Self-adjoint operator on arvau

Wikipedia / Self-adjoint operator

In mathematics, a self-adjoint operator on a complex vector space with inner product is a linear map to itself) that is its own adjoint is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix is a Hermitian matrix, i.e., equal to its conjugate transpose. By the finite-dimensional spectral theorem has an orthonormal basis relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.

Self-adjoint operators are used in functional analysis and quantum mechanics. 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. Of particular significance is the Hamiltonian defined by

which as an observable corresponds to the total energy in a real potential field. Differential operators are an important class of unbounded operators.

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 if and only if they are unitarily equivalent to real-valued multiplication operators. 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. This is explained below in more detail.

Definitions

be a Hilbert space an unbounded (i.e. not necessarily bounded) linear operator with a dense domain is finite-dimensional for every linear operator on a finite-dimensional space.

The graph is said to extend

be conjugate linear on the second argument. The adjoint operator such that

The densely defined is called symmetric (or Hermitian is symmetric if and only if

, symmetric operators are always closable is the graph of an operator).. Hence,

for symmetric operators and

for closed symmetric operators.

is called self-adjoint, i.e.,

is said to be essentially self-adjoint is essentially self-adjoint if it has a unique self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator.

In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.

Bounded self-adjoint operators

be a Hilbert space a symmetric operator. According to Hellinger–Toeplitz theorem is necessarily bounded. A bounded operator is self-adjoint if

can be written in the complex are bounded self-adjoint operators.

Alternatively, every positive bounded linear operator is complex.

Properties

has the following properties:

is invertible if the image
The operator norm
is an eigenvalue; the eigenvalues are real and the corresponding eigenvectors are orthogonal.

is a compact self-adjoint operator and corresponding normalized eigenvector.

Spectrum of self-adjoint operators

be an unbounded operator. The resolvent set (or regular set is defined as

being bijective. The spectrum is defined as the complement

consists exclusively of (complex) eigenvalues.), though non-self-adjoint operators with real spectrum exist as well. For bounded (normal) operators, however, the spectrum is real if and only if the operator is self-adjoint. This implies, for example, that a non-self-adjoint operator with real spectrum is necessarily unbounded.

By the Cauchy–Schwarz inequality,

is called bounded below.

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, after replacing the usual Kronecker delta by a Dirac delta function.

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

be a σ-finite measure space a measurable function, defined by

where

is called a multiplication operator. Any multiplication operator is a self-adjoint operator.

, respectively, are unitarily equivalent if and only if there is a unitary transformation such that:

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 operators. is just the essential range.

More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".

Functional calculus

One application of the spectral theorem is to define a functional calculus.

is the Hamiltonian operator such that:

has no normalizable eigenvectors).

It has been customary to introduce the following notation

is the indicator function. 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:

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,

with

where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvectors ΨE. Such a notation is purely formal. The resolution of the identity (sometimes called projection-valued measures. 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 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 , the theorem is referred to as resolution of unity:

is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix, 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).

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 L2[0,1] and the differential operator

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. 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.

The operator A can be seen to have a 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 . The same can then be said for A.

Pure point spectrum

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.

Example. The Hamiltonian for the harmonic oscillator has a quadratic potential V, that is

This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. 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.

Symmetric vs self-adjoint operators

Although the distinction between a symmetric operator and a (essentially) self-adjoint operator is subtle, it is important since self-adjointness is the hypothesis in the spectral theorem. Here we discuss some concrete examples of the distinction.

Boundary conditions

In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying boundary conditions (the space of square-integrable functions on the interval [0,1]). 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 choose

then A is not symmetric (because the boundary terms in the integration by parts do not vanish).

If we choose

then using integration by parts, one can easily verify that A is symmetric. This operator is not essentially self-adjoint, however, basically because we have specified too many boundary conditions on the domain of A, which makes the domain of the adjoint too big (see also the example below).

Specifically, with the above choice of domain for A of A is

of A is

That is to say, the domain of the closure has the same boundary conditions as the domain of A itself, just a less stringent smoothness assumption. Meanwhile, since there are "too many" boundary conditions on A.

Since the domain of the closure and the domain of the adjoint do not agree, A is the same as the domain of the adjoint of A is not self-adjoint, which by definition means that A is not essentially self-adjoint.

The problem with the preceding example is that we imposed too many boundary conditions on the domain of A. A better choice of domain would be to use periodic boundary conditions:

With this domain, A is essentially self-adjoint.

, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions), A has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A. Thus, in this case finding a domain such that A is self-adjoint is a compromise: the domain has to be small enough so that A.

Schrödinger operators with singular potentials

A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from Schrödinger operators in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator

is not essentially self-adjoint on the space of smooth, rapidly decaying functions. potential escapes to infinity in finite time. This operator does not have a unique is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension.)

on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely

has eigenvectors with pure imaginary eigenvalues,.

Non-self-adjoint operators in quantum mechanics

In quantum mechanics, observables correspond to self-adjoint operators. By Stone's theorem on one-parameter unitary groups, self-adjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. 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.

Example., defined initially on smooth compactly supported functions, is essentially self-adjoint for but not for .

: The classical particle escapes to infinity in finite time.

Example. of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin.

Examples

A symmetric operator that is not essentially self-adjoint

and the differential operator

defined on the space of continuously differentiable complex-valued functions on [0,1], satisfying the boundary conditions

Then D is a symmetric operator as can be shown by integration by parts. The spaces N+, N− (defined below) are given respectively by the distributionalal solutions to the equation

which are in L2[0, 1]. One can show that each one of these solution spaces is 1-dimensional, generated by the functions x → e−x and x → ex respectively. This shows that D is not essentially self-adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings N+ → N−, which in this case happens to be the unit circle T.

is symmetric. If we replaced the boundary conditions given above by the single boundary condition

then D would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of D.

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.

Constant-coefficient operators

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. Thus

and

We also use the notation

Then the operator P(D) defined on the space of infinitely differentiable functions of compact support on Rn by

is essentially self-adjoint on L2(Rn).

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. P is a linear operator

Corresponding to P there is another differential operator, the formal adjoint of P

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.

Uniform multiplicity

We first define uniform multiplicity:

Definition. 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.

This representation is unique in the following sense: For any two such representations of the same A, the corresponding measures are equivalent in the sense that they have the same sets of measure 0.

Direct integrals

The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces:

Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of μ is determined almost everywhere with respect to μ. is the spectral multiplicity function of the operator.

We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral multiplicity functions agree almost everywhere with respect to the measure in the direct integral.

Example: structure of the Laplacian

The Laplacian on Rn is the operator

As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the negative of the Laplacian −Δ since as an operator it is non-negative; (see elliptic operator).