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:
“The spirit of the world, the great calm presence of the creator, comes not forth to the sorceries of opium or of wine. The sublime vision comes to the pure and simple soul in a clean and chaste body.”
—Ralph Waldo Emerson (1803–1882)
“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)