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:

    “I
    Am a pure acetylene
    Virgin
    Attended by roses,
    By kisses, by cherubim,
    By whatever these pink things mean.”
    —Sylvia Plath (1932–1963)

    “The realization that he is white in a black country, and respected for it, is the turning point in the expatriate’s career. He can either forget it, or capitalize on it. Most choose the latter.”
    —Paul Theroux (b. 1941)