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:
“I am too pure for you or anyone.”
—Sylvia Plath (1932–1963)
“Here I swear, and as I break my oath may ... eternity blast me, here I swear that never will I forgive Christianity! It is the only point on which I allow myself to encourage revenge.... Oh, how I wish I were the Antichrist, that it were mine to crush the Demon; to hurl him to his native Hell never to rise again—I expect to gratify some of this insatiable feeling in Poetry.”
—Percy Bysshe Shelley (1792–1822)