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 first duty of a lecturer—to hand you after an hour’s discourse a nugget of pure truth to wrap up between the pages of your notebooks and keep on the mantlepiece for ever.”
—Virginia Woolf (1882–1941)
“I am less disposed to think of a West Point education as requisite for this business than I was at first. Good sense and energy are the qualities required.”
—Rutherford Birchard Hayes (1822–1893)