User:Brews ohare/Identity

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Resolutions of the identity

Given a complete orthonormal basis set of functions {\varphi_n} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as:

f = \sum_{n=1}^\infty \ \alpha_n \varphi_n \ .

The coefficients {αn} are found as:

\alpha_n = \langle \varphi_n,\ f \rangle \ ,

which may be represented by the notation:

\alpha_n =  \varphi_n^\dagger \ f  \ ,

a form of the bra-ket notation of Dirac.[1] Adopting this notation, the expansion of f takes the dyadic form:[2]

f = \left( \sum_{n=1}^\infty \ \varphi_n \varphi_n^\dagger \right) \ f.

The expression:

I = \sum_{n=1}^\infty \ \varphi_n \varphi_n^\dagger,

is called a resolution of the identity I. When the Hilbert space is the space L2(D) of square-integrable functions on a domain D, the quantity:

\varphi_n \varphi_n^\dagger,

is an integral operator, and the expression for f can be rewritten as:

f(x) = \int_D\, d \xi \ w(\xi) \left( \sum_{n=1}^\infty \ \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi)\ ,

where allowance is made that a weighting function w(x) occurs in the inner product. The right-hand side converges to f in the L2 sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write the inner product of f with the δ-function as:

f(x) = \int \ d\xi\ w(\xi)\ \delta(x-\xi) f (\xi),

resulting in the representation of the delta function:

\delta(x-\xi) =\sum_{n=1}^\infty \ \varphi_n (x) \varphi_n^*(\xi).

With a suitable rigged Hilbert space (Φ,L2(D),Φ) where Φ⊂L2(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φn. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.

Example: This formalism encompasses much of generalized Fourier series. For example, the Fourier-Bessel series:

f(x) = \sum_{n=0}^\infty c_n J_0(\lambda_n x/b),

for x in the range 0 ≤ xb, where the {λn} are the zeros of the zero-order Bessel function J0, with coefficients:

=\frac{2}{b^2J_1(\lambda_n)^2}\int_0^b \ \xi \ d\xi \ f(\xi)\ J_0\left(\frac{\lambda_n \xi}{b}\right) \ ,

converges in the norm Lw2(0, b) with weight w = x.[3] The basis functions satisfy the orthonormality condition based upon the weight function w(x) = x:

\int_0^1 J_0(x \lambda_m)\,J_0(x \lambda_n)\,x\,dx
= \frac{\delta_{mn}}{2} [J_1(\lambda_n)]^2 \ .

If the coefficient expression is substituted back into the series expansion, the result is:

f(x) =\int_0^b \ d\xi \  \xi  f(\xi)\ \left(  \sum_{n=0}^\infty \frac{2}{b^2J_1(\lambda_n)^2}\ J_0(\lambda_n \xi / b) J_0(\lambda_n x/b)\right) \ ,

which may be viewed as the inner product of f with the δ-function, based upon the weight function w(x) = x:

f(x) =\int_0^b \ d\xi \  \xi  f(\xi) \delta(x-\xi) = \langle f(\xi), \ \delta(x-\xi)\rangle \ ,

resulting in the representation of the delta function as:

\delta(x-\xi) =\ \sum_{n=0}^\infty \  \frac{2}{b^2J_1(\lambda_n)^2}
J_0 ( \lambda_n x / b)  J_0 ( \lambda_n \xi/b) \ .


  1. ^ The development of this section in bra-ket notation is found in Frank S. Levin (2002). "Coordinate-space wave functions and completeness". An introduction to quantum theory. Cambridge University Press. pp. 109ff. ISBN 0521598419. 
  2. ^ Howard Ted Davis, Kendall T. Thomson (2000). "Perfect operators". Linear algebra and linear operators in engineering with applications in Mathematica. Academic Press. pp. 344ff. ISBN 012206349X. 
  3. ^ Gerald B Folland (1999). Fourier analysis and its applications (Reprint of 1992 Brooks/Cole ed.). AMS Bookstore. p. 148. ISBN 0821847902.