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:
“A dramatist is one who believes that the pure event, an action involving human beings, is more arresting than any comment that can be made upon it.”
—Thornton Wilder (1897–1975)
“All Coolidge had to do in 1924 was to keep his mean trap shut, to be elected. All Harding had to do in 1920 was repeat “Avoid foreign entanglements.” All Hoover had to do in 1928 was to endorse Coolidge. All Roosevelt had to do in 1932 was to point to Hoover.”
—Robert E. Sherwood (1896–1955)