Self-adjoint Operator - Pure Point Spectrum

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:

    We are born with luck
    which is to say with gold in our mouth.
    As new and smooth as a grape,
    as pure as a pond in Alaska,
    as good as the stem of a green bean
    we are born and that ought to be enough....
    Anne Sexton (1928–1974)

    I look upon it as a Point of Morality, to be obliged by those who endeavour to oblige me.
    Richard Steele (1672–1729)