Energy eigenvalue
= Energy eigenvalue
An energy eigenvalue is an <eigenvalue> of the <Hamiltonian operator> $H$: there is a nonzero <vector> $\psi$ in its <operator domain> $D(H)$ with
$$
H\psi=E\psi.
$$
For a <self-adjoint operator>, $E$ is real. Normalizing $\psi$ produces an <energy eigenstate>; an ideal <energy measurement> then returns $E$ with probability one. The associated <energy eigenspace> includes all vectors with this same <energy eigenvalue>, including the zero vector, which represents no <quantum state>. Different <energy eigenvalues> have orthogonal <energy eigenspaces>. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf[Teschl, Mathematical Methods in Quantum Mechanics, Lemma 2.20]{ref}.
The <energy eigenvalues> form the <point spectrum>, which can be only part of the <spectrum>. The term <discrete spectrum> more specifically means isolated <eigenvalues> of finite multiplicity. The usual fixed-nucleus <hydrogen atom> model has discrete negative <bound state> <energy eigenvalues> together with a continuum at nonnegative <energy>. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf[Teschl, Mathematical Methods in Quantum Mechanics, sections 6.4 and 10.2]{ref}. A <quantum system> need not have a ladder of discrete <energy eigenvalues>: a whole-line <free particle> has <continuous spectrum>. A <plane wave> used for a <free particle> is a generalized <eigenfunction> of the <Hamiltonian operator>, rather than a normalized vector in the physical <Hilbert space>. https://live.ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/61bc31b8d8bf0680c322733910a71aa0_MIT8_05F13_Chap_01.pdf[Barton Zwiebach, MIT 8.05 Wave Mechanics, sections 2 and 4]{ref}.
For the <infinite square well> $0<x<L$, the <Dirichlet boundary conditions> give
$$
\phi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}{L},\qquad E_n=\frac{\hbar^2\pi^2 n^2}{2mL^2},\qquad n=1,2,\ldots.
$$
Substitution into $H\phi_n=E_n\phi_n$ and <wavefunction normalization> verify these statements directly. Here $E_1$ is the <ground-state energy>, even though its index is one; the labeling convention does not determine which <quantum state> is lowest. Equal populations of $\phi_1$ and $\phi_2$ have <energy> <expectation value> $(E_1+E_2)/2=5E_1/2$, but an <energy measurement> returns $E_1$ or $4E_1$, each with probability $1/2$. An average of allowed <energy eigenvalues> need not itself be an <energy eigenvalue>.