Green Functions and Density of States

The density of states counts how many independent states live near an energy. Green functions reveal it because their singularities sit exactly at spectral values. In the most compact form,

$$\rho(E) = \operatorname{Tr}\delta(E-H) = -\frac{1}{\pi} \operatorname{Im} \operatorname{Tr} \frac{1}{E+i0-H}.$$

The first expression is state counting. The second expression is the same state counting read from the boundary value of the resolvent. This page explains that relation and how it is used.

For compact formulas independent of Green-function derivations, see the reference entry Density of States.

Density of States

For a finite Hamiltonian with eigenvalues $E_n$, including degeneracies by repeating labels, the exact density of states is the distribution

$$\rho(E) = \sum_n\delta(E-E_n).$$

Equivalently, if $N(E)$ counts states with energies below $E$,

$$N(E) = \operatorname{Tr}\Theta(E-H),$$

then

$$\rho(E) = \frac{dN}{dE} = \operatorname{Tr}\delta(E-H).$$

In finite systems, $\rho(E)$ is a sum of delta functions. In large systems, thermodynamic limits, coarse graining, or explicit broadening often turn it into a smooth density. The normalization must always be stated: total density of states, per volume, per unit cell, per spin species, per band, or per internal degeneracy.

Trace of the Resolvent

Let

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

In an eigenbasis,

$$R(z) = \sum_n \frac{|n\rangle\langle n|} {z-E_n}.$$

Taking the trace gives

$$\operatorname{Tr}R(z) = \sum_n \frac{1}{z-E_n}.$$

The trace of the resolvent is therefore a meromorphic function for a finite system. Its poles are the energy levels, and its residues count degeneracy. This is the analytic form of state counting.

More generally, the spectral theorem gives

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

so a trace or matrix element of $R(z)$ is a transform of the spectral measure.

Imaginary Part of the Green Function

The retarded boundary value is

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

The distribution identity

$$\frac{1}{x+i0} = \operatorname{PV}\frac{1}{x} - i\pi\delta(x)$$

implies

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

Taking the trace gives the density-of-states formula:

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

This is the same relation introduced in Spectral Representation of Green Functions, now emphasized as a practical state-counting tool.

Broadening and Numerical Spectra

In calculations one often keeps a small positive $\eta$ instead of taking the exact boundary value:

$$R_\eta(E) = \frac{1}{E+i\eta-H}.$$

Then each exact delta function becomes a Lorentzian:

$$-\frac{1}{\pi} \operatorname{Im} \frac{1}{E+i\eta-E_n} = \frac{1}{\pi} \frac{\eta} {(E-E_n)^2+\eta^2}.$$

Thus

$$\rho_\eta(E) = -\frac{1}{\pi} \operatorname{Im} \operatorname{Tr}R_\eta(E)$$

is a broadened density of states. This is useful for plots and finite-size numerics, but it is not the exact density of states unless the broadening prescription is part of the physical model, for example through finite lifetime effects.

Local Density of States

In a position representation, the diagonal Green-function kernel gives the local density of states:

$$\rho(\mathbf x,E) = \langle\mathbf x|\delta(E-H)|\mathbf x\rangle.$$

Using the retarded Green function,

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

The total density of states is recovered by integrating over position:

$$\rho(E) = \int d^d x\, \rho(\mathbf x,E),$$

with the correct boundary, volume, and normalization conventions. In lattice models, the analogous statement is a site trace:

$$\rho(E) = \sum_i \rho_i(E), \qquad \rho_i(E) = -\frac{1}{\pi} \operatorname{Im}G^R_{ii}(E).$$

Local density of states is central in tunneling spectroscopy, impurity problems, mesoscopic systems, and numerical Green-function methods.

Bound and Continuous Spectra

For isolated bound states, the density of states contains delta-function peaks:

$$\rho_{\rm bound}(E) = \sum_b d_b\,\delta(E-E_b),$$

where $d_b$ is the degeneracy. In the resolvent, these appear as poles.

For continuous spectra, the density is supported on intervals or bands. In the resolvent, this appears as a branch cut or as a limiting boundary value along the real axis. The imaginary part of the retarded Green function measures the discontinuity across that cut:

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

This is why density of states, spectral functions, scattering thresholds, and band edges are all read from analytic structure. For the distinction between density of states, operator spectral functions, and measured intensity, see Spectral Functions.

Worked Example: Free Particle

For a spinless free particle in $d$ spatial dimensions,

$$E_{\mathbf k} = \frac{\hbar^2k^2}{2m}.$$

In a large box, the density of states per volume is

$$\frac{\rho_d(E)}{V} = \int\frac{d^d k}{(2\pi)^d}\, \delta\left( E-\frac{\hbar^2k^2}{2m} \right).$$

For $E\gt0$, using spherical coordinates in momentum space gives

$$\frac{\rho_d(E)}{V} = \frac{S_{d-1}}{2(2\pi)^d} \left( \frac{2m}{\hbar^2} \right)^{d/2} E^{d/2-1},$$

where

$$S_{d-1} = \frac{2\pi^{d/2}}{\Gamma(d/2)}$$

is the area of the unit sphere in $d$ dimensions. The dimension dependence is physically important:

Dimension	Spinless free-particle behavior
$d=1$	$\rho(E)/L\propto E^{-1/2}$
$d=2$	$\rho(E)/A$ is constant
$d=3$	$\rho(E)/V\propto E^{1/2}$

The Green-function formula gives the same answer:

$$\frac{\rho_d(E)}{V} = -\frac{1}{\pi} \operatorname{Im} \int\frac{d^d k}{(2\pi)^d} \frac{1}{E+i0-E_{\mathbf k}}.$$

The imaginary part turns the denominator into the delta function on the energy shell.

Applications Preview

Density of states appears whenever a calculation needs to count available final states or spectral weight:

• In scattering and transitions, Density of States in Transition Rates explains how the final-state measure combines with matrix elements and normalization conventions.
• In condensed matter, band structure produces van Hove singularities where energy shells have critical points.
• In many-body theory, spectral functions generalize density-of-states ideas to interacting excitations.
• In open or disordered systems, local density of states measures how strongly states are available near a location.
• In scattering theory, changes in density of states can be related to phase shifts and resonances, with the detailed derivation belonging to scattering theory.

The key habit is to ask what trace, normalization, and Hilbert-space sector are being counted.

Common Mistakes

• Forgetting that $\rho(E)$ is a distribution before smoothing or taking a thermodynamic limit.
• Mixing total density of states with density per volume, per unit cell, or per spin.
• Taking the imaginary part of the wrong Green function or using the wrong $i0$ sign.
• Treating numerical broadening as a physical lifetime without justification.
• Assuming the local density of states is uniform in an inhomogeneous system.
• Applying a smooth energy-shell formula at a critical point where $\nabla_{\mathbf k}E=0$.

Cross-Links

• Spectral Representation of Green Functions
• Energy Green Function
• Retarded and Advanced Green Functions
• Resolvent Operator
• Spectral Functions
• Formula Sheet
• Green Function Table
• Density of States
• Density of States: First Encounter
• Density of States in Transition Rates
• Fermi Golden Rule
• Green Functions

References

• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. Starting from a discrete spectral decomposition, derive $\rho(E)=-(1/\pi)\operatorname{Im}\operatorname{Tr}(E+i0-H)^{-1}$.

Solution

For a discrete spectrum,

$$\operatorname{Tr}(E+i0-H)^{-1} = \sum_n \frac{1}{E+i0-E_n}.$$

Using

$$\operatorname{Im} \frac{1}{x+i0} = -\pi\delta(x),$$

one finds

$$-\frac{1}{\pi} \operatorname{Im} \operatorname{Tr}(E+i0-H)^{-1} = \sum_n\delta(E-E_n) = \rho(E).$$

2. Show that finite $\eta$ broadens a delta function into a Lorentzian.

Solution

For real $x$,

$$\frac{1}{x+i\eta} = \frac{x-i\eta}{x^2+\eta^2}.$$

Therefore

$$-\frac{1}{\pi} \operatorname{Im} \frac{1}{x+i\eta} = \frac{1}{\pi} \frac{\eta}{x^2+\eta^2}.$$

With $x=E-E_n$, this is the Lorentzian approximation to $\delta(E-E_n)$ as $\eta\to0^+$.

3. Derive the spinless three-dimensional free-particle density of states per volume.

Solution

Start from

$$\frac{\rho_3(E)}{V} = \int\frac{d^3k}{(2\pi)^3} \delta\left( E-\frac{\hbar^2k^2}{2m} \right).$$

In spherical coordinates,

$$\frac{\rho_3(E)}{V} = \frac{4\pi}{(2\pi)^3} \int_0^\infty dk\,k^2 \delta(E-ak^2), \qquad a=\frac{\hbar^2}{2m}.$$

The root is $k_E=\sqrt{E/a}$, and

$$\delta(E-ak^2) = \frac{\delta(k-k_E)}{2ak_E}$$

for $k\ge0$. Hence

$$\frac{\rho_3(E)}{V} = \frac{1}{4\pi^2} \left( \frac{2m}{\hbar^2} \right)^{3/2} \sqrt{E},$$

for $E\gt0$, before spin degeneracy factors.

4. Show that the local density of states integrates to the total density of states.

Solution

By definition,

$$\rho(\mathbf x,E) = \langle\mathbf x|\delta(E-H)|\mathbf x\rangle.$$

Integrating over position and using the coordinate resolution of identity gives

$$\int d^d x\,\rho(\mathbf x,E) = \int d^d x\, \langle\mathbf x|\delta(E-H)|\mathbf x\rangle = \operatorname{Tr}\delta(E-H) = \rho(E).$$

The same argument becomes a sum over sites or basis states in a lattice model.