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 pure work implies the disappearance of the poet as speaker, who hands over to the words.”
—Stéphane Mallarmé (1842–1898)
“You should go to picture-galleries and museums of sculpture to be acted upon, and not to express or try to form your own perfectly futile opinion. It makes no difference to you or the world what you may think of any work of art. That is not the question; the point is how it affects you. The picture is the judge of your capacity, not you of its excellence; the world has long ago passed its judgment upon it, and now it is for the work to estimate you.”
—Anna C. Brackett (1836–1911)