Editing Self-adjoint operator (section)

== Spectral theorem ==
{{Main|Spectral theorem}}

In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the classic sense ''or'' some continuous analog thereof. In the case of the [[momentum operator]] <math display="inline">P = -i\frac{d}{dx}</math>, for example, physicists would say that the eigenvectors are the functions <math>f_p(x) := e^{ipx}</math>, which are clearly not in the Hilbert space <math>L^2(\mathbb{R})</math>. (Physicists would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "generalized eigenvectors" form an "orthonormal basis in the continuous sense" for <math>L^2(\mathbb{R})</math>, after replacing the usual [[Kronecker delta]] <math>\delta_{i,j}</math> by a [[Dirac delta function]] <math>\delta\left(p - p'\right)</math>.{{sfn|Hall|2013|pp=123-130|ps=}} 

Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general <math>L^2</math> function to be expressed as a "superposition" (i.e., integral) of the functions <math>e^{ipx}</math>, even though these functions are not in <math>L^2</math>. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by <math>p</math>, where <math>p</math> is the variable of the Fourier transform.

The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. 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.

=== Multiplication operator form of the spectral theorem ===
Firstly, let <math>(X, \Sigma, \mu)</math> be a [[Measure space#Important classes of measure spaces|σ-finite measure space]] and <math>h : X \to \mathbb{R}</math> a [[measurable function]] on <math>X</math>. Then the operator <math>T_h : \operatorname{Dom}T_h \to L^2(X,\mu)</math>, defined by
: <math>T_h \psi(x) = h(x)\psi(x), \quad \forall \psi \in \operatorname{Dom}T_h,</math>
where
: <math>\operatorname{Dom}T_h := \left\{\psi\in L^2(X,\mu) \;|\; h\psi \in L^2(X,\mu)\right\},</math>
is called a '''[[multiplication operator]]'''.{{sfn|Hall|2013|p=207|ps=}} Any multiplication operator is a self-adjoint operator.{{sfn | Akhiezer | 1981 | p=152|ps=}}

Secondly, two operators <math>A</math> and <math>B</math> with dense domains <math>\operatorname{Dom}A \subseteq H_1</math> and <math>\operatorname{Dom}B \subseteq H_2</math> in Hilbert spaces <math>H_1</math> and <math>H_2</math>, respectively, are '''unitarily equivalent''' if and only if there is a [[unitary transformation]] <math>U: H_1 \to H_2</math> such that:{{sfn | Akhiezer | 1981 | pp=115-116|ps=}}
* <math>U\operatorname{Dom}A = \operatorname{Dom}B,</math>
* <math> U A U^{-1} \xi = B \xi, \quad \forall \xi \in \operatorname{Dom}B. </math>
If unitarily equivalent <math>A</math> and <math>B</math> are bounded, then <math>\|A\|_{H_1}=\|B\|_{H_2}</math>; if <math>A</math> is self-adjoint, then so is <math>B</math>. 

{{math theorem|Any self-adjoint operator <math>A</math> on a [[separable space|separable]] Hilbert space is unitarily equivalent to a multiplication operator, i.e.,{{sfn|Hall|2013|pp=127,207|ps=}}
: <math>UAU^{-1}\psi(x) = h(x)\psi(x), \quad \forall \psi \in U\operatorname{Dom}(A)</math>
}}
The spectral theorem holds for both bounded and unbounded self-adjoint operators. Proof of the latter follows by reduction to the spectral theorem for [[unitary operator]]s.<ref>{{harvnb|Hall|2013}} Section 10.4</ref> We might note that if <math>T</math> is multiplication by <math>h</math>, then the spectrum of <math>T</math> is just the [[essential range]] of <math>h</math>.

More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".{{sfn|Hall|2013|pp=144-147,206-207|ps=}}

=== 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>

=== Formulation in the physics literature ===
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]],
: <math>f(H) = \int dE \left| \Psi_E \rangle f(E) \langle \Psi_E \right|</math>
with
: <math>H \left|\Psi_E\right\rangle = E \left|\Psi_E\right\rangle</math>

where the integral runs over the whole spectrum of ''H''. The notation suggests that ''H'' is diagonalized by the eigenvectors Ψ<sub>''E''</sub>. Such a notation is purely [[Formal calculation|formal]]. The resolution of the identity (sometimes called [[projection-valued measure]]s) formally resembles the rank-1 projections <math>\left|\Psi_E\right\rangle \left\langle\Psi_E\right|</math>. In the Dirac notation, (projective) measurements are described via [[eigenvalues]] and [[eigenstates]], both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the [[spectral measure]] of <math>|\Psi \rangle</math>, if the system is prepared in <math>|\Psi \rangle</math> prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable [[rigged Hilbert space]].

If {{nowrap|1=''f'' = 1}}, the theorem is referred to as resolution of unity:
: <math>I = \int dE \left|\Psi_E\right\rangle \left\langle\Psi_E\right|</math>

In the case <math>H_\text{eff} = H - i\Gamma</math> is the sum of an Hermitian ''H'' and a skew-Hermitian (see [[skew-Hermitian matrix]]) operator <math> -i\Gamma</math>, one defines the [[biorthogonal system|biorthogonal]] basis set
: <math>H^*_\text{eff} \left|\Psi_E^*\right\rangle = E^* \left|\Psi_E^*\right\rangle</math>
and write the spectral theorem as:
: <math>f\left(H_\text{eff}\right) = \int dE \left|\Psi_E\right\rangle f(E) \left\langle\Psi_E^*\right|</math>

(See ''[[Feshbach–Fano partitioning]]'' for the context where such operators appear in [[scattering theory]]).

=== Formulation for symmetric operators ===

The [[spectral theorem]] applies only to self-adjoint operators, and not in general to symmetric operators. 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. Consider the complex Hilbert space ''L''<sup>2</sup>[0,1] and the [[differential operator]]
: <math>A = -\frac{d^2}{dx^2}</math>
with <math>\mathrm{Dom}(A)</math> consisting of all complex-valued infinitely [[differentiable function|differentiable]] functions ''f'' on [0, 1] satisfying the boundary conditions 
: <math>f(0) = f(1) = 0.</math>

Then [[integration by parts]] of the inner product shows that ''A'' is symmetric.<ref group=nb>The reader is invited to perform integration by parts twice and verify that the given boundary conditions for <math>\operatorname{Dom}(A)</math> ensure that the boundary terms in the integration by parts vanish.</ref> The eigenfunctions of ''A'' are the sinusoids
: <math>f_n(x) =  \sin(n \pi x) \qquad  n= 1, 2, \ldots</math>
with the real eigenvalues ''n''<sup>2</sup>π<sup>2</sup>; the well-known orthogonality of the sine functions follows as a consequence of ''A'' being symmetric.

The operator ''A'' can be seen to have a [[compact operator on Hilbert space|compact]] inverse, meaning that the corresponding differential equation ''Af'' = ''g'' is solved by some integral (and therefore compact) operator ''G''. The compact symmetric operator ''G'' then has a countable family of eigenvectors which are complete in {{math|''L''<sup>2</sup>}}. The same can then be said for ''A''.

=== Pure point spectrum ===
{{distinguish|Discrete spectrum (mathematics)}}
A self-adjoint operator ''A'' on ''H'' has pure [[point spectrum]] if and only if ''H'' has an orthonormal basis {''e<sub>i</sub>''}<sub>''i'' ∈ I</sub> consisting of eigenvectors for ''A''.

'''Example'''. The Hamiltonian for the harmonic oscillator has a quadratic potential ''V'', that is
: <math>-\Delta  + |x|^2.</math>

This Hamiltonian has pure point spectrum; this is typical for bound state [[Hamiltonian (quantum mechanics)|Hamiltonians]] in quantum mechanics.{{clarify|reason=The above example is from classical mechanics|date=November 2023}}{{sfn | Ruelle | 1969|ps=}} 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.