Spectral Representation of Green's Functions

The spectral representation is the working engine of modern condensed-matter and many-body physics, because the one quantity you can actually measure is the density of states — the discontinuity of $G$ across its cut. A scanning tunnelling microscope does exactly equation (8) experimentally: the tunnelling conductance $\mathrm{d}I/\mathrm{d}V$ at position $x$ and bias $V$ is proportional to the local density of states $\rho(x,eV)=\tfrac1\pi\,\mathrm{Im}\,G(x,x;eV+i0^+)$, so an STM map is a direct image of $|\phi_n(x)|^2$ summed over energy. In mesoscopic transport the recursive Green's function method and the non-equilibrium Green's function (NEGF) formalism compute conductance through nanostructures by building the resolvent of a tight-binding Hamiltonian slice by slice; the Landauer–Büttiker conductance is a trace of products of these $G$'s. Dynamical mean-field theory (DMFT) attacks strongly correlated materials by solving self-consistently for the interacting single-particle Green's function, whose spectral function $A(\omega)=\tfrac1\pi\,\mathrm{Im}\,G(\omega)$ shows quasiparticle poles turning into incoherent continua as correlations grow. And in lattice quantum chromodynamics, the same Källén–Lehmann spectral representation is inverted numerically — an ill-posed problem tackled with maximum-entropy and Bayesian methods — to pull hadron masses and transport coefficients out of Euclidean correlators. In every case the physics lives in the analytic structure: a sharp pole is a stable particle or bound state, a broadened pole is a resonance or quasiparticle, and a cut is a continuum.

Think of $\mathcal{L}$ as a matrix you have already diagonalized. Its eigenvectors are the eigenfunctions $\phi_n$, its eigenvalues the numbers $\lambda_n$. Inverting $\mathcal{L}-\lambda$ is then trivial in this basis: each eigenvalue $\lambda_n$ maps to $1/(\lambda_n-\lambda)$. Rotate that diagonal inverse back into position space and you have the Green's function, $G=\sum_n \phi_n(x)\phi_n^*(x')/(\lambda_n-\lambda)$ — the literal infinite-dimensional analogue of the finite-matrix identity $(A-\lambda I)^{-1}=\sum_n v_n v_n^\dagger/(\lambda_n-\lambda)$. The spectral parameter $\lambda$ is a probe you sweep; wherever it lands on an eigenvalue, one denominator vanishes and $G$ diverges. That divergence, seen in the complex plane, is a pole, and the first figure is the map of where those poles and cuts sit.

The completeness relation and the Green's function are two readings of the same projectors. Add up every eigenprojector $\phi_n(x)\phi_n^*(x')$ with no weight and you get $\delta(x-x')$, the identity resolved in position space. Add up the same projectors each weighted by $1/(\lambda_n-\lambda)$ and you get $G$. A single contour that encircles the whole spectrum and sums the residues carries you from one to the other. When the spectrum turns continuous the discrete spikes smear into a branch cut, and the jump in $G$ across that cut — its imaginary part just above the real axis — is the density of states, the number of modes per unit $\lambda$. The second figure makes that density concrete for the simplest possible case, the free particle on a line.

The starting point is a definition: the Green's function is the kernel that inverts $\mathcal{L}-\lambda$. Feeding it a point source $\delta(x-x')$ returns the response of the operator to a unit disturbance at $x'$, so $G$ solves the inhomogeneous equation with a delta on the right.

where: $\mathcal{L}$ — self-adjoint differential operator (e.g. Sturm–Liouville); $\lambda$ — complex spectral parameter; $G$ — Green's function, the kernel of $(\mathcal{L}-\lambda)^{-1}$; $\delta(x-x')$ — Dirac delta, the identity kernel. Reading: $G$ is the resolvent written in position space, and the whole article studies its behaviour as $\lambda$ moves through the complex plane. Everything spectral rests on the eigenvalue problem for $\mathcal{L}$ under its boundary conditions. Self-adjointness forces the eigenvalues $\lambda_n$ to be real and the eigenfunctions to be orthonormal with respect to the Sturm–Liouville weight $w(x)$.

where: $\phi_n$ — $n$-th eigenfunction obeying the boundary conditions; $\lambda_n$ — $n$-th real eigenvalue (real because $\mathcal{L}=\mathcal{L}^\dagger$); $w(x)$ — Sturm–Liouville weight. Reading: real eigenvalues are what will place the poles of $G$ on the real $\lambda$-axis. For a regular Sturm–Liouville problem the eigenfunctions are complete: they resolve the identity. This closure relation is the single fact that makes the eigenfunction expansion of $G$ legitimate.

where: the left side is the resolution of the identity in position space. Reading: summing every eigenprojector with unit weight rebuilds the delta — this is $G$'s completeness twin, recovered later by a contour. Combining the definition with completeness gives the headline result. The Green's function is the eigenprojector sum weighted by the inverse eigenvalue gap — poles wherever $\lambda$ meets an eigenvalue.

where: $\lambda_n-\lambda$ — the gap from the probe $\lambda$ to the $n$-th eigenvalue; it vanishes at a pole. Reading: near $\lambda\to\lambda_n$ a single term dominates and the system rings in the mode $\phi_n$. The same statement, stripped of the position labels, is an operator identity: the resolvent is a weighted sum of orthogonal eigenprojectors, manifestly analytic in $\lambda$ everywhere except on the spectrum.

where: $|\phi_n\rangle\langle\phi_n|$ — orthogonal projector onto the $n$-th eigenspace; $\hat G(\lambda)$ — resolvent operator, analytic off the spectrum. Reading: this is the operator form the whole chapter turns on; equation (4) is its position-space kernel. The residue of $G$ at a pole is not just a number but a whole projector kernel. It reads off the eigenfunction responsible for the resonance at $\lambda_n$.

where: $\operatorname*{Res}$ — residue in the complex $\lambda$-plane; the minus sign comes from $1/(\lambda_n-\lambda)=-1/(\lambda-\lambda_n)$. Reading: the analytic data (a residue) equals the spectral data (an eigenprojector) — the article's central identity. Sum those residues over a contour enclosing the entire spectrum and the weights disappear, returning completeness. This closes the loop between $G$ and the delta.

where: $C$ — a large contour encircling the whole spectrum, traversed counterclockwise. Reading: each residue contributes $-\phi_n\phi_n^*$, and the $-1/(2\pi i)$ undoes the sign and factor, leaving the closure relation (3). When the spectrum is continuous the poles fuse into a branch cut and the useful quantity is the discontinuity across it: the local density of states. Approaching the real axis from above and taking the imaginary part counts the modes.

where: $\rho(\lambda)$ — local (spectral) density of states at $\lambda$; $i0^+$ — infinitesimal approach from the upper half-plane. Reading: with the convention $(\mathcal{L}-\lambda)G=\delta$ used here, $\mathrm{Im}\,[\,\lambda_n-(\lambda+i0)\,]^{-1}=\pi\,\delta(\lambda-\lambda_n)\ge0$, so $\rho=\sum_n|\phi_n|^2\delta(\lambda-\lambda_n)$ is manifestly non-negative. (In terms of the physicists' retarded resolvent $G^+=(\lambda-\mathcal{L}+i0)^{-1}=-G$, the same statement is the familiar $\rho=-\tfrac1\pi\,\mathrm{Im}\,G^+$.) Finally, before diagonalizing anything, $G$ can be built directly from two solutions of the homogeneous equation — one obeying each boundary condition — glued at $x=x'$. This is the classical Sturm–Liouville form.

where: $y_<,y_>$ — solutions of $(\mathcal{L}-\lambda)y=0$ obeying the left/right boundary conditions; $x_<,x_>$ — the lesser/greater of $x,x'$; $p(x)$ — the coefficient of the highest derivative in Sturm–Liouville form; $W[y_<,y_>]=y_<y_>'-y_<'y_>$ — their Wronskian. Reading: Abel's theorem makes $p\,W$ constant in $x$; the explicit minus sign makes $G$ positive for the box below, and $W(\lambda)=0$ is exactly the pole condition.

The entire character of $G$ — sum, integral, or both — is set by the nature of the spectrum.

1 · Both forms of G for a string, checked to a number (confidence builder)

Take $\mathcal{L}=-\mathrm{d}^2/\mathrm{d}x^2$ on $[0,1]$ with Dirichlet conditions $u(0)=u(1)=0$, and probe at $\lambda=0$. Build $G(x,x';0)$ as a closed form and as an eigenfunction sum, then evaluate both at the midpoint $x=x'=\tfrac12$.

Step 1. Closed form. The homogeneous equation is $-y''=0$, so $y_<=x$ (obeys $y(0)=0$) and $y_>=1-x$ (obeys $y(1)=0$); here $p=1$ and the Wronskian is $W[y_<,y_>]=y_<y_>'-y_<'y_>=x(-1)-(1)(1-x)=-1$. Equation (9) gives $G(x,x';0)=-\dfrac{x_<(1-x_>)}{(1)(-1)}=\mathbf{x_<(1-x_>)}$, positive as a Green's function of $-\partial_x^2$ must be.

Step 2. Eigenfunction sum. The normalized eigenfunctions are $\phi_n(x)=\sqrt{2}\,\sin(n\pi x)$ with eigenvalues $\lambda_n=n^2\pi^2$. Equation (4) at $\lambda=0$ reads $G(x,x';0)=\sum_{n=1}^\infty \dfrac{2\sin(n\pi x)\sin(n\pi x')}{n^2\pi^2}$.

Step 3. Evaluate at the midpoint. Closed form: $G=\tfrac12(1-\tfrac12)=\mathbf{1/4}$. Sum: $\sin(n\pi/2)=0$ for even $n$ and $\pm1$ for odd $n$, so only odd terms survive, $G=\sum_{n\,\text{odd}}\dfrac{2}{n^2\pi^2}=\dfrac{2}{\pi^2}\sum_{n\,\text{odd}}\dfrac{1}{n^2}=\dfrac{2}{\pi^2}\cdot\dfrac{\pi^2}{8}=\mathbf{1/4}$, using $\sum_{n\,\text{odd}}1/n^2=\pi^2/8$.

Step 4. The two constructions — one built from two solutions and a jump, the other from an infinite spectral series — agree exactly. Both give G(½,½;0) = 1/4, a numerical instance of the equivalence theorem in Derivation 2.

2 · Driven string: resonance and the Fredholm alternative

Solve $(\mathcal{L}-\lambda)u=f$ on the same string, expanding in the eigenbasis, and follow what happens as the drive $\lambda$ is tuned toward an eigenvalue $\lambda_m=m^2\pi^2$.

Step 1. Apply the resolvent: $u(x)=\int_0^1 G(x,x';\lambda)f(x')\,\mathrm{d}x'=\sum_n \dfrac{\langle\phi_n,f\rangle}{\lambda_n-\lambda}\,\phi_n(x)$, where $\langle\phi_n,f\rangle=\int_0^1\phi_n^* f\,\mathrm{d}x$. This is the generalized Fourier solution: project the source onto each mode, divide by the gap, sum.

Step 2. Tune toward resonance. As $\lambda\to\lambda_m$ the single term $n=m$ has a vanishing denominator, so $u\sim \dfrac{\langle\phi_m,f\rangle}{\lambda_m-\lambda}\phi_m(x)$ diverges — the string is being driven at one of its natural frequencies and the response runs away, dominated by the shape $\phi_m$.

Step 3. The escape clause. The divergence is avoided only if the source has no overlap with the resonant mode, $\langle\phi_m,f\rangle=0$. This is the Fredholm alternative: at $\lambda=\lambda_m$ exactly, $(\mathcal{L}-\lambda_m)u=f$ is solvable if and only if $f\perp\phi_m$.

Step 4. When it is solvable, drop the singular term and use the modified (reduced) Green's function $G_{\text{red}}=\sum_{n\neq m}\dfrac{\phi_n(x)\phi_n^*(x')}{\lambda_n-\lambda_m}$; the solution is then fixed only up to an arbitrary multiple of $\phi_m$, since $\phi_m$ solves the homogeneous equation. Concretely, if $f(x)=\sin(2\pi x)=\phi_2/\sqrt2$ and we drive at $\lambda_1=\pi^2$, then $\langle\phi_1,f\rangle\propto\int_0^1\sin(\pi x)\sin(2\pi x)\,\mathrm{d}x=0$, so the first-mode resonance is silent and the response stays finite.

The spectral representation is not one person's theorem; it is the meeting point of two traditions — the nineteenth-century boundary-value theory of Green, Sturm and Liouville, and the twentieth-century operator theory of Hilbert, Weyl and von Neumann — fused into a single object, the resolvent.

Texts. Stone & Goldbart, Mathematics for Physics, chapters on Green's functions and the spectral theorem · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Green's-function chapter · Hassani, Mathematical Physics, Sturm–Liouville and Green's-function chapters · Byron & Fuller, Mathematics of Classical and Quantum Physics, §§5–7 · Courant & Hilbert, Methods of Mathematical Physics, Vol. I · Reed & Simon, Methods of Modern Mathematical Physics, Vol. I · Economou, Green's Functions in Quantum Physics (density of states and the free resolvent). Two sign conventions coexist in the literature: this article fixes $(\mathcal{L}-\lambda)G=\delta$ throughout, giving spectral sum $\sum\phi_n\phi_n^*/(\lambda_n-\lambda)$, residue $-\phi_n\phi_n^*$, and $\rho=+\tfrac1\pi\mathrm{Im}\,G$; the alternative $(\lambda-\mathcal{L})G=\delta$ flips every sign and gives the familiar $\rho=-\tfrac1\pi\mathrm{Im}\,G^+$. The free 1D resolvent is $i\,e^{i\sqrt\lambda|x-x'|}/(2\sqrt\lambda)$ and the van Hove density of states is $\rho=1/(2\pi\sqrt\lambda)$; the box example confirms both forms of $G$ give $1/4$ at the midpoint via $\sum_{\text{odd}}n^{-2}=\pi^2/8$, and the free-line contour integral gives $\rho(1)=0.159$, $\rho(4)=0.080$.

Prerequisites. Linear Second-Order ODEs · Two-Point Boundary-Value Problems · The Wronskian and Linear Independence · Variation of Parameters. Leads to. Density of states · Scattering theory and the resolvent · Propagators in quantum field theory.

Ask the tutor: Want to see the spectral representation built explicitly for your own operator — a box, a drum, a hydrogen atom, or a tight-binding chain — with its poles, cut, and density of states worked out?