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:
“Genius resembles a bell; in order to ring it must be suspended into pure air, and when a foreign body touches it, its joyful tone is silenced.”
—Franz Grillparzer (1791–1872)
“What we are, that only can we see. All that Adam had, all that Caesar could, you have and can do. Adam called his house, heaven and earth; Caesar called his house, Rome; you perhaps call yours, a cobbler’s trade; a hundred acres of ploughed land; or a scholar’s garret. Yet line for line and point for point, your dominion is as great as theirs, though without fine names. Build, therefore, your own world.”
—Ralph Waldo Emerson (1803–1882)