Spectral Theorem

Full treatment. Read the Spectral Theorem for Unbounded Self-Adjoint Operators for the multiplication representation, proof architecture, continuous spectrum, solved exercises, and failure cases.

Statement

For every self-adjoint $A$ there is a unique PVM $E_A$ on $\mathbb R$ such that

$$A=\int_{\mathbb R}\lambda\,dE_A(\lambda)$$

on

$$D(A) = \left\{ \psi: \int_{\mathbb R}\lambda^2 \,d\langle\psi,E_A(\lambda)\psi\rangle<\infty \right\}.$$

For a Borel function $f$,

$$f(A)=\int f(\lambda)\,dE_A(\lambda),$$

with domain determined by the same formula using $|f(\lambda)|^2$.

Check before use

• $A$ is self-adjoint, not merely symmetric.
• Unbounded spectral integrals have a declared domain.
• $E_A(\{\lambda\})\ne0$ exactly for eigenvalues; continuous spectrum need not have normalizable eigenvectors.
• Bounded functions give bounded everywhere-defined operators; unbounded functions generally do not.

Quick consequence

For a normalized $\psi$,

$$\Pr_\psi(A\in\Delta) = \langle\psi,E_A(\Delta)\psi\rangle.$$

This gives sharp-event probabilities but does not specify a measurement instrument or state update.

References

• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.