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:
“... that’s what living happens to be ... the physiological denial of reverence and good manners and Christianity.... At your age one’s quite old enough to know what the essence of life really is. Shamelessness, that’s all; pure shamelessness.”
—Aldous Huxley (1894–1963)
“The women’s liberation movement at this point in history makes the American Communist Party of the 1930s look like a monolith.”
—Nora Ephron (b. 1941)