Pure Point Spectrum
A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A.
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 sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse.
Read more about this topic: Self-adjoint Operator
Famous quotes containing the words pure and/or point:
“The whole fauna of human fantasies, their marine vegetation, drifts and luxuriates in the dimly lit zones of human activity, as though plaiting thick tresses of darkness. Here, too, appear the lighthouses of the mind, with their outward resemblance to less pure symbols. The gateway to mystery swings open at the touch of human weakness and we have entered the realms of darkness. One false step, one slurred syllable together reveal a man’s thoughts.”
—Louis Aragon (1897–1982)
“I am the kind of a woman that likes to move on mentally from point to point, and I like for my man to be there way ahead of me. Then if he is strong and honest, it goes on from there. Good looks are not essential, just extra added attraction.”
—Zora Neale Hurston (1891–1960)