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.

Helpful background. Spectra and the Spectral Theorem explain eigenvalues and continuous spectral measures. Complex Analysis supplies poles and boundary limits.

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.

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;
• periodic or antiperiodic imaginary-time conditions for the appropriate bosonic or fermionic thermal correlators.

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 regularizes the inverse and, in numerical work, broadens spectral peaks. Its effect on nearby levels should be checked rather than interpreted automatically as a physical lifetime.

Boundary Values on the Spectrum

At a real energy in the continuous spectrum, suitable matrix elements or weighted-space limits may admit 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. Infinite-volume traces and coincident-point kernels can diverge; the finite-system formula must then be replaced by a suitable density or regularized quantity.

Page Map

Question	Explanation	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

Read the inverse-operator and resolvent pages first, then choose spectral analysis, scattering, or response according to your calculation.

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. Continue to Scattering Theory for 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.

A two-level example

For $H=\operatorname{diag}(0,\Delta)$ with $\Delta>0$,

$$R(z)=\begin{pmatrix}1/z&0\\0&1/(z-\Delta)\end{pmatrix}.$$

At $z=E+i\eta$ with $\eta>0$, the broadened state-counting function is

$$\rho_\eta(E)=\frac{1}{\pi}\left[ \frac{\eta}{E^2+\eta^2} +\frac{\eta}{(E-\Delta)^2+\eta^2}\right].$$

Each Lorentzian has unit area. As $\eta\to0^+$, the peaks approach delta functions at the two energies. If $\eta$ is comparable to the level spacing, they overlap; an apparent smooth band need not mean the exact spectrum is continuous. See Density of States for the general construction.

Exercises

Integrate $\rho_\eta(E)$ over the whole real line. How does the answer change if the upper energy level has degeneracy three?

Solution

Each normalized Lorentzian integrates to one, so the integral is two. A threefold upper level contributes a projector of trace three and therefore a Lorentzian with three times the weight. The integral becomes four. Broadening changes resolution, but with this normalization it preserves the total number of finite-dimensional states.

References

• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006, doi:10.1007/3-540-28841-4.
• 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.
• 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.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.