Self-adjoint Operator

Self-adjoint operators are used in Functional Analysis and Quantum Mechanics . In quantum mechanics their importance lies in the fact that in the Dirac - Von Neumann formulation of quantum mechanics, physical Observable s such as Position , Momentum , Angular Momentum and Spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian : $H \psi = - rac{\hbar^2}{2 m} \Delta \psi + V \psi$ which as an observable corresponds to the total energy of a particle of mass ''m'' in a potential field ''V''. The structure of self-adjoint operators on infinite dimensional Hilbert spaces essentially resembles the finite dimensional case, that is to say, operators are self adjoint iff they are unitarily equivalent to real-valued multiplication operators. The result also holds for the unbounded case, provided one is careful. An everywhere defined and self adjoint operator is necessarily bounded. This means one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail. Differential operators are typical examples of unbounded operators. SYMMETRIC OPERATORS A partially defined Linear Operator ''A'' on a Hilbert space ''H'' is called symmetric Iff : $\langle Ax \mid y angle = \lang x \mid Ay ang$ for all elements ''x'' and ''y'' in the domain of ''A''. This usage is fairly standard in the functional analysis literature. By the Hellinger-Toeplitz Theorem , a symmetric ''everywhere defined'' operator is Bounded . Bounded symmetric operators are also called Hermitian. The previous definition agrees with the one for matrices given in the introduction to this article, if we take as ''H'' the Hilbert space C^''n'' with the standard dot product and interpret a square matrix as a linear operator on this Hilbert space. It is however much more general as there are important infinite-dimensional Hilbert spaces. The Spectrum of any bounded symmetric operator is real; in particular all its eigenvalues are real, although a symmetric operator may not have any eigenvalues. A general version of the Spectral Theorem which also applies to bounded symmetric operators is stated below. If the set of eigenvalues for a symmetric operator is non empty, and the eigenvalues are nondegenerate, then it follows from the definition that eigenvectors corresponding to distinct eigenvalues are orthogonal. Contrary to what is sometimes claimed in introductory physics textbooks, it is possible for symmetric operators to have no eigenvalues at all (although the ''spectrum'' of any self adjoint operator, bounded or otherwise is nonempty). The example below illustrates the special case when an (unbounded) symmetric operator does have a set of eigenvalues which constitute a Hilbert space basis. The operator ''A'' below can be seen to have a compact "inverse," meaning that the corresponding differential equation ''A'' ''f'' = ''g'' is solved by some integral, therefore compact, operator $G$ . The compact self adjoint operator ''G'' then has a countable family of eigenvectors which are complete in $L^2$ . The same can then be said for ''A''. Example. Consider the complex Hilbert space L² {Link without Title} and the Differential Operator : $A = - rac{d^2}{dx^2}$ defined on the subspace consisting of all complex-valued infinitely Differentiable functions ''f'' on {Link without Title} with the boundary conditions: : $f(0) = f(1) = 0 \quad$ Then Integration By Parts shows that ''A'' is symmetric. Its eigenfunctions are the sinusoids : $f_n(x) = \sin(n \pi x) \quad n= 1,2, \ldots$ with the real eigenvalues ''n''²π²; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric. We consider generalizations of this operator below. SELF-ADJOINT OPERATORS is defined as follows: The domain of ''A''--- consists of vectors ''x'' in ''H'' such that :: $y \mapsto \langle x \mid A y angle$ : (which is a densely defined ''linear'' map) is a continuous linear functional. By continuity and density of the domain of ''A'', it extends to a unique continuous linear functional on all of ''H''. By the Riesz Representation Theorem for linear functionals, if ''x'' is in the domain of ''A''---, there is a unique vector ''z'' in ''H'' such that :: $\langle x \mid A y angle = \langle z \mid y angle \quad orall y \in \operatorname{dom} A$ ''x''. It can be shown that the dependence of ''z'' on ''x'' is linear. Notice that it is the denseness of the domain of the operator, along with the uniqueness part of Riesz representation, that ensures the adjoint operator is well defined. Geometric interpretation There is a useful Geometric al way of looking at the adjoint of an operator ''A'' on ''H'' as follows: we consider the graph G(''A'') of ''A'' defined by : $\operatorname{G}(A) = \{(\xi, A \xi): \xi \in \operatorname{dom}(A)\} \subseteq H \oplus H .$ Theorem. Let J be the Symplectic Mapping : $H \oplus H ightarrow H \oplus H$ given by : $\operatorname{J}: (\xi, \eta) \mapsto (-\eta, \xi).$ is the Orthogonal Complement of JG(''A''): ) = (\operatorname{J}\operatorname{G}(A))^\perp = \{
	Where The Integral Runs Over The Whole Spectrum Of ''H'' The Notation Suggests That ''H'' Is Diagonalized By The Eigenvalues &Psi<sub>''E''</sub> Such A Notation Is Purely	"http://wwwinformationdelightinfo/encyclopedia/entry/formal" class="copylinks">Formal One can see the similarity between Dirac's notation and the previous section The resolution of the identity(sometimes called projection valued measures) formally resembles the rank-1 projections <math> \Psi_{E} angle \langle \Psi_{E} </math>
	In The Dirac Notation, (projective) Measurements Are Described Via	"http://wwwinformationdelightinfo/encyclopedia/entry/eigenvalues" class="copylinks">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 angle </math>, if the system is prepared in <math> \Psi angle </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
	:<math> \langle \xi \eta Angle \mathrm{graph}	\langle \xi \eta angle + \langle A^\xi A^ \eta angle </math>

where ''P''_dist is the distributional extension of ''P''.

We next give the example of differential operators with Constant Coefficient s. Let

:

P(ec{x}) = \sum_\alpha c_\alpha x^\alpha

be a polynomial on R^''n'' with ''real'' coefficients, where α ranges over a (finite) set of Multi-indices . Thus

:

\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)

and

:

x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}.

We also use the notation:

of ''P'' is a restriction of the distributional extension of the formal adjoint. Specifically:

= \{u \in L^2(M): P^{\mathrm{---form}}u

SPECTRAL MULTIPLICITY THEORY

The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. 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. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the '' Hahn - Hellinger theory of spectral multiplicity''.

We first define ''uniform multiplicity'':

Definition. A self-adjoint operator ''A'' has uniform multiplicity ''n'' where ''n'' is such that 1 ≤ ''n'' ≤ ω
iff ''A'' is unitarily equivalent to the operator M_''f'' of multiplication by the function ''f''(λ) = λ on

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	CATEGORIES ABOUT SELF-ADJOINT OPERATOR

	operator theory

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Self-adjoint Operator