Continuous spectrum


Continuous spectrum

The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.

If H is a topological vector space and A:H \to H is a linear map, the spectrum of A is the set of complex numbers λ such that A - \lambda I : H \to H is not invertible. We divide the spectrum depending on why this is not invertible.

If A − λI is not injective, we say that λ is in the point spectrum of A. Elements of the point spectrum are called eigenvalues of A and non-zero elements of the null space of A − λI are known as eigenvectors of A. Thus λ is an eigenvalue of A if and only if there is a non-zero vector v \in H such that Av = λv.

If A − λI does not have closed range, but the range is dense in H, we say that λ is in the continuous spectrum of A. The union of the point spectrum and the continuous spectrum is known as the set of generalized eigenvalues. Thus λ is a generalized eigenvalue of A if and only if there is a sequence of vectors {vn}, bounded away from zero, such that  Av_n - \lambda v_n \to 0 .

Finally, if A − λI does not have closed range, and its range is not dense in H, we say that λ is in the residual spectrum of A.

Quantum mechanical interpretations

The position operator usually has a continuous spectrum, much like the momentum operator in an infinite space. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems, specially bound states, tend to have a discrete (quantized) spectrum -- that is where the name quantum mechanics comes from. However 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. The formal scattering theory has a strong overlap with the theory of continuous spectra.

The quantum harmonic oscillator and the hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom, it has both continuous as well as discrete part of the spectrum; the continuous part represents the ionized atom.


References

See also

Related physical concepts:

Mathematically rigorous point of view:


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