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For a differentiable radial factor and integer $n\ge 0$, the coordinate polynomial obeys $L_3\psi =n\hslash \psi$ and $L_{+}\psi =0$. The identity $L^2=L_{-}{L}_{+}+L_3^2+\hslash L_3$ gives the displayed angular eigenvalue. Lowering gives $L_{-}\psi =-2n\hslash z(x+iy{)}^{n-1}f(r)$, a nonzero eigenfunction with the same angular eigenvalue when $n>0$ and the original state is nonzero. At $n=0$ lowering gives the zero function, which is not an eigenfunction. Negative powers satisfy local differential identities off the axis but are not regular angular states.

For $-\Delta +I$ on the unit square with zero Dirichlet boundary conditions, the continuous eigenvalues are ${\lambda }_{m,n}=1+{\pi }^2(m^2+n^2)$, whereas the mesh-$1/N$ five-point Laplacian gives ${\Lambda }_{m,n}=1+4N^2[{\sin }^2(m\pi /(2N))+{\sin }^2(n\pi /(2N))]$. The inequality $\sin z\le z$ yields ${\Lambda }_{m,n}\le {\lambda }_{m,n}$. For fixed indices, Taylor's theorem gives an error of order $N^{-2}$; for indices $m=n=N-1$, division by $N^2$ gives the positive limit $2{\pi }^2-8$. Thus an error estimate for fixed smooth eigenfunctions need not be uniform over all grid-resolvable eigenfunctions.

For a measurable dynamical map $F$, the Koopman operator acts on observables by $K_{F}g=g\circ F$. If $F$ is nonsingular, it defines an operator on equivalence classes of measurable functions.

For a Cauchy-Euler equation, logarithmic coordinates turn scale changes in $x>0$ into translations in $t$. With $Y(t)=y(e^t)$, the chain rule gives $x{y}^{\prime }=Y^{\prime }$ and $x^2{y}^{\prime \prime }=Y^{\prime \prime }-Y^{\prime }$. Consequently the Sturm-Liouville operator $-x^2{y}^{\prime \prime }-x{y}^{\prime }-cy$ becomes $-Y^{\prime \prime }-cY$. Its natural weighted inner product has measure $dx/x=dt$, so this change of variable also preserves the norm when the transformed problem uses ordinary integration in $t$. On $1<x<e^{\ell }$ with Dirichlet boundary conditions, the orthonormal eigenfunctions are $\sqrt{2/\ell }\sin (n\pi \log x/\ell )$ and the eigenvalues are $(n\pi /\ell {)}^2-c$, for $n\ge 1$.

For $x>0$, write $t=\log x$ and $y=xv(t)$. The chain rule gives $y^{\prime }=v+v_t$, $y^{\prime \prime }=x^{-1}(v_t+v_{tt})$, and thereforeOn $1<x<e^{\ell }$ with homogeneous Dirichlet boundary conditions, the eigenfunctions are $x\sin (n\pi \log x/\ell )$ and the eigenvalues are $-(n\pi /\ell {)}^2$. The weighted inner product with weight $x^{-3}$ becomes ${\int }_0^{\ell }\overline{v}wdt$, so their squared norms are $\ell /2$ and their orthogonality is exactly that of the Fourier sine series. The weight is the factor that makes this conjugacy preserve the inner product.

In 1926, Erwin Schrödinger constructed arrays from differential operators and complete orthogonal function systems, showing how their products reproduce the array calculus. His paper connected wave mechanics with matrix mechanics; the translation's sections 2–3 develop the product and commutation rules.^[ref]

The modern proof is a change of orthonormal basis. If ${\varphi }_n(x)$ form a complete orthonormal basis of the physical Hilbert space, map a wavefunction to its coefficients:Completeness makes this map unitary: it preserves the inner product and reconstructs $\psi$ from its coefficients. Inserting the complete orthonormal basis proves the matrix multiplication rule. The Schrödinger equation becomesFor a time-independent Hamiltonian operator, the Heisenberg picture transfers the same evolution to the observables, preserving measurement predictions. Unitary equivalence requires matching operator domains as well as formal entries. A finite selection of eigenfunctions gives a compression, not an exact replacement of the full Hilbert space; a continuous spectrum needs its spectral integrals. This equivalence concerns representations of specified operators, and does not remove the ordering question when quantizing an arbitrary classical expression, as Weyl ordering illustrates.

Mercer's theorem says that a continuous symmetric positive-semidefinite kernel on a compact domain has an orthonormal eigenfunction expansionwith nonnegative eigenvalues and absolute uniform convergence under its standard hypotheses.

The mode shape is the fixed spatial eigenfunction multiplying the temporal factor of a normal mode. It is determined by the governing differential equation and boundary conditions, and may be normalized by a chosen amplitude or energy convention.

For $u_t=u_{xx}+\beta u_x$ with $u_x(0,t)=g(t)$ and $u_x(L,t)=h(t)$, two applications of integration by parts against a Neumann eigenfunction give the boundary forcing $e^{\beta L}{\varphi }_k(L)h-{\varphi }_k(0)g$. Solving the resulting modal linear differential equation givesThe endpoint derivatives are interior limits: the short-time singularity prevents evaluating each homogeneous boundary derivative before the time integral and infinite sum. The constant eigenfunction gives the weighted-mass law ${\partial }_t{\int }_0^L{e}^{\beta x}udx=e^{\beta L}h-g$.

For the wave equation ${\psi }_{tt}=c^2{\psi }_{xx}$ on $0<x<L$, impose homogeneous Neumann boundary conditions ${\psi }_x(0,t)={\psi }_x(L,t)=0$, where $c,L>0$. Separation of variables gives $X^{\prime \prime }+\lambda X=0$ and $T^{\prime \prime }+c^2\lambda T=0$. Integration by parts gives ${\int }_0^L(X^{\prime }{)}^{2}dx=\lambda {\int }_0^L{X}^{2}dx$, so every nontrivial spatial eigenfunction has $\lambda \ge 0$.

For $\lambda =0$, $X$ is constant and $T=A_0+B_{0}t$. For $\lambda >0$, the boundary conditions select ${\lambda }_n=(n\pi /L{)}^2$ and $X_n=\cos (n\pi x/L)$. The superposition principle gives the displayed expansion when the initial data have sufficient regularity. The constant zero mode must be retained: homogeneous Neumann boundary conditions do not force the spatial average of the displacement to vanish.

With zero initial velocity, all $B_n$ vanish, including $B_0$. Initial displacement $a+b\cos (n\pi x/L)$ then evolves as $a+b\cos (cn\pi t/L)\cos (n\pi x/L)$, a standing wave above a constant offset.

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.

A Neumann eigenfunction is an eigenfunction of a spatial differential operator whose normal derivative vanishes on the boundary. A Sturm-Liouville eigenfunction expansion in these functions gives a heat kernel for homogeneous Neumann boundary conditions.

With the weighted inner product ${\int }_{-1}^1\overline{f}gdx/\sqrt{1-x^2}$, the Chebyshev differential equation has the Sturm-Liouville form with coefficients $p=\sqrt{1-x^2}$, $w=(1-x^2{)}^{-1/2}$ and $q=0$. The substitution $x=\cos \theta$ converts its operator to $-d^2/d{\theta }^2$ on $(0,\pi )$ and its inner product to the ordinary one in $d\theta$. The natural endpoint conditions $p{y}^{\prime }\to 0$ become zero derivative conditions at $\theta =0,\pi$.

Consequently the eigenfunctions are the Chebyshev polynomials $T_n(x)=\cos (n\arccos x)$ with eigenvalues $n^2$, $n=0,1,2,\dots$. Their squared norms are $\pi$ for $n=0$ and $\pi /2$ otherwise. In particular $T_1=x$ and $T_3=4x^3-3x$ are orthogonal. The endpoints of the original differential expression are singular; specifying these transformed Neumann boundary conditions defines its self-adjoint realization. The Dirichlet eigenfunctions of the Chebyshev differential operator use a different endpoint condition and have sine rather than cosine profiles.

For $L>0$ and $t>0$, the Sturm-Liouville eigenfunction expansion for $u_t=u_{xx}$ on $(0,L)$ with homogeneous Neumann boundary conditions givesThe constant eigenfunction mode preserves the spatial integral. This is the $\beta =0$ limit of the weighted Neumann heat kernel with constant drift, and its boundary inputs have the signs in the Neumann boundary-forcing formula.

A nodeless real eigenfunction has no zero in the interior of the domain. For a one-dimensional confining Schrödinger operator, the nodeless theorem for a one-dimensional ground state identifies the ground state as the eigenfunction that can be chosen strictly positive. Normalizable excited states have nodes.

Suppose $p(x)=c(x-a)+O((x-a{)}^2)$ with $c>0$, and the other coefficients in $(p{y}^{\prime }{)}^{\prime }+(\lambda w-q)y=0$ extend continuously to $a$, with $w(a)>0$. A bounded eigenfunction has $(p{y}^{\prime }{)}^{\prime }=O(1)$, hence $p{y}^{\prime }=C+O(x-a)$. A nonzero $C$ would give $y^{\prime }\sim C/[c(x-a)]$ and an unbounded logarithmic singularity. Thus $C=0$, and the regular branch has bounded first derivative. The second local branch carries a logarithm. The title condition excludes that branch.

For two regular eigenfunctions, $p(\overline{u}{v}^{\prime }-\overline{u^{\prime }}v)\to 0$. This removes the singular-end contribution to the Lagrange identity for a Sturm-Liouville operator. Together with a separated self-adjoint condition at the other endpoint, it selects the regular self-adjoint differential operator realization. The precise operator domain uses the maximal differential-operator domain and the no-log endpoint condition; boundedness of classical eigenfunctions is its convenient local expression, not a replacement for specifying an operator domain.

For $-y^{\prime \prime }=\lambda y$, positive eigenvalues have eigenfunctions $\sin (q_{n}x)$, where $\tan q_n=\alpha q_n$ and $q_n>0$. Their squared norms are $N_n=1/2-\sin (2q_n)/(4q_n)$. A zero eigenvalue occurs precisely at $\alpha =1$, with eigenfunction $x$. A negative eigenvalue occurs precisely at $0<\alpha <1$: it is $-{\kappa }^2$ with eigenfunction $\sinh (\kappa x)$, where $\tanh \kappa =\alpha \kappa$. There is exactly one such $\kappa$, since $\tanh \kappa /\kappa$ decreases strictly from $1$ to $0$. The integration by parts identityand the Cauchy-Schwarz inequality show positivity for $\alpha >1$ and nonnegativity at equality. The complete expansion of a square-integrable function therefore includes the extra term $3x{\int }_0^{1}tf(t)dt$ when $\alpha =1$, or $\sinh (\kappa x){\int }_0^{1}f(t)\sinh (\kappa t)dt/[\sinh (2\kappa )/(4\kappa )-1/2]$ when $\alpha <1$, in addition to ${\sum }_n[{\int }_0^{1}f(t)\sin (q_{n}t)dt/N_n]\sin (q_{n}x)$. Omitting that extra mode makes the system incomplete. In particular, $f=x$ at $\alpha =1$ is orthogonal to every positive sine mode and is an immediate counterexample to a sine-only expansion.

Let $L=w^{-1}[-(d/dx)(pd/dx)+q]$ be the regular self-adjoint operator of a Sturm-Liouville problem, with eigenfunctions orthonormal for the weighted inner product $\int w\overline{u}v$. For $\mu$ outside its spectrum, the displayed kernel satisfies $(L_x-\mu )G_{\mu }=\delta (x-\xi )$ with an ordinary, unweighted Dirac delta. Indeed weighted completeness gives $w(\xi ){\sum }_n{y}_n(x)y_n(\xi {)}^{\ast }=\delta (x-\xi )$. Thus $u(x)=\int G_{\mu }(x,\xi )f(\xi )d\xi$ solves $(L-\mu )u=f$. A kernel integrated against $w(\xi )d\xi$ instead omits the prefactor. For real coefficients, the weighted reciprocity relation is $G_{\mu }(x,\xi )/w(\xi )=G_{\mu }(\xi ,x)/w(x)$ when the spectral parameter is real and outside the spectrum. A zero eigenvalue prevents inversion at $\mu =0$ without a forcing compatibility condition or a projected inverse.

A normal mode, also called an eigenmode, is a pattern of free oscillation in a linear system whose relative displacements stay fixed while a single temporal factor changes their size and sign. For an undamped stable vibrational mode,Here $\varphi$ is the mode shape, $A$ sets the amplitude, $\omega$ is the eigenfrequency expressed as an angular frequency, and $\phi$ is the oscillation phase. Every snapshot is a scalar multiple of the same mode shape; neighbouring regions may move in opposite directions. Normal modes of a double pendulum show that normal modes also describe coupled discrete systems, beyond strings. Their shapes are eigenvectors, rather than spatial eigenfunctions. ^[ref]

For the ideal wave equation on a string, take a uniform string of length $L$ with constant tension $\tau$ and linear mass density $\mu$. Assume small transverse slope, negligible bending stiffness, no damping, and fixed endpoints. Its wave speed is $c=\sqrt{\tau /\mu }$, and the nonzero normal modes areThe angular frequency ${\omega }_n$ is in radians per second; the frequency $f_n$ is in hertz. The wavelength is $2L/n$. The mode shape has $n$ loops separated by nodes of a standing wave, including both fixed endpoints. ^[ref]

As a worked example, choose $L=1\mathrm{m}$, $\tau =1\mathrm{N}$, and $\mu =1\mathrm{kg}{\mathrm{m}}^{-1}$. Then $c=1\mathrm{m}{\mathrm{s}}^{-1}$ and $f_1=0.5\mathrm{Hz}$, $f_2=1\mathrm{Hz}$, $f_3=1.5\mathrm{Hz}$. Their oscillation periods are $T_1=2\mathrm{s}$, $T_2=1\mathrm{s}$, and $T_3=2/3\mathrm{s}$. These are deliberately simple model parameters, rather than measurements of a musical instrument. The plot divides displacement by $A$ so that the three mode shapes can be compared, and uses each mode's own oscillation period for its snapshots.

For $\phi =0$, at $t=T_n/4$ every point has zero displacement, but most points have nonzero velocity. Only zeros of the mode shape are permanent nodes of a standing wave. At $t=T_n/2$ the original mode shape has reversed sign, and at $t=T_n$ it returns. The synchronized conversion between kinetic energy and potential energy is shown in normal-mode energy exchange.

The short derivation uses separation of variables: substituting $u=\varphi q$ into the wave equation gives ${\varphi }^{\prime \prime }+k^2\varphi =0$ and $q^{\prime \prime }+c^2{k}^{2}q=0$. The fixed-end boundary conditions remove the cosine spatial term and require $\sin (kL)=0$, hence $k=n\pi /L$ and ${\omega }_n=ck$. Different geometries and boundary conditions give different eigenfunctions; a sine mode shape is a property of this uniform fixed string, not a universal form of an eigenmode.

The usefulness is that a complicated vibration becomes a collection of harmonic oscillators. For sufficiently regular fixed-end initial data, the Fourier sine series givesThe initial displacement and velocity determine $a_n$ and $b_n$. This is the superposition principle in a Fourier sine basis, and modal energy distribution measures how strongly each normal mode is excited. The striking simplification is that each component keeps its identity while their sum can continually change shape. ^[ref]

These independent normal modes belong to the linear model. At larger slopes, changing tension introduces nonlinear effects and can couple the modes. A linearly damped string still admits a Fourier sine basis, but its temporal factors decay and need not oscillate; its energy dissipation identity for a linearly damped string replaces conservation of energy. More general damping need not leave undamped mode shapes independent.

For $L=d^2/d{x}^2$ on $(0,\pi )$ with zero endpoints, the zero-parameter Green function is $-x_{<}(\pi -x_{>})/\pi$. Its eigenfunctions are $\sqrt{2/\pi }\sin (nx)$ with eigenvalues $-n^2$. At $x=\xi =\pi /2$, the kernel is $-\pi /4$ and its expansion contains precisely the odd reciprocal squares, giving the displayed identity.