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:
“Examinations, sir, are pure humbug from beginning to end. If a man is a gentleman, he knows quite enough, and if he is not a gentleman, whatever he knows is bad for him.”
—Oscar Wilde (1854–1900)
“Zhivago: It seems you bombed the wrong village.
Strelnikov: They always say that. And what does it matter? A village betrays us, a village is burned. The point made.
Zhivago: Your point. Their village.”
—Robert Bolt (1924–1995)