Spectral Theory - Resolution of The Identity

Resolution of The Identity

This section continues in the rough and ready manner of the above section using the bra-ket notation, and glossing over the many important and fascinating details of a rigorous treatment. A rigorous mathematical treatment may be found in various references.

Using the bra-ket notation of the above section, the identity operator may be written as:

where it is supposed as above that { } are a basis and the { } a reciprocal basis for the space satisfying the relation:

This expression of the identity operation is called a representation or a resolution of the identity., This formal representation satisfies the basic property of the identity:

valid for every positive integer n.

Applying the resolution of the identity to any function in the space , one obtains:

which is the generalized Fourier expansion of ψ in terms of the basis functions { e_i }.

Given some operator equation of the form:

with h in the space, this equation can be solved in the above basis through the formal manipulations:

which converts the operator equation to a matrix equation determining the unknown coefficients c_j in terms of the generalized Fourier coefficients of h and the matrix elements = of the operator O.

The role of spectral theory arises in establishing the nature and existence of the basis and the reciprocal basis. In particular, the basis might consist of the eigenfunctions of some linear operator L:

with the { λ_i } the eigenvalues of L from the spectrum of L. Then the resolution of the identity above provides the dyad expansion of L: