the Hermite and Laguerre functions
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These are the eigenfunction families that live on infinite domains — the half-line and the whole real line — where sines and cosines no longer fit because there is no finite interval to wrap around. They are the natural modes of two of the most important problems in physics: the quantum harmonic oscillator and the hydrogen atom.
The Hermite functions are the eigenfunctions of the singular Sturm-Liouville problem on the whole line behind the quantum oscillator: -u'' + x^2 u = lambda u, with eigenvalues lambda = 2n + 1, n = 0, 1, 2, .... The bounded solutions are H_n(x) e^(-x^2/2), where H_n are the Hermite polynomials, orthogonal on (-infinity, infinity) with weight w(x) = e^(-x^2) (a Gaussian). The Laguerre functions arise on the half-line [0, infinity) — for instance from the radial part of the hydrogen atom — with eigenfunctions built from the Laguerre polynomials L_n, orthogonal with weight w(x) = e^(-x) (or x^k e^(-x) for the associated version). In both cases the decaying weight is what tames the infinite domain and makes the integrals converge.
What makes them special is that the Sturm-Liouville machinery still works on an infinite interval: real eigenvalues, weighted orthogonality, completeness, an eigenfunction expansion. They are the reason the energy levels of the quantum oscillator are evenly spaced (lambda = 2n + 1) and why hydrogen has its discrete spectrum. Beyond physics, Hermite functions are the eigenfunctions of the Fourier transform itself, and Gauss-Hermite and Gauss-Laguerre quadrature use their zeros for highly accurate numerical integration against those weights.
The quantum harmonic oscillator -psi'' + x^2 psi = E psi has bounded solutions only at E = 2n + 1: psi_n(x) = H_n(x) e^(-x^2/2), with H_0 = 1, H_1 = 2x, H_2 = 4x^2 - 2. The ground state psi_0 = e^(-x^2/2) is a nodeless Gaussian (n = 0, zero interior zeros — the oscillation theorem again), and each higher state adds one node.
Hermite functions are the energy eigenstates of the quantum harmonic oscillator.
Mind the two conventions for Hermite polynomials — the 'physicists' H_n (weight e^(-x^2)) and the 'probabilists' He_n (weight e^(-x^2/2)) differ by scaling; mixing them gives wrong normalizations. And on these infinite domains the eigenfunctions still decay (via the Gaussian or exponential factor), but boundedness alone, not a boundary value, is what selects them.