This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space.
The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case.
With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces.
Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case.
Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have "eigenvectors" that are not actually in the Hilbert space in question.
More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".
The family of projection operators E(λ) is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved: In quantum mechanics, Dirac notation is used as combined expression for both the spectral theorem and the Borel functional calculus.
That is, if H is self-adjoint and f is a Borel function, with where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvectors ΨE.
In the Dirac notation, (projective) measurements are described via eigenvalues and eigenstates, both purely formal objects.
, one defines the biorthogonal basis set and write the spectral theorem as: (See Feshbach–Fano partitioning for the context where such operators appear in scattering theory).
Nevertheless, we can at this point give a simple example of a symmetric (specifically, an essentially self-adjoint) operator that has an orthonormal basis of eigenvectors.
consisting of all complex-valued infinitely differentiable functions f on [0, 1] satisfying the boundary conditions Then integration by parts of the inner product shows that A is symmetric.
[nb 1] The eigenfunctions of A are the sinusoids with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of A being symmetric.
A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A.
[clarification needed][23] As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.
Let us define a momentum operator A on this space by the usual formula, setting the Planck constant to 1: We must now specify a domain for A, which amounts to choosing boundary conditions.
If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A, the functions
In one dimension, for example, the operator is not essentially self-adjoint on the space of smooth, rapidly decaying functions.
However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric.
In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.
of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions.
This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators P on an open set M. They are determined by the unitary maps between the eigenvalue spaces where Pdist is the distributional extension of P. We next give the example of differential operators with constant coefficients.
Let be a polynomial on Rn with real coefficients, where α ranges over a (finite) set of multi-indices.
Then F*P(D)F is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function P. More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support.
If M is an open subset of Rn where aα are (not necessarily constant) infinitely differentiable functions.
This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators A and B are unitarily equivalent.
A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator Mf of multiplication by the function f(λ) = λ on where Hn is a Hilbert space of dimension n. The domain of Mf consists of vector-valued functions ψ on R such that Non-negative countably additive measures μ, ν are mutually singular if and only if they are supported on disjoint Borel sets.
Theorem — Let A be a self-adjoint operator on a separable Hilbert space H. Then there is an ω sequence of countably additive finite measures on R (some of which may be identically 0)
The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces: Theorem — [35] Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ ↦ λ on