Continuous spectrum

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Mathematical formulation

In mathematics and physics, the spectrum of an operator, in particular self adjoint or normal operators, can be classified via its spectral measures. Let A be a bounded self adjoint operator on some Hilbert space [H], we now make use of continuous functional calculus

[f \rightarrow f(A)]

, where f is a continuous function defined on [\sigma(A)], the spectrum of A. For a fixed ψ in [H], we notice that

[f \rightarrow \langle \psi, f(A) \psi \rangle]

is a positive linear functional on [C(\sigma(A))]. It then follows immediately from Riesz-Markov representation theorem that there exists a unique Radon measure μ on [\sigma(A)] such that

[\int_ f(x) d \mu(x) = \langle \psi, f(A) \psi \rangle].

This measure is called the spectral measure associated to ψ. Now consider the subspace, [G] containing only and all vectors whose spectral measures are absolutely continuous with respect to the Lebesgue measure. The spectrum of the restriction of A to this subspace is then called the absolutely continuous spectrum of A. Classified in this way, the other parts of the spectrum are the discrete spectrum and singular spectrum.

Equivalently, for each ψ, we can decompose its spectral measure according to Lebesgue-Radon-Nikodym. The direct sum of the spectral subspaces corresponding to the continuous parts of the spectral measures is then [G].

We have no difficulty in extending to the unbounded self-adjoint case since Riesz-Markov holds for locally compact Hausdorff spaces.

In quantum mechanics, observables are self adjoint operators and their spectra are the possible outcomes of measurements (the singular spectrum correspond to physically impossible outcomes). A typical example of a quantum mechanical observable which has purely continuous spectrum is the position operator of a particle moving on a line. Also, since the momentum operator is unitarily equivalent to the position operator, via the Fourier transform, they have the same spectrum. Purely continuous spectrum of a physical observable correspond to free states of a system. On the other hand, the discrete spectrum correspond to bound states.

Use in non-mathematical context

A physicist not interested in mathematical rigor would think of continuous spectrum as "a continuous set of eigenvalues" (one can justify this notion using theory of operators on Rigged Hilbert spaces). Experimentally, computing the spectra or cross sections associated with scattering experiments (like for instance high resolution electron energy loss spectroscopy) usually requires the computation of the non quantized or continuous spectrum (density of states) of the Hamiltonian. This is particularly true when broad resonances or strong background scattering is observed. The branch of quantum mechanics concerned with these scattering events is referred to as scattering theory.

Continuous spectrum

Mathematical formulation

Use in non-mathematical context

See also