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:

    “What if all ponds were shallow? Would it not react on the minds of men? I am thankful that this pond was made deep and pure for a symbol. While men believe in the infinite some ponds will be thought to be bottomless.”
    —Henry David Thoreau (1817–1862)

    “I have come to a still, but not a deep center,
    A point outside the glittering current;
    My eyes stare at the bottom of a river,”
    —Theodore Roethke (1908–1963)