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:
“Nothing is accidental in the universe—this is one of my Laws of Physics—except the entire universe itself, which is Pure Accident, pure divinity.”
—Joyce Carol Oates (b. 1938)
“The point is, that the function of the novel seems to be changing; it has become an outpost of journalism; we read novels for information about areas of life we don’t know—Nigeria, South Africa, the American army, a coal-mining village, coteries in Chelsea, etc. We read to find out what is going on. One novel in five hundred or a thousand has the quality a novel should have to make it a novel—the quality of philosophy.”
—Doris Lessing (b. 1919)