What does it mean that the spectrum is discrete?

Question

Let $T$ be some operator on a Hilbert space. What does it mean that the spectrum is discrete?

Answer 1

"Discrete" here means "not continuous". If the spectrum of an operator is "all integers" then its spectrum is discrete. If the spectrum of an operator is "all real numbers from 0 to 2, inclusive" then its spectrum is continuous, not discrete.

For example, the operator $\frac{d^2}{dx^2}$, on the interval [0, 1], has eigenvalue equation $\frac{d^2f}{dx^2}= \alpha f$ with boundary conditions f(0)= 0, f(1)= 0.

In order to have non trivial solution to that equation, the eigenvalues, $\alpha$ must be $(n\pi)^2$ for some integer n and the corresponding eigenvectors are $sin(n\pi x)$. The spectrum is the set of all numbers of the form $(n\pi)^2$ for n a positive integer. The spectrum, the set of all eigenvalues, is "discrete".