---
tag: PJWY3JJQQ7
kind: concept
name: Discrete spectrum
msc: [47A10]
sources:
  - url: https://en.wikipedia.org/wiki/Discrete_spectrum_(mathematics)
same_as:
  wikidata: Q97359748
  wikipedia: Discrete spectrum (mathematics)
---

## Summary

The discrete spectrum of an operator consists of its normal eigenvalues: isolated points of the spectrum whose Riesz projector has finite rank.
Such an eigenvalue behaves like an eigenvalue of a matrix, with finite algebraic multiplicity and a finite-dimensional generalised eigenspace.
For a self-adjoint operator on a Hilbert space, the spectrum is the disjoint union of the discrete spectrum and the essential spectrum.

## Explanations

### L1 human

Some points of the spectrum look exactly like matrix eigenvalues: they are isolated, and only finitely many directions belong to them.
These points form the discrete spectrum.
In quantum mechanics, the discrete spectrum of the energy operator gives the bound states, such as the energy levels of the hydrogen atom.
For the identity on an infinite-dimensional space, $1$ is isolated, but infinitely many directions belong to it, so it is not in the discrete spectrum.