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:
“Some have asked if the stock of men could not be improved,—if they could not be bred as cattle. Let Love be purified, and all the rest will follow. A pure love is thus, indeed, the panacea for all the ills of the world.”
—Henry David Thoreau (1817–1862)
“The trouble with Reason is that it becomes meaningless at the exact point where it refuses to act.”
—Bernard Devoto (1897–1955)