Spectral and Pseudospectral Methods

Spectral methods approximate a wavefunction with global basis functions. Pseudospectral methods evaluate differential equations or products at interpolation nodes. For smooth, well-resolved problems they can converge much faster with resolution than a fixed-order finite-difference stencil, but boundary mismatch, singularities, and aliasing remove that advantage. Here “spectral” describes the spatial approximation; it does not mean that every resulting eigenvalue is accurate.

Required background. Basis choice explains coefficient metrics, and boundary conditions identify the operator domain. Helpful background. Split-operator propagation explains Fourier grids used in time evolution.

Fourier differentiation on a periodic interval

Let $H=-\alpha\partial_x^2+V(x)$, with $\alpha=\hbar^2/(2m)$, on a periodic interval of length $L$. Sample $x_j=jL/N$, $j=0,\ldots,N-1$; the endpoint $L$ is not an additional independent point. With a unitary discrete Fourier transform $F$, define represented wave numbers $k_n=2\pi n/L$ in the transform’s frequency order. Then

$$T=F^\dagger\operatorname{diag}(\alpha k_n^2)F, \qquad H_{\mathrm{coll}}=T+\operatorname{diag}(V(x_j)).$$

For real sampled $V$, this matrix is Hermitian in the uniform grid metric, proportional to the identity. Applying $T$ uses two FFTs and a diagonal multiplication. No dense kinetic matrix is needed. Distinguish physical wavefunction samples $\psi_j$ from normalized discrete coefficients $c_j=\sqrt{L/N}\,\psi_j$.

For $V=0$, each represented Fourier mode is an exact eigenvector of this discretization with energy $\alpha k_n^2$. On $L=2\pi$, the mode $e^{3ix}$ has eigenvalue $9\alpha$ whenever it is unambiguously represented. This checks differentiation of a single represented mode. A sharply localized wave packet generally needs many modes, so it also requires a bandwidth check.

For even $N$, the Nyquist mode requires a consistent convention. Its second derivative is unambiguous in this kinetic construction; a real-grid first derivative has additional symmetry considerations. Do not create the Laplacian by squaring an arbitrary first-derivative implementation that has zeroed its Nyquist multiplier.

Projection and collocation make different potential matrices

A Fourier Galerkin method computes matrix elements of $V$ projected onto the retained basis. Multiplying sampled values and transforming back instead computes a discrete convolution whose frequency indices wrap modulo $N$. These operations coincide only when the relevant products are represented accurately by the quadrature.

For example, on an $N=8$ grid, a sampled product $e^{3ix}e^{3ix}=e^{6ix}$ is indistinguishable from $e^{-2ix}$. A well-converged eigensolver cannot detect this aliasing error from its residual. Increase spatial resolution, inspect Fourier coefficient tails, and, when forming products, evaluate them on a sufficiently enlarged grid before projecting back. The necessary padding depends on the product bandwidth; a single slogan about dealiasing is not a universal rule for arbitrary potentials or nonlinearities.

Polynomial collocation on a finite interval

For nonperiodic problems, Chebyshev–Lobatto nodes $\xi_j=\cos(j\pi/N)$ cluster near the endpoints. If $\ell_j$ are the corresponding cardinal polynomials, the differentiation matrix is

$$D_{ij}=\ell_j'(\xi_i).$$

Map $x=(a+b)/2+(b-a)\xi/2$. The second derivative is represented by $[2/(b-a)]^2D^2$. For homogeneous Dirichlet data, form the full second derivative first and then retain its interior rows and columns. Squaring the already-truncated first derivative generally gives a different operator because the intermediate derivative does not satisfy the wavefunction’s boundary condition.

Unlike a variational stiffness matrix, a raw collocation matrix need not be Euclidean Hermitian. Do not pass it to a Hermitian eigensolver merely because the continuum Hamiltonian is self-adjoint. Verify the discrete metric and symmetry or use a solver suitable for the matrix actually constructed. Finite elements offer a different route when retaining the weak-form variational structure is central.

What spectral convergence requires

Trefethen develops the relation between function regularity and spectral approximation. For an analytic solution with compatible boundaries, coefficient tails often decrease geometrically until other errors dominate. Finite differentiability generally produces algebraic decay. A discontinuity causes oscillatory interpolation errors; increasing polynomial degree alone does not turn that problem into an analytic one.

Separate three checks: enlarge the physical domain when approximating an infinite line; increase the number of modes at fixed domain; and tighten the algebraic solver tolerance. Compare low-energy states first, inspect wavefunctions as well as energies, and monitor the last retained coefficients. A residual measures the discrete problem, whereas changing the representation tests its approximation to the continuum.

Exercises

Explain why a localized Gaussian on a periodic interval may stop showing rapid Fourier convergence even though a Gaussian is analytic on the real line.

Solution

The periodically extended restriction must match at the endpoints, including derivatives. A small value at both boundaries does not guarantee exact derivative matching. The resulting periodic seam limits asymptotic convergence. Enlarging the domain reduces the seam error; increasing $N$ at a fixed domain eventually resolves that error rather than eliminating it.

On $N=8$ equally spaced points on $[0,2\pi)$, which represented frequency has the same samples as $e^{5ix}$? Why does the kinetic operator then give the wrong continuum energy for that unrepresented mode?

Solution

The samples equal those of $e^{-3ix}$ because $e^{8ix_j}=1$. The grid kinetic operator assigns $9\alpha$, whereas the continuum frequency $5$ has energy $25\alpha$. Sampling alone has lost the information needed to distinguish them.

References

• Boyd, John P. Chebyshev and Fourier Spectral Methods. 2nd ed. Dover, 2001.
• Trefethen, Lloyd N. Spectral Methods in MATLAB. SIAM, 2000. Author’s book page and programs.