Green's Functions for Ordinary Differential Equations

The ODE Green's function is the seed of the propagators and resolvents that organize modern physics. In quantum field theory the Feynman propagator is literally the Green's function of the Klein–Gordon or Dirac operator, and every Feynman diagram is a product of such propagators tied at vertices; lattice QCD extracts hadron masses from the decay of quark propagators on a discretized spacetime. In condensed-matter physics the many-body Green's function obeys Dyson's equation $G=G_0+G_0\Sigma G$ — the analogue of Lippmann–Schwinger with the self-energy $\Sigma$ in place of the scattering potential — and the $GW$ approximation built on it now predicts band gaps to tenths of an electronvolt in materials databases. The non-equilibrium Green's function formalism is the workhorse of nanoscale device simulation, modelling transport through sub-10-nanometre transistors by solving coupled Green's-function and Poisson equations self-consistently, while dynamical mean-field theory attacks strongly correlated systems — high-temperature superconductors, Mott insulators — through a local Green's function that flags metal-to-insulator transitions band theory misses. In every case the one-dimensional lessons — jump conditions, poles as spectrum, reciprocity, the Fredholm alternative — survive intact.

Picture a taut string pinned at both ends, pressed down at an interior point $\xi$ with a needle. It straightens into two segments — solving the source-free equation $-u''=0$ everywhere the needle is not — meeting at a sharp kink under the needle. That triangular tent is the Green's function, and every feature is dictated by three demands: it must vanish at the supports (boundary conditions), be a homogeneous solution on each side of the needle (straight segments), and kink just steeply enough that the second derivative supplies a unit delta at $\xi$ (the jump condition). The first figure makes the corner and its unit jump one picture.

Two homogeneous solutions do all the work. Call $y_1$ the one that obeys the left boundary and $y_2$ the right; the Green's function is $y_1$ left of the source and $y_2$ right, glued at $\xi$. The source never changes the shape of either piece — only their relative weighting and the size of the corner. Superposition finishes the story: an arbitrary load is a dense comb of point presses, so the total displacement is the integral of the tents weighted by the load. The second figure shows this assembly and delivers a hard number — a uniform unit load turns the family of tents into a parabola with a definite peak.

Reciprocity is the last idea the picture carries. For a self-adjoint operator the tent produced at $x$ by a source at $\xi$ equals the tent produced at $\xi$ by a source at $x$: $G(x,\xi)=G(\xi,x)$, the same symmetry that makes mutual inductances, acoustic transfer functions, and antenna gain patterns identical when source and receiver swap. Driven near a natural frequency, the spectral sum $\sum_n \phi_n(x)\phi_n(\xi)/(\lambda-\lambda_n)$ shows one term ballooning: $G$ diverges, announcing a resonance and the failure of an ordinary inverse.

Everything begins with a single definition: the Green's function is what the operator returns when fed a unit point source. The subscript on $\mathcal{L}_x$ is not decoration — the differentiation acts on the field point $x$, treating the source point $\xi$ as a fixed parameter.

where: $\mathcal{L}_x$ — linear differential operator acting on $x$; $G(x,\xi)$ — Green's function, the response at $x$ to a unit source at $\xi$; $\delta(x-\xi)$ — Dirac delta, the unit point source. Reading: $G$ solves the homogeneous problem away from $\xi$ and absorbs the delta through a controlled singularity there. The reason $G$ is worth computing is that it inverts the operator once and for all. Superposing the point responses against the actual source recovers the full solution, and the boundary conditions ride along for free because every $G(\cdot,\xi)$ already obeys them.

where: $u(x)$ — solution of $\mathcal{L}u=f$ obeying the boundary conditions; $f(\xi)$ — prescribed source; $[a,b]$ — domain. Reading: the differential equation has become a definite integral, with $G$ the kernel that represents $\mathcal{L}^{-1}$. The structure is cleanest when $\mathcal{L}$ is self-adjoint — which any real second-order operator becomes once its coefficients are absorbed into Sturm–Liouville form. Here $p$ multiplies the leading derivative and sets the scale of the jump, while $q$ plays the role of a potential.

where: $p(x)$ — positive coefficient of the leading derivative; $q(x)$ — zeroth-order (potential-like) coefficient. Reading: in this form the operator is formally self-adjoint, which is what makes $G$ symmetric and its eigenfunctions orthogonal. For a second-order operator the whole Green's function collapses to one closed form: the two homogeneous solutions multiplied together — smaller argument into $y_1$, larger into $y_2$ — divided by a single constant.

where: $y_1$ — homogeneous solution obeying the left boundary at $a$; $y_2$ — obeying the right boundary at $b$; $W(\xi)=y_1y_2'-y_1'y_2$ — the Wronskian. Reading: the numerator is symmetric under $x\leftrightarrow\xi$, so $G$ is manifestly reciprocal, and the denominator (below) is a genuine constant. The number that normalizes $G$ comes from integrating the defining equation across the source. Only the highest-derivative term survives the shrinking interval, so it carries the entire delta, forcing a jump in $p\,G'$.

where: $\xi^{\pm}$ — approach to the source from above and below. Reading: with $G$ continuous, the first derivative jumps by exactly $1/p(\xi)$ — large $p$ a gentle kink, small $p$ a sharp one. The denominator $p\,W$ looks as though it should vary with $\xi$, but Abel's identity for the self-adjoint form kills its derivative, so one evaluation fixes it everywhere.

where: $W(x)$ — Wronskian of the two homogeneous solutions; $p(x)$ — Sturm–Liouville leading coefficient. Reading: $(pW)'=0$, so the constant in equation (4) is evaluated at whichever point is convenient. When the eigenfunctions are known, $G$ has a second face — a bilinear sum over the spectrum, each mode weighted inversely by its distance from the driving parameter, so the sum diverges the moment $\lambda$ meets an eigenvalue.

where: $\phi_n$ — orthonormal eigenfunctions, $\mathcal{L}\phi_n=\lambda_n\phi_n$; $\lambda_n$ — eigenvalues; $\lambda$ — spectral parameter in $(\mathcal{L}-\lambda)G=-\delta$. Reading: this is the resolvent $(\mathcal{L}-\lambda)^{-1}$ in the eigenbasis, its poles the spectrum. Finally, self-adjointness delivers reciprocity as an exact theorem, not a lucky symmetry of one example.

where: $G(x,\xi)$ — Green's function of a self-adjoint operator with self-adjoint boundary conditions. Reading: swapping source and observation point leaves the response unchanged — the content of every reciprocity theorem.

The single object $G$ specializes to a menagerie of named kernels depending on the operator and its boundary conditions:

1 · The 1D Poisson Green's function (fully worked)

Construct $G(x,\xi)$ for $-u''=f$ on $[0,1]$ with $u(0)=u(1)=0$, evaluate it at a sample point, and verify continuity, the jump, and symmetry.

Step 1. Here $\mathcal{L}=-\mathrm{d}^2/\mathrm{d}x^2$, so $p=1$, $q=0$. The homogeneous equation $-y''=0$ has linear solutions; pick the boundary-adapted pair $y_1(x)=x$ (vanishes at $0$) and $y_2(x)=1-x$ (vanishes at $1$). Their Wronskian is $W=x(-1)-(1)(1-x)=-1$, so $pW=-1$, constant as Abel promised.

Step 2. Construct directly from continuity and the jump $[-G']_{\xi^-}^{\xi^+}=1$ (minding the overall minus in $\mathcal{L}$): this gives $G(x,\xi)=x(1-\xi)$ for $x\le\xi$ and $\xi(1-x)$ for $x\ge\xi$, i.e. the compact $G(x,\xi)=\min(x,\xi)\,[1-\max(x,\xi)]$.

Step 3. Evaluate at $x=0.3,\ \xi=0.6$: since $x<\xi$, $G=x(1-\xi)=0.3\times0.4=\mathbf{0.12}$. Swapping arguments, $G(0.6,0.3)$ has $x>\xi$, so $G=\xi(1-x)=0.3\times0.4=\mathbf{0.12}$ — identical, confirming reciprocity numerically.

Step 4. Check the jump. Left slope $\partial_xG=1-\xi$; right slope $-\xi$; jump $\partial_xG(\xi^+)-\partial_xG(\xi^-)=-\xi-(1-\xi)=\mathbf{-1}$, and since $-G''=\delta$ this is $[-G']=+1$ as required. At $\xi=0.4$ the tent peaks at $\xi(1-\xi)=\mathbf{0.24}$ with slopes $+0.6$ and $-0.4$ — precisely the first figure. $G=\min(x,\xi)[1-\max(x,\xi)]$, symmetric, with a unit derivative jump.

2 · Solving a real source with the kernel

Use the Green's function of Example 1 to solve $-u''=1$ on $[0,1]$ (a uniformly loaded string) and confirm the result independently.

Step 1. Superpose via equation (2), splitting at the source so each piece uses the correct branch: $u(x)=\int_0^1 G(x,\xi)\,\mathrm{d}\xi=\int_0^x \xi(1-x)\,\mathrm{d}\xi+\int_x^1 x(1-\xi)\,\mathrm{d}\xi$.

Step 2. The first integral is $(1-x)\tfrac{x^2}{2}$, the second $x\tfrac{(1-x)^2}{2}$. Adding and factoring $\tfrac{x(1-x)}{2}[x+(1-x)]$ collapses to $u(x)=\tfrac{1}{2}x(1-x)$, a downward parabola pinned at both ends.

Step 3. Verify directly: $u=\tfrac12(x-x^2)$ gives $u''=-1$, so $-u''=1$ as required, and $u(0)=u(1)=0$ holds. The maximum sag is at the centre, $u(\tfrac12)=\tfrac12\cdot\tfrac12\cdot\tfrac12=\mathbf{0.125=1/8}$ — the green curve and peak of the second figure. The kernel reproduced the exact parabolic deflection, peak $1/8$, in one integration.

3 · The retarded oscillator (impulse and step response)

Find the causal Green's function of $\mathcal{L}=\mathrm{d}^2/\mathrm{d}t^2+\omega^2$, then use it to compute the response of an initially resting oscillator to a suddenly applied constant force, with $\omega=2\ \mathrm{s}^{-1}$.

Step 1. Solve $G''+\omega^2G=\delta(t-\tau)$ with $G=0$ for $t<\tau$ (causality). For $t>\tau$, $G=C\cos\omega(t-\tau)+D\sin\omega(t-\tau)$; continuity at $t=\tau$ forces $C=0$ and the unit jump in $G'$ (here $p=1$) forces $D\omega=1$. Hence $G(t,\tau)=\dfrac{\sin[\omega(t-\tau)]}{\omega}\,\theta(t-\tau)$, an oscillation that switches on at the impulse with onset slope $G'(\tau^+)=1$.

Step 2. Drive with a unit step force $f(t)=F_0\,\theta(t)$, $F_0=1$, from rest. Convolve: $u(t)=\int_0^t \dfrac{\sin\omega(t-\tau)}{\omega}F_0\,\mathrm{d}\tau=\dfrac{F_0}{\omega}\int_0^t\sin\omega s\,\mathrm{d}s=\dfrac{F_0}{\omega^2}\big(1-\cos\omega t\big)$.

Step 3. Read the physics. A slowly applied $F_0$ gives static deflection $F_0/\omega^2=1/4=\mathbf{0.25}$; the dynamic response oscillates about it between $0$ and $2F_0/\omega^2$, so a suddenly applied load overshoots to twice the static value — dynamic amplification factor exactly $\mathbf{2}$. With $\omega=2$ the peak $u=\mathbf{0.5}$ first occurs at $\omega t=\pi$, i.e. $t=\pi/2\approx1.57\ \mathrm{s}$. $u(t)=\tfrac14(1-\cos2t)$, static $0.25$, peak $0.5$, amplification factor 2.

The Green's function carries the name of a man who had almost no formal education and whose masterwork sold to a few dozen subscribers before vanishing for two decades — that it now underlies quantum field theory is one of the great improbabilities in the history of mathematics.

Texts. Stone and Goldbart, Mathematics for Physics, Ch. 5 (Green's Functions) · Arfken, Weber and Harris, Mathematical Methods for Physicists, 7th ed., Ch. 10 · Hassani, Mathematical Physics, 2nd ed., Ch. 21 · Byron and Fuller, Mathematics of Classical and Quantum Physics, Ch. 7 · Riley, Hobson and Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 17 · Courant and Hilbert, Methods of Mathematical Physics, Vol. I · Morse and Feshbach, Methods of Theoretical Physics, Part I, Ch. 7. All equations are standard results. Key checks: the jump $[pG']=1$ follows from integrating $\mathcal{L}G=\delta$ across $\xi$; the 1D Poisson kernel $G=\min(x,\xi)[1-\max(x,\xi)]$ satisfies $-G''=\delta$, is symmetric, and peaks at $\xi(1-\xi)=0.24$ when $\xi=0.4$; integrating that kernel against $f=1$ recovers $u=\tfrac12 x(1-x)$ with maximum $1/8$; the retarded oscillator kernel $\sin[\omega(t-\tau)]/\omega\cdot\theta(t-\tau)$ yields the step response $\tfrac{F_0}{\omega^2}(1-\cos\omega t)$ with amplification factor 2; and Abel's identity $(pW)'=0$ underpins the modified-Green's-function construction. No numerical constants beyond exact analytic forms are asserted.

Builds on: Sturm–Liouville Theory · Dirac Delta Distribution · Wronskian and Linear Independence · Second-Order Linear ODEs. Leads to: WKB Approximation · Connection Formulae and Turning Points · Bohr–Sommerfeld Quantization · Asymptotic Series and Poincaré Expansions.

Ask the tutor: Name any operator — a damped oscillator, a Bessel equation, the finite-well Schrödinger problem — and I will build its Green's function with you from the jump condition to the full superposed solution.