Variational eigenvalue principles

For a Hermitian matrix A and nonzero vector x, the Rayleigh quotient is x*Ax divided by x*x. Expanding x in an orthonormal eigenbasis expresses the quotient as an average of the eigenvalues with nonnegative weights summing to one. It lies between the smallest and largest eigenvalues. Equality at the smallest value occurs precisely for vectors in its eigenspace. This statement tolerates degeneracy: a unique preferred ground-state vector is not required.

For an unbounded self-adjoint operator bounded below, a closed quadratic form extends this idea to a form domain V that can be larger than D(A). For the Dirichlet Laplacian, the form integrates |u′|² and its domain is H1_0, while the operator domain additionally requires a second weak derivative in L2. Admissible variational trial functions must belong to the form domain. A discontinuous or boundary-violating trial function cannot be rescued by a small numerical quotient computed from an inconsistent discretization.

Assume a self-adjoint operator is bounded below and has compact resolvent on an infinite-dimensional Hilbert space. Its eigenvalues can then be listed increasingly with multiplicity and tend to positive infinity. In this discrete setting the kth eigenvalue is the infimum, over all k-dimensional subspaces of the form domain, of the largest Rayleigh quotient in that subspace. Every k-dimensional space must contain a vector orthogonal to the first k−1 eigenvectors, preventing its worst quotient from falling below λ_k.

Choosing one finite trial space restricts the available subspaces and therefore gives Ritz values that are upper bounds for the corresponding exact eigenvalues. Enlarging nested conforming trial spaces cannot increase those Ritz values. Convergence to the true eigenvalues requires the trial spaces to approximate the form domain appropriately. Compactness is a convenient sufficient setting for a fully discrete sequence. With essential spectrum, related min–max statements need a different formulation and may reach a spectral threshold rather than produce another isolated eigenvalue.

A trial energy above the exact ground energy is guaranteed by the variational principle, but the principle alone does not tell how far above it lies. A small residual can add information. For a normalized vector x in D(A) and real μ, the spectral theorem gives distance from μ to the spectrum at most norm((A−μI)x). This locates some spectral value near μ; identifying which eigenvalue and bounding an eigenvector error additionally require spectral separation or other information.

Numerical quadrature, nonconforming spaces, and roundoff can alter the computed form or inner product, so an observed floating-point Ritz value is not automatically a rigorous mathematical upper bound. One needs controlled integration and arithmetic for certified bounds. The practical workflow is to check admissibility, state the min–max hypotheses, refine the trial space, and inspect residuals or independent bounds. The interactive finite sine family helps suggest trial shapes, but it computes neither the quadratic form nor a variational eigenvalue certificate.