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:
“Like most vigorous-minded men, seeing that there was no stopping-place between dogma and negation, he preferred to accept dogma. Of all weaknesses he most disliked timed and half-hearted faith. He would rather have jumped at once to Strong’s pure denial, than yield an inch to the argument that a mystery was to be paltered with because it could not be explained.”
—Henry Brooks Adams (1838–1918)
“There are neither good nor bad subjects. From the point of view of pure Art, you could almost establish it as an axiom that the subject is irrelevant, style itself being an absolute manner of seeing things.”
—Gustave Flaubert (1821–1880)