Editing Self-adjoint operator (section)

=== Functional calculus ===
One application of the spectral theorem is to define a [[Borel functional calculus|functional calculus]]. That is, if <math>f</math> is a function on the real line and <math>T</math> is a self-adjoint operator, we wish to define the operator <math>f(T)</math>. The spectral theorem shows that if <math>T</math> is represented as the operator of multiplication by <math>h</math>, then <math>f(T)</math> is the operator of multiplication by the composition <math>f \circ h</math>.

One example from quantum mechanics is the case where <math>T</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian operator]] <math>\hat{H}</math>. If <math>\hat{H}</math> has a true orthonormal basis of eigenvectors <math>e_j</math> with eigenvalues <math>\lambda_j</math>, then <math>f(\hat{H}) := e^{-it\hat{H}/\hbar}</math> can be defined as the unique bounded operator with eigenvalues <math>f(\lambda_j) := e^{-it\lambda_j/\hbar}</math> such that:
: <math>f(\hat{H}) e_j = f(\lambda_j)e_j.</math>

The goal of functional calculus is to extend this idea to the case where <math>T</math> has continuous spectrum (i.e. where <math>T</math> has no normalizable eigenvectors). 

It has been customary to introduce the following notation
: <math>\operatorname{E}(\lambda) = \mathbf{1}_{(-\infty, \lambda]} (T)</math>
where <math>\mathbf{1}_{(-\infty, \lambda]}</math> is the [[indicator function]] of the interval <math>(-\infty, \lambda]</math>. The family of projection operators E(λ) is called [[Borel functional calculus#Resolution of the identity|'''resolution of the identity''']] for ''T''. Moreover, the following [[Stieltjes integral]] representation for ''T'' can be proved:
: <math> T = \int_{-\infty}^{+\infty} \lambda d \operatorname{E}(\lambda).</math>