Courant-Hilbert Minimax Principle for Eigenvalues

The Courant-Hilbert minimax principle serves as a foundational technique in functional analysis to derive the eigenvalues of differential operators. It finds extensive applications in quantum mechanics and other fields.

Compact Self-Adjoint Operators

For a compact, self-adjoint operator $A$ on a Hilbert space $H$, eigenvalues can be characterized variationaly:

\max_{S_k} \min_{x \in S_k, \|x\|=1} (Ax, x) = \lambda_k^\downarrow

\min_{S_{k-1}} \max_{x \in S_{k-1}^\perp, \|x\|=1} (Ax, x) = \lambda_k^\downarrow

Unbounded Self-Adjoint Operators

For more general unbounded self-adjoint operators with a continuous spectrum:

E_n = \min_{\psi_1, \ldots, \psi_n} \max\{\langle \psi, A\psi \rangle : \psi \in \text{span}(\psi_1, \ldots, \psi_n), \|\psi\| = 1\}

E_n = \max_{\psi_1, \ldots, \psi_{n-1}} \min\{\langle \psi, A\psi \rangle : \psi \perp \psi_1, \ldots, \psi_{n-1}, \|\psi\| = 1\}

General Formulation

The nth eigenvalue $\lambda_n$ of an operator is given by:

\min\{\max\{R_A(x) | x \in U \text{ and } x \neq 0\} | \dim(U) = k\} \geq \lambda_k

\min\{\max\{R_A(x) | x \in U \text{ and } x \neq 0\} | \dim(U) = k\} \leq \lambda_k

Employing the Rayleigh quotient within strategically chosen subspaces is pivotal in applying this principle to find eigenvalues, a crucial concept in the calculus of variations and mathematical physics.