On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional spectral theorem such operators have an orthonormal basis in which the operator can be represented as a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product or dot product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... For the square matrix section, see square matrix. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematics, an orthonormal basis of an inner product space V(i. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Concept A is a (strict) generalization of concept B if and only if: every instance of concept B is also an instance of concept A; and there are instances of concept A which are not instances of concept B. Equivalently, A is a generalization of B if B is a... A concept is an abstract, idea, notion, or entity that serves to designate a category or class of entities, events, phenomena or relations between them. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

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 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 Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... A simple introduction to this subject is provided in Basics of quantum mechanics. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... A separate article covers Saint John Neumann, the American priest. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... The word position can have one of a billion meanings. ... In physics, momentum is the product of the mass and velocity of an object. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... The terms spin and SPIN have several meanings, including those primarily discussed as spinning: For spin in sub-atomic physics, see spin (physics) For the stalled aircraft maneuver or any of several forms of loss of control in aircraft, see spin (flight) For the periodical, see Spin Magazine For the... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

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 In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

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. In functional analysis, a branch of mathematics, the Hellinger-Toeplitz theorem states that an everywhere defined symmetric operator on a Hilbert space is bounded. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...

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 Cⁿ 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. In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ...

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 L². The same can then be said for A. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

Example. Consider the complex Hilbert space L²[0,1] and the differential operator In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

defined on the subspace consisting of all complex-valued infinitely differentiable functions f on [0,1] with the boundary conditions: In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

Then integration by parts shows that A is symmetric. Its eigenfunctions are the sinusoids In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

with the real eigenvalues n²π²; 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

Given a densely defined linear operator A on H, its adjoint A* is defined as follows:

• The domain of A* consists of vectors x in H such that

(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

This vector z is defined to be 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. There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

Geometric interpretation

There is a useful geometrical way of looking at the adjoint of an operator A on H as follows: we consider the graph G(A) of A defined by Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...

Theorem. Let J be the symplectic mapping In mathematics, a symplectic matrix is a 2n×2n matrix M (whose entries are typically either real or complex) satisfying the condition where MT denotes the transpose of M and Ω is the 2n×2n skew-symmetric matrix Here In is the n×n identity matrix. ...

given by

Then the graph of A* is the orthogonal complement of JG(A): In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...

A densely defined operator A is symmetric iff ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

where the subset notation is understood to mean An operator A is self-adjoint iff A = A ^* ; that is, iff G(A) = G(A ^* ).

Example. Consider the complex Hilbert space L²(R), and the operator which multiplies a given function by x:

Af(x) = xf(x)

The domain of A is the space of all L² functions for which the right-hand-side is square-integrable. A is a symmetric operator without any eigenvalues and eigenfunctions. In fact it turns out that the operator is self-adjoint, as follows from the theory outlined below.

As we will see later, self-adjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces.

Spectral theorem

Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent iff there is a unitary operator U:H → K such that In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ...

• U maps dom A bijectively onto dom B,

A multiplication operator is defined as follows: Let (X,Σ,μ) be a countably additive measure space and f a real-valued measurable function on X. An operator T of the form In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. ... In mathematics, a measure is a function that assigns a number, e. ...

whose domain is the space of ψ for which the right-hand side above is in L² is called a multiplication operator.

Theorem. Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.

This version of the spectral theorem for self-adjoint operators can be proved by reduction to the spectral theorem for unitary operators. This reduction uses the Cayley transform for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the essential range of f.

Borel functional calculus

Given the representation of T as a multiplication operator, it is then easy to explain how the Borel functional calculus should operate: If h is a bounded real-valued Borel function on R, then h(T) is the operator of multiplication by the composition . In order for this to be well-defined, we need to show that it is the unique operation on bounded real-valued Borel functions satisfying a number of conditions. In functional analysis, the Borel functional calculus is a functional calculus (i. ...

For the bounded case, an alternative way of obtaining the Borel functional calculus is the following: First pass from polynomial to continuous functional calculus using the Stone-Weierstrass theorem. The use the Riesz-Markov theorem to pass from integration on continuous functions to spectral measures. In operator theory and C*-algebra theory the continuous functional calculus allows applications of continuous functions to normal elements of a associates to a normal element C*-algebra. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

Resolution of the identity

It has been customary to introduce the following notation

where denotes the function which is identically 1 on the interval . The family of projection operators E_T(λ) is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved: The Stieltjes integral provides a direct way of (numerically) defining an integral of the type without first having to convert it to and then integrating this converted form by means of a pre-existing, non-Stieltjes integration method. ...

The notion of convergence of the integral above can be provided by various operator topologies(e.g. the weak operator topology). In more modern treatments, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus.

Formulation in the physics literature

In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the Borel functional calculus using Dirac notation as follows: In functional analysis, the Borel functional calculus is a functional calculus (i. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

If H Hermitian (the name for self-adjoint in the physics literature) and f is a Borel function, In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

with

where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvalues Ψ_E. Such a notation here is purely formal, but can be made rigorous using the concept of a rigged Hilbert space. Formal - relating to form. ... In mathematics, a rigged Hilbert space is a construction designed to link the distribution (test function) and square-integrable aspects of functional analysis. ...

If f=1, the theorem is referred to as resolution of unity:

In the case H_eff = H − iΓ is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix) operator − iΓ, one defines the biorthogonal basis set In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... In mathematics, a biorthogonal system in a pair of topological vector spaces E and F that are in duality is a pair of indexed subsets vi in E and wi in F such that <vi,wj> = δij with the Kronecker delta. ...

and write the spectral theorem as:

(See Feshbach-Fano partitioning method for the context where such operators appears in scattering theory). In quantum mechanics, and in particular in scattering theory, the Feshbach- Fano method is a method for separating (partitionning) the resonant and the background components of the wave function and therefore of the associated quantities like cross sections or phase shift. ... Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ...

Extensions of symmetric operators

In quantum mechanics, observables are given by self adjoint operators, and only self adjoint operators can be exponentiated to obtain the (unitary) time evolution operator. However, in practice, confronted by a physical problem, one usually can only write down a Hamiltonian that is symmetric at first. Thus the following question is of interest: if an operator A on the Hilbert space H is symmetric, when does it have self-adjoint extensions? One answer is provided by the Cayley transform of a self-adjoint operator and the deficiency indices. (We should note here that it is often of technical convenience to deal with closed operators. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.) Cayley transform maps upper half plane to open unit disk In complex analysis, the Cayley transform is the map The Cayley transform is a linear fractional transformation. ...

Theorem. Suppose A is a symmetric operator. Then there is a unique partially defined linear operator

such that

Here, ran and dom denote the range and the domain, respectively. W(A) is isometric on its domain. Moreover, the range of 1 - W(A) is dense in H. In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ... In topology and related areas of mathematics a subset A of a topological space X is called dense (in X) if the only closed subset of X containing A is X itself. ...

Conversely, given any partially defined operator U which is isometric on its domain (which is not necessarily closed) and such that 1 - U is dense, there is a (unique) operator S(U)

such that

The operator S(U) is densely defined and symmetric.

The mappings W and S are inverses of each other.

The mapping W is called the Cayley transform. It associates a partially defined isometry to any symmetric densely-defined operator. Note that the mappings W and S are monotone: This means that if B is a symmetric operator that extends the densely defined symmetric operator A, then W(B) extends W(A), and similarly for S. In functional analysis a partial isometry is a linear map W between Hilbert spaces H, K such that there is a closed vector subspace H1 of H such that W restricted to H1 is an isometric map and W restricted to the orthogonal complement of H1 is zero. ... Monotone refers to one of: a steady chant like way of speaking on one pitch, or a monotonic function Monotone Records, a record label [1] Monotone (software), revision control software [2] This is a disambiguation page: a list of articles associated with the same title. ...

Theorem. A necessary and sufficient condition for A to be self-adjoint is that its Cayley transform W(A) be unitary.

This immediately gives us a necessary and sufficient condition for A to have a self-adjoint extension, as follows:

Theorem. A necessary and sufficient condition for A to have a self adjoint extension is that W(A) have a unitary extension.

A partially defined isometric operator V on a Hilbert space H has a unique isometric extension to the norm closure of dom(V). A partially defined isometric operator with closed domain is called a partial isometry. In functional analysis a partial isometry is a linear map W between Hilbert spaces H, K such that there is a closed vector subspace H1 of H such that W restricted to H1 is an isometric map and W restricted to the orthogonal complement of H1 is zero. ...

Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range: In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of an inner product space V is the set of all vectors in V that are orthogonal to every vector in W, i. ...

Theorem. A partial isometry V has a unitary extension iff the deficiency indices are identical. Moreover, V has a unique unitary extension iff the both deficiency indices are zero.

We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. An operator which has a unique self-adjoint extension is said to be essentially self-adjoint. Such operators have a well-defined Borel functional calculus. Symmetric operators which are not essentially self-adjoint may still have a canonical self-adjoint extension. Such is the case for non-negative symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of the Laplacian operator), so the issue of essential adjointness for these operators is less critical. In functional analysis, the Borel functional calculus is a functional calculus (i. ... Canonical is an adjective derived from canon. ... In functional analysis, the Friedrichs extension is a canonical self-adjoint extension of a non-negative densely defined symmetric operator. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

Von Neumann's formulas

Suppose A is symmetric; any symmetric extension of A is a restriction of A*; Indeed if B is symmetric

Theorem. Suppose A is a densely defined symmetric operator. Let

Then

and

where the decomposition is orthogonal relative to the graph inner product of dom(A*):

These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference.

Examples

We first consider the differential operator

defined on the space of complex-valued C^∞ functions on [0,1] vanishing near 0 and 1. D is a symmetric operator as can be shown by integration by parts. The spaces N₊, N₋ are given respectively by the distributional solutions to the equation In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...

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^{i x} and x → e^{- i x} 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

which in this case happens to be the unit circle T.

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 P_dist 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. Thus The notion of multi-indices simplifies formula used in the calculus of several variables, partial differential equations and the theory of distributions by generalising the concept of an integer index to an array of indices. ...

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).

Theorem. Let P a polynomial function on Rⁿ with real coefficients, F the Fourier transform considered as a unitary map L²(Rⁿ) → L²(Rⁿ). 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 a_α 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 In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

Theorem. The operator theoretic adjoint P* of P is a restriction of the distributional extension of the formal adjoint. Specifically:

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. Hans Hahn (1879 - 1934) was an Austrian mathematician who made many contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. ...

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

where H_n is a Hilbert space of dimension n. The domain of M_f consists of vector-valued functions ψ on R such that

Non-negative countably additive measures μ, ν are mutually singular iff 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)

such that the measures are pairwise singular and A is unitarily equivalent to the operator of multiplication by the function f(λ) = λ on

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.

The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces: In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. ...

Theorem. Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ → λ on

The measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable family {H_x}_x is determined almost everywhere with respect to μ.

Example: structure of the Laplacian

The Laplacian on Rⁿis 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). In mathematics, an Elliptic operator is a major type of differential operator P defined on spaces of complex-valued functions, or some more general function-like objects, such that the coefficients of the highest-order derivatives satisfy a positivity condition. ...

Theorem. If n=1, the - Δ has uniform multiplicity mult=2, otherwise - Δ has uniform multiplicity mult=ω. Morover, the measure μ_mult is Borel measure on [0, ∞).

Pure point spectrum

The spectrum of a self adjoint operator can be classified via its spectral measures. The support of the discrete spectral measures is called point(or discrete) spectrum. Similarly, the support of the absolutely continuous(resp. singular) spectral measures is the absolutely continuous(resp. singular) spectrum. A simple, though important class of operators are those with pure point spectrum. A self-adjoint operator A on H has pure point spectrum iff H has an orthonormal basis {e_i}_{i ∈ I} consisting of eigenvectors for A. In mathematics, projection-valued measures are used to express results in spectral theory. ... In mathematics and physics, discrete spectrum of an operator on Hilbert space is the part of the spectrum which corresponds to discrete spectral measures. ... In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. ...

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 sufficent condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact "inverse". The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

References

• N.I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (two volumes), Pitman, 1981.

• K. Yosida, Functional Analysis, Academic Press, 1965.

• M. Reed and B. Simon, Methods of Mathematical Physics vol 2, Academic Press, 1972.