested Routes

First systematic pass

Read What Is a Green Function?, Resolvent Operator, and Energy Green Function in order. Then use Retarded and Advanced Green Functions to see how a local inverse equation acquires physical meaning through support and boundary values.

Spectral-analysis route

After the resolvent operator, read Spectral Representation and Green Functions and Density of States. Pair them with the Spectral Theorem to distinguish isolated eigenvalue poles, continuous spectral measure, and finite broadening.

Scattering route

Read the energy Green function and its outgoing free-particle examples, then continue to the Lippmann–Schwinger Preview. The full scattering chapter owns existence assumptions, asymptotic states, the $T$ matrix, cross sections, and practical solution methods.

Response route

Read retarded and advanced functions before Green Functions and Response Preview. Retarded support is necessary for causal linear response, but the response function also contains a commutator and a reference state. A one-particle evolution kernel is not automatically a susceptibility.

QFT bridge

Finish with Green Functions from QM to QFT and Correlation Functions in Path Integrals. Keep operator ordering, vacuum or thermal state, field content, and normalization explicit. The QFT use of “propagator” is related to, but not identical with, the kernel of one-particle time evolution.

Prescription Checklist

Before using a Green function, state

1. the operator being inverted and its domain;
2. the source equation and delta-function normalization;
3. the coordinate measure or basis normalization;
4. the complex spectral parameter and sign convention;
5. the boundary, asymptotic, support, or ordering prescription;
6. the Fourier or Laplace transform convention;
7. whether the spectrum is discrete, continuous, or mixed;
8. whether traces and coincident-point limits require regularization;
9. whether a finite $\eta$ is numerical broadening or a limiting prescription;
10. whether the object is a one-particle inverse, a response correlator, or a QFT propagator.

Common Distinctions

Object	Defining structure	Main warning
resolvent	$(zI-H)^{-1}$ off the spectrum	singular boundary values need a prescription
energy Green kernel	$\langle x\rvert R(z)\lvert x'\rangle$	depends on measure, domain, and asymptotics
retarded Green operator	future-supported inverse	temporal causality does not imply relativistic light-cone support
advanced Green operator	past-supported inverse	useful analytically, not the usual initial-value response
propagator kernel	matrix element of $U(t,t_0)$	evolves states and is not itself a resolvent
response function	retarded commutator in a specified state	requires observables and an ensemble or state
QFT Feynman propagator	time-ordered field correlator	depends on field theory, state, and ordering conventions

Common Mistakes

• Writing “the Green function” without naming the operator and prescription. The inverse equation is incomplete by itself.
• Using $(E-H)^{-1}$ directly on the spectrum. Boundary values or a different generalized interpretation are required.
• Dropping the sign of $i0$. Incoming and outgoing, retarded and advanced solutions are then conflated.
• Treating a finite $\eta$ as harmless. It broadens spectral features and may hide nearby poles or gaps.
• Reading every singularity as a normalizable bound state. Continua produce cuts and boundary-value discontinuities.
• Comparing formulas before reconciling Fourier conventions. Factors of $i$, $\hbar$, and $2\pi$ can differ legitimately.
• Taking a trace or coincident-point limit without checking convergence. Ultraviolet or volume divergences may require regularization or normalization per volume.
• Equating a one-particle retarded inverse with a many-body response function. The latter contains an operator commutator and a chosen state.
• Using the Lippmann–Schwinger equation without its boundary label. The $+$ and $-$ states encode different asymptotic conditions.
• Importing relativistic causal claims into nonrelativistic kernels. The theories organize locality differently.

Cross-Links

• Propagators and Kernels
• Spectral Theorem
• Complex Analysis Essentials
• Path Integral Formulation
• Green Function Table
• Lippmann–Schwinger Equation
• Correlation Functions
• From Propagators in QM to Propagators in QFT

References

• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006, doi:10.1007/3-540-28841-4.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer, 2000, doi:10.1007/978-1-4757-5714-9.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, revised ed., Academic Press, 1980.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I,” Journal of the Physical Society of Japan 12, 570–586 (1957), doi:10.1143/JPSJ.12.570.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.