Spectral Decomposition¶

The spectral decomposition expresses the Green's function as a sum over eigenfunctions of the spatial operator. Each eigenfunction is an independent mode that decays at its own characteristic rate, turning a single parabolic PDE into a family of uncoupled ODEs. This is the PDE analog of diagonalizing a matrix, and in finance it produces the term-structure modes that organize long-maturity pricing.

This page is the eigenstructure lens on Green's functions; the operator/PDE lens is in Green's Function for Parabolic PDEs, the probability lens in Transition Density as Green's Function, and the geometry lens in Free vs Bounded Domains.

The Eigenvalue Problem¶

On a bounded domain \([a, b]\) with homogeneous boundary conditions, the generator \(\mathcal{L}\) has an eigenvalue problem

\[ \mathcal{L}\phi_n = -\lambda_n\,\phi_n, \qquad \text{boundary conditions on } [a,b]. \]

The minus sign makes \(\lambda_n > 0\) for dissipative operators. When \(\mathcal{L}\) is self-adjoint in the relevant inner product, spectral theory guarantees:

Real, ordered eigenvalues: \(0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \to \infty\).
Orthonormal eigenfunctions: \(\int_a^b \phi_m\,\phi_n\,dx = \delta_{mn}\).
Completeness: any \(f \in L^2([a,b])\) expands as \(f(x) = \sum_n c_n\,\phi_n(x)\) with \(c_n = \int f\phi_n\,dy\).

Spectral Expansion of the Green's Function¶

Expanding the delta initial condition \(\delta(x-y) = \sum_n \phi_n(y)\phi_n(x)\) and evolving each mode by \(e^{-\lambda_n t}\) gives the central formula:

\[ \boxed{\,G(t, x; s, y) = \sum_{n=1}^{\infty} e^{-\lambda_n(t-s)}\,\phi_n(x)\,\phi_n(y)\,} \]

Each term has three factors:

\(\phi_n(x)\) -- spatial shape of mode \(n\) at the observation point,
\(\phi_n(y)\) -- coupling of the source to mode \(n\),
\(e^{-\lambda_n(t-s)}\) -- temporal decay of mode \(n\).

Higher modes decay faster, so for large \(t-s\) only the lowest mode survives:

\[ G(t, x; 0, y) \sim e^{-\lambda_1 t}\,\phi_1(x)\,\phi_1(y). \]

Relation to separation of variables

The ansatz \(u(t,x) = T(t)X(x)\) in \(\partial_t u = \mathcal{L}u\) gives \(T'/T = \mathcal{L}X/X = -\lambda\), so the eigenvalue problem is exactly the spatial half of separation of variables. The spectral theorem guarantees the construction is complete.

Think of each eigenfunction as an independent channel. Mode \(n\) has its own amplitude (set by the initial data), its own shape \(\phi_n\), and its own lifetime \(1/\lambda_n\). Evolution is just exponential decay applied mode-by-mode.

Two convergence regimes matter:

Long time, \(t - s\) large: \(e^{-\lambda_n(t-s)}\) suppresses high modes strongly, and a handful of terms give extreme accuracy. The error after truncating at \(N\) terms is \(O(e^{-\lambda_{N+1}(t-s)})\).
Short time, \(t - s\) small: the series must reproduce a near-delta function, all high frequencies contribute, and convergence is slow. The short-time regime is better served by the method-of-images representation (see Free vs Bounded Domains).

The spectral gap \(\lambda_2 - \lambda_1\) controls how quickly the solution collapses onto the first mode.

Example: Heat Equation on [0, L]¶

For \(\mathcal{L} = \tfrac12\partial_{xx}\) on \([0,L]\) with Dirichlet conditions \(\phi(0)=\phi(L)=0\):

\[ \phi_n(x) = \sqrt{\tfrac{2}{L}}\sin\!\left(\tfrac{n\pi x}{L}\right), \qquad \lambda_n = \tfrac{n^2\pi^2}{2L^2}, \]

so

\[ G(t,x;0,y) = \frac{2}{L}\sum_{n=1}^{\infty} e^{-n^2\pi^2 t/(2L^2)}\sin\!\left(\tfrac{n\pi x}{L}\right)\sin\!\left(\tfrac{n\pi y}{L}\right). \]

With Neumann conditions \(\phi'(0)=\phi'(L)=0\), the same \(\lambda_n\) appear for \(n\geq 1\), plus a zero eigenvalue \(\lambda_0 = 0\) with constant eigenfunction \(\phi_0 = 1/\sqrt{L}\). The zero mode does not decay, reflecting conservation of total probability with reflecting boundaries.

Non-Self-Adjoint Operators¶

For generators with drift, \(\mathcal{L} = \mu(x)\partial_x + \tfrac12\sigma^2(x)\partial_{xx}\), the operator is not self-adjoint in \(L^2\). Two standard remedies.

Biorthogonal expansion: introduce adjoint eigenfunctions \(\psi_n\) with \(\mathcal{L}^*\psi_n = -\lambda_n\psi_n\), normalized so \(\int \phi_m\psi_n\,dx = \delta_{mn}\). Then

\[ G(t,x;s,y) = \sum_n e^{-\lambda_n(t-s)}\,\phi_n(x)\,\psi_n(y). \]

Weighted inner product: if \(\mathcal{L}\) is self-adjoint in \(L^2(m\,dx)\) for some speed measure \(m\), the expansion uses \(\phi_n\) orthonormal under that weight. This is the preferred route whenever a natural stationary measure exists.

Ornstein-Uhlenbeck: Hermite modes

Recall (see § Transition Density as Green's Function): the Ornstein-Uhlenbeck transition density is explicitly Gaussian. Here we focus on the eigenstructure. For \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\) on \(\mathbb{R}\), the stationary density \(m(x) \propto e^{-\kappa x^2/\sigma^2}\) makes \(\mathcal{L}\) self-adjoint in \(L^2(m\,dx)\). The eigenfunctions are the Hermite polynomials \(H_n\) with eigenvalues \(\lambda_n = n\kappa\). The spectral gap is exactly the mean-reversion rate \(\kappa\), and higher Hermite modes decay in integer multiples of it.

Finance Application: Term Structure Modes¶

In a short-rate model with generator \(\mathcal{L}\) and discrete spectrum, the zero-coupon bond price admits a spectral representation

\[ P(t, r; T) = \sum_{n=0}^{\infty} a_n(r)\,e^{-\lambda_n (T-t)}, \]

where the \(\lambda_n\) combine the generator's eigenvalues with the discounting. Each \(a_n(r)\,e^{-\lambda_n(T-t)}\) is a yield-curve mode: a fixed spatial shape in \(r\) whose contribution decays at a maturity-specific rate.

The yield \(Y(\tau) = -\log P / \tau\) for long \(\tau\) is dominated by the smallest eigenvalue:

\[ Y(\tau) \to \lambda_0 \quad \text{as } \tau \to \infty. \]

This is the long rate of the term structure -- a direct spectral quantity. In empirical studies, the first few eigenmodes give the standard "level, slope, curvature" decomposition of yield-curve movements, with spectral gaps controlling how quickly curvature shocks revert relative to slope shocks.

Summary¶

Aspect	Description
Eigenvalue problem	\(\mathcal{L}\phi_n = -\lambda_n\phi_n\) with BCs
Green's function	\(\sum_n e^{-\lambda_n(t-s)}\phi_n(x)\phi_n(y)\)
Long-time limit	Dominated by smallest \(\lambda_1\)
Spectral gap	\(\lambda_2 - \lambda_1\) sets approach-to-equilibrium rate
Non-self-adjoint	Biorthogonal or weighted expansion
Finance	Yield-curve modes, long rate = smallest eigenvalue

Exercises¶

Exercise 1. For \(\mathcal{L} = \tfrac12\partial_{xx}\) on \([0,\pi]\) with Dirichlet conditions, verify that \(\phi_n(x) = \sqrt{2/\pi}\sin(nx)\) are eigenfunctions with eigenvalues \(\lambda_n = n^2/2\), and check orthonormality.

Solution to Exercise 1

Solve \(\phi'' = -2\lambda\phi\) with \(\phi(0)=\phi(\pi)=0\): the general solution \(A\sin(\sqrt{2\lambda}x) + B\cos(\sqrt{2\lambda}x)\) reduces to \(A\sin(\sqrt{2\lambda}x)\) via \(\phi(0)=0\), and \(\phi(\pi)=0\) forces \(\sqrt{2\lambda} = n\), giving \(\lambda_n = n^2/2\).

Normalization: \(\int_0^\pi \sin^2(nx)\,dx = \pi/2\), so \(\phi_n(x) = \sqrt{2/\pi}\sin(nx)\). For \(m\neq n\), \(\int_0^\pi \sin(mx)\sin(nx)\,dx = \tfrac12\int_0^\pi[\cos((m-n)x) - \cos((m+n)x)]\,dx = 0\). Combined with the normalization, \(\int \phi_m\phi_n\,dx = \delta_{mn}\).

Exercise 2. For the heat equation on \([0,1]\) with Dirichlet conditions, write the spectral Green's function and compute the first three terms at \(t=1\). Estimate how many terms are needed for \(10^{-6}\) accuracy at \(t=1\) versus \(t=0.01\).

Solution to Exercise 2

With \(L=1\), \(\lambda_n = n^2\pi^2/2\) and \(\phi_n = \sqrt{2}\sin(n\pi x)\), so

\[ G(t,x;0,y) = 2\sum_{n=1}^\infty e^{-n^2\pi^2 t/2}\sin(n\pi x)\sin(n\pi y). \]

At \(t=1\): \(e^{-\pi^2/2}\approx 7.2\times 10^{-3}\), \(e^{-2\pi^2}\approx 2.7\times 10^{-9}\), \(e^{-9\pi^2/2}\approx 5.2\times 10^{-20}\). Two terms already give \(\sim 10^{-9}\) accuracy; one term beats \(10^{-6}\).

At \(t=0.01\): \(e^{-n^2\pi^2/200}\approx e^{-0.0493 n^2}\), decaying much more slowly in \(n\). Roughly \(e^{-0.0493 n^2} < 10^{-6}\) requires \(n^2 > 280\), i.e. about \(n\geq 17\). Short time needs many more modes -- the image series is better there.

Exercise 3. Explain why \(\lambda_1\) controls the long-time decay rate of the Dirichlet Green's function, and relate \(\lambda_1\) to the maturity decay of a double-barrier knock-out option on a corridor of log-width \(L\).

Solution to Exercise 3

For large \(t\) every term in \(\sum_n e^{-\lambda_n t}\phi_n(x)\phi_n(y)\) is dominated by the slowest decay, \(e^{-\lambda_1 t}\), with subleading terms suppressed by \(e^{-(\lambda_2-\lambda_1)t}\). Thus

\[ G(t,x;0,y) \sim e^{-\lambda_1 t}\phi_1(x)\phi_1(y). \]

Double-barrier option: the log-price lives on a corridor of width \(L = \log(B_u/B_l)\), with Dirichlet Green's function on that interval. The principal eigenvalue is \(\lambda_1 = \pi^2/(2L^2)\). For large maturity \(T\), the knock-out price decays like \(e^{-(\lambda_1 + r)T}\): the factor \(\lambda_1\) is the probabilistic leakage rate through the barriers, and \(r\) is discounting. Narrower corridors mean larger \(\lambda_1\) and faster decay.

Exercise 4. For the Sturm-Liouville problem \(-\tfrac12\phi'' = \lambda\phi\) on \([0,1]\) with Neumann conditions, find all eigenpairs. Why does a \(\lambda_0 = 0\) mode appear, and what does it imply about the long-time limit of the Green's function?

Solution to Exercise 4

The general solution \(A\cos(\sqrt{2\lambda}x) + B\sin(\sqrt{2\lambda}x)\) subject to \(\phi'(0)=0\) forces \(B=0\), and \(\phi'(1)=0\) requires \(\sin(\sqrt{2\lambda})=0\), giving \(\sqrt{2\lambda} = n\pi\) for \(n = 0,1,2,\dots\). Hence \(\lambda_n = n^2\pi^2/2\) with \(\phi_0 = 1\) and \(\phi_n = \sqrt{2}\cos(n\pi x)\) for \(n\geq 1\).

\(\lambda_0 = 0\) arises because the constant function satisfies both \(\phi''=0\) and the Neumann conditions -- Neumann boundaries do not pin the value anywhere, so a nonzero constant mode is admissible.

As \(t\to\infty\), all \(n\geq 1\) modes decay exponentially and \(G \to 1\) (uniform density on \([0,1]\)). Total probability is conserved because the zero mode does not decay: reflecting boundaries keep mass inside, and the limit is the equilibrium distribution.

Exercise 5. A short-rate model has generator with discrete spectrum \(\{\lambda_n\}_{n\geq 0}\). Starting from the spectral expansion of bond prices \(P(t,r;T) = \sum_n a_n(r)e^{-\lambda_n(T-t)}\), show that the long-maturity yield \(Y(\tau) = -\log P/\tau\) converges to \(\lambda_0\) as \(\tau\to\infty\). What financial quantity is this?

Solution to Exercise 5

For large \(\tau = T-t\), the term with smallest \(\lambda_n\) dominates:

\[ P(t,r;T) \sim a_0(r)\,e^{-\lambda_0\tau}\left[1 + \tfrac{a_1(r)}{a_0(r)}e^{-(\lambda_1-\lambda_0)\tau} + \cdots\right]. \]

Taking logs and dividing by \(\tau\),

\[ Y(\tau) = -\frac{\log P}{\tau} = \lambda_0 - \frac{\log a_0(r)}{\tau} + O(e^{-(\lambda_1-\lambda_0)\tau}/\tau) \to \lambda_0. \]

Financially this is the long rate of the term structure: the asymptotic yield common to all starting states \(r\). The spectral gap \(\lambda_1 - \lambda_0\) sets how quickly the yield curve flattens to this long rate.

Exercise 6. For an Ornstein-Uhlenbeck process \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\), state the eigenfunctions and eigenvalues in the weighted space \(L^2(m\,dx)\) with Gaussian stationary weight. Using only this eigenstructure (no explicit Gaussian formula), argue that the spectral gap equals the mean-reversion rate.

Solution to Exercise 6

With stationary density \(m(x) = \sqrt{\kappa/\pi\sigma^2}\,\exp(-\kappa x^2/\sigma^2)\), the generator \(\mathcal{L} = -\kappa x\partial_x + \tfrac12\sigma^2\partial_{xx}\) is self-adjoint in \(L^2(m\,dx)\). The eigenfunctions are the Hermite polynomials \(H_n\) (orthonormalized against \(m\)), with eigenvalues \(\lambda_n = n\kappa\) for \(n = 0,1,2,\dots\). The \(n=0\) mode is the constant function (the stationary density itself under the weight).

The spectral gap is \(\lambda_1 - \lambda_0 = \kappa - 0 = \kappa\). Any mean-zero perturbation of the stationary density lives in the span of \(H_n\) for \(n\geq 1\), and the slowest such mode decays as \(e^{-\kappa t}\). So any deviation from equilibrium relaxes at rate \(\kappa\) -- exactly the mean-reversion rate. The eigenstructure alone reproduces the familiar OU relaxation timescale without computing any densities.