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Write $\mathcal{A}y=-(p{y}^{\prime }{)}^{\prime }+qy$ and $\mathcal{A}{y}_n={\lambda }_{n}r{y}_n$, where $r$ is the positive weight. Normalize the real eigenfunctions using $\int r{y}_m{y}_n={\delta }_{mn}$. For $\mu$ outside the spectrum, applying $\mathcal{A}-\mu r$ to the displayed Green function expansion gives $r(x){\sum }_n{y}_n(x)y_n(\xi )=\delta (x-\xi )$ by weighted completeness. This is the kernel with respect to ordinary source integration $d\xi$; using the weighted measure would change its normalization.

Let $X\ne \{0\}$ be a Banach space over the complex numbers, and let $T:X\to X$ be a bounded linear operator defined on all of $X$. Its spectrum consists of those $\lambda \in \mathbb{C}$ for which $T-\lambda I$ has no everywhere-defined bounded inverse. Its complement is the resolvent set of an operator $\rho (T)$. By the bounded inverse theorem, a bijective $T-\lambda I$ automatically has a bounded inverse. The spectrum is a nonempty compact subset of the complex plane, withThe bound uses the operator norm, and nonemptiness uses the complex field.^[ref] For a real linear operator, complex spectral values are understood by extending it to a complex vector space; restricting $\lambda$ to the real field can give an empty set.

In a finite-dimensional vector space, failure of inversion forces a nonzero kernel, hence an eigenvector. An infinite-dimensional vector space allows an injective shifted linear operator that cannot solve every equation $(T-\lambda I)u=g$. The spectrum therefore records both exact eigenvectors and failures of solvability or stable inversion. Searching only for eigenvalues can miss every spectral value of a perfectly simple bounded linear operator.

Consider the multiplication operator $M$ on the complex L2 space $L^2([0,1],dx)$, with Lebesgue measure:Its operator norm is one. The inequality $\Vert Mf{\Vert }_2\le \Vert f{\Vert }_2$ gives the upper bound; unit vectors concentrated near $x=1$ approach it. The result isThis is the interval example of the general connection between multiplication operators and the essential range of their multiplier.^[ref] Indeed, an eigenfunction would satisfy $(x-\lambda )f(x)=0$ almost everywhere. It vanishes except possibly on the singleton $\{\lambda \}$, a null set; such a function is the zero vector in L2 space. Concentrating at one point cannot produce a nonzero L2 function.

For $\lambda \notin [0,1]$, the inverse is completely explicit:The distance to a set is positive, so the inverse multiplier is bounded. Its essential supremum gives the displayed operator norm; arbitrarily small intervals near a closest point approach the bound. Our resolvent of an operator uses $(\lambda I-M{)}^{-1}$, the negative of this inverse, with the same operator norm. For $\lambda =-1/4$, $5/4$, and $1/2+i/10$, the inverse norms are respectively $4$, $4$, and $10$. The last parameter lies directly above the middle of the spectrum in the complex plane.

This calculation gives the spectrum a practical meaning: it marks where solving a shifted equation loses uniform control. At $\lambda =1/2+i\eta$, the largest amplification of an input error is $1/|\eta |$. Changing $\eta$ from $0.1$ to $0.01$ increases that factor from ten to one hundred. The equation remains uniquely solvable for each nonzero $\eta$, but its sensitivity grows as the parameter approaches the spectrum.

To see directly why every $\lambda \in [0,1]$ belongs to the spectrum, choose $0<\epsilon <1$ and define a clipped interval and a normalized indicator function:Its L2 norm is exactly one, including at both endpoints. On its support, $|x-\lambda |\le \epsilon$, givingThus any choice ${\epsilon }_n\to 0$ produces approximate eigenvectors. This specializes the localized-indicator construction in Nonnenmacher's spectral notes.^[ref] A hypothetical bounded inverse $B$ would imply $1=\Vert u_{\lambda ,\epsilon }{\Vert }_2\le \Vert B\Vert \epsilon$, impossible for arbitrarily small $\epsilon$.

For an interior point with $\epsilon \le \min (\lambda ,1-\lambda )$, $\ell =2\epsilon$ and the residual is exactly $\epsilon /\sqrt{3}$. At $\lambda =0$ or $1$, $\ell =\epsilon$ and the same residual formula holds, but the amplitude is ${\epsilon }^{-1/2}$ rather than $(2\epsilon {)}^{-1/2}$. At $\lambda =1/2$, taking $\epsilon =0.2,0.1,0.05$ gives amplitudes approximately $1.5811,2.2361,3.1623$ and residuals $0.11547,0.057735,0.028868$. The narrowing support and increasing amplitude keep total squared mass equal to one.

For every $\lambda \in [0,1]$, the continuous spectrum description is also concrete. Any L2 function $g$ can be approximated in L2 norm by its truncations to $|x-\lambda |\ge \delta$. Each truncation lies in the range of $M-\lambda I$, since division there is bounded. The range is therefore dense. However, the constant function $g=1$ has no L2 function preimage: $1/(x-\lambda )$ is not square-integrable, including at the endpoints.

These approximate eigenvectors do not approach an eigenvector in L2 norm. Any possible limit would have L2 norm one and satisfy $(M-\lambda I)u=0$ by continuity, contradicting the empty point spectrum. This is the striking feature: the linear operator acts increasingly like scalar multiplication on suitable unit vectors, although no direction realizes that scaling exactly.

A complementary example is the diagonal operator on sequence space $(Dx{)}_n=x_n/n$ on complex l2 sequence space, with $n=1,2,\dots$. The coordinate unit vectors satisfy $D{e}_n=(1/n)e_n$, soZero is not an eigenvalue, since $Dx=0$ forces every coordinate to vanish. Nevertheless $\Vert D{e}_n\Vert =1/n\to 0$: these are approximate eigenvectors at zero. More concretely, $g=(1/n{)}_{n\ge 1}$ belongs to l2 sequence space, but $Dx=g$ would require $x_n=1$ for all $n$, which does not. Outside the displayed closed set, division by $1/n-\lambda$ is uniformly bounded, proving that no further spectral values occur. Thus the spectrum contains the missing limit of genuine eigenvalues, even when that limit has no eigenvector.

With the nonnegative Laplace-Beltrami operator, a frequency $w$ in the dual lattice gives the eigenfunction $e^{2\pi i⟨w,x⟩}$ and eigenvalue $4{\pi }^2|w{|}^2$. Its multiplicity is the number of dual vectors with that norm. Fourier series prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced Gram matrix, proving spectral rigidity.

For the unit $S^d$ and the nonnegative Laplace-Beltrami operator, the eigenvalues are $\ell (\ell +d-1)$, with multiplicities $\left(\genfrac{}{}{0ex}{}{\ell +d}{d}\right)-\left(\genfrac{}{}{0ex}{}{\ell +d-2}{d}\right)$ for $\ell \ge 0$. The eigenfunctions are restrictions of degree-$\ell$ harmonic polynomials. The harmonic decomposition of homogeneous polynomials and the Stone-Weierstrass theorem prove completeness.

A spherical harmonic is an eigenfunction of the Laplacian on the unit sphere:

In a square with impermeable walls and background topographic potential-vorticity gradient $\beta$, the linear shallow-water quasi-geostrophic potential vorticity equation is ${\partial }_t({\nabla }^2-L_D^{-2})\psi +\beta {\psi }_x=0$. For a nonzero-frequency normal mode, setting $\hat{\psi }=\psi e^{i\beta x/(2\omega )}$ gives a Helmholtz equation with homogeneous wall data. The eigenfunctions are $\hat{\psi }=\sin (m\pi x/L)\sin (n\pi y/L)$, with $m,n\ge 1$, and the displayed frequencies. The positive and negative frequency representations are complex conjugates for a real wave.

For a self-adjoint operator with two normalized orthogonal eigenvectors $u_0,u_1$ and $E_0<E_1$, the Rayleigh quotient of ${\psi }_t=u_1+t{u}_0$ is $(E_1+t^2{E}_0)/(1+t^2)$. It is stationary at $t=0$ but has second derivative $2(E_0-E_1)<0$ there. Thus an exact excited eigenfunction can be a maximum within a trial family. A constrained minimum requires exclusion of lower states, as in an excited-state variational constraint.

Let $B[u,v]={\int }_U{a}^{ij}{\partial }_{i}u{\partial }_{j}v+cuv$ on $H_0^1(U)$ for a bounded smooth domain, smooth symmetric uniformly positive coefficients, and bounded real $c$. If $B[u,u]>0$ for every nonzero $u$, then $B[u,u]\ge \alpha \Vert u{\Vert }_{H^1}^2$ for some $\alpha >0$. If this failed, choose $\Vert u_j{\Vert }_{H^1}=1$ with $B[u_j,u_j]\to 0$. The Rellich-Kondrachov compactness theorem gives a subsequence weakly converging in $H_0^1$ and strongly in $L^2$. The principal energy has weak lower semicontinuity, and the potential term converges, so strict positivity forces the limit to be zero. Uniform ellipticity then gives $\Vert D{u}_j{\Vert }_2\to 0$, contradicting normalization. Merely being a positive semidefinite operator is insufficient: $-\Delta -{\lambda }_1$ has a nonzero Dirichlet eigenfunction of zero energy.

An eigenfunction solving a Sturm-Liouville problem, including its prescribed boundary conditions.

A second-order differential equation in the displayed divergence form. Under the usual regular spectral assumptions, $p>0$, $w>0$, the coefficients have suitable regularity, and separated self-adjoint boundary conditions specify a Sturm-Liouville problem. Multiplying by $y$ and using zero endpoint values gives the Rayleigh quotient $\lambda =\int (p{y}^{\prime 2}+q{y}^2)dx/\int w{y}^{2}dx$ for a nonzero real eigenfunction. Sign conventions for the potential and eigenvalue vary; they must be read from the stated equation.

A regular Sturm-Liouville eigenvalue problem has the formon a bounded interval with self-adjoint boundary conditions and positive weight $w$.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

For $R=4{\pi }^2+{ϵ}^2$, fixed roll phase, $T={ϵ}^{2}t$ and normalization $\psi =ϵA(T)\cos (\pi x)\sin (\pi z)+\cdots$, the Landau amplitude equation isThe mean-temperature correction in weakly nonlinear Darcy convection supplies the cubic negative feedback. Projection of the third-order equations onto the critical eigenfunction via the Fredholm solvability condition for a self-adjoint operator fixes both coefficients. The stable nonzero amplitudes are $\pm 2/\pi$ within the chosen phase, giving a supercritical pitchfork bifurcation in this real-amplitude reduction.

For diffusion starting at the centre of $(-L,L)$ with absorbing endpoints, only even-parity spatial eigenfunctions contribute. Integrating their cosine expansion gives the displayed series. The leading late-time term has prefactor $4/\pi$ and decay rate ${\pi }^{2}D/(4L^2)$; its first relative correction is $-\frac{1}{3}{e}^{-2{\pi }^{2}Dt/L^2}$.

For a positive quadratic bending operator $K$ with a real orthonormal basis of eigenfunctions, the equipartition theorem gives $C(x,y)=k_{B}T{\sum }_n{W}_n(x)W_n(y)/{\mu }_n$, where $K{W}_n={\mu }_n{W}_n$. This is the inverse-operator Green function multiplied by Boltzmann constant and temperature. Unconstrained zero-energy filament modes prevent a normalizable canonical ensemble, and negative modes signal an unstable quadratic model.

For both ends clamped and $K=A{\partial }_x^4$, the diagonal is $C(x,x)=k_{B}T{x}^3(L-x{)}^3/(3A{L}^3)$. This differs from a clamped-free tip variance. The center variance is $k_{B}T{L}^3/(192A)$.

For a time-independent Hamiltonian operator $H$, an energy eigenstate satisfiesFor a one-dimensional particle of mass $m$ in a real potential energy $U(x)$,Its full time-dependent wavefunction is $\Psi (x,t)=\psi (x)e^{-iEt/\hslash }$: substitution in the Schrödinger equation checks both the minus sign and the factor $\hslash$. Thus an energy eigenstate has stationary position probability density, even though its quantum phase evolves. The boundary conditions and operator domain are part of this eigenvalue problem, selecting physically allowed solutions; the infinite square well and Radial Schrodinger equation for the hydrogen atom give complementary confined examples.

For a complete discrete orthonormal basis of energy eigenstates, linearity gives $\Psi (x,t)={\sum }_n{c}_n{\psi }_n(x)e^{-i{E}_{n}t/\hslash }$, with ${\sum }_n|c_n{|}^2=1$. Cross terms in $|\Psi {|}^2$ oscillate at angular frequencies $(E_n-E_m)/\hslash$. A continuous spectrum requires the corresponding spectral integrals; for a free particle on the whole line, individual plane waves are generalized eigenfunctions, and a normalized Gaussian wave packet is their integral quantum superposition. Adding a spatially constant real $C$ to $U$ shifts every energy eigenvalue by $C$ and multiplies the full state by $e^{-iCt/\hslash }$, a global phase; energy differences, position probabilities and probability current are unchanged.

The normalized eigenfunction ${\psi }_0(x)=\mathrm{sech}x/\sqrt{2}$ has zero eigenvalue under the Sine-Gordon kink fluctuation operator. It is proportional to ${\partial }_x{\varphi }_K$ and comes from shifting the collective coordinate of the kink. Its frequency is zero, so it contributes no oscillator zero-point energy; it must be handled separately from a Gaussian functional determinant.

For a regular real mode between rigid boundaries, multiply the Taylor–Goldstein equation by its vertical structure and integrate to get the stationary quotient $k^2=\int ({\ell }^2{\varphi }^2-{\varphi }_z^2)dz/\int {\varphi }^{2}dz$. Along a differentiable dispersion branch, stationarity removes the variation of the eigenfunction, giving $2k=(\partial I/\partial c)dc/dk$. Since ${\partial }_c{\ell }^2=N^2/(U-c{)}^3+{\ell }^2/(U-c)$, the displayed group velocity follows from $d(kc)/dk$. Critical levels and vanishing branch derivatives require a separate limiting analysis.

Vector normalization converts a nonzero vector in a normed vector space into a unit vector. Absolute homogeneity of a norm givesThe direction is unchanged by multiplying by a positive real scalar. For $c>0$, $\Vert cv\Vert =c\Vert v\Vert$ and $\widehat{cv}=\hat{v}$; for a negative real scalar, $\widehat{cv}=-\hat{v}$. Thus $(3,4)$ and $(6,8)$ normalize to the same unit vector, while $(-3,-4)$ normalizes to its negative. This decomposition is useful for specifying a direction independently of a force, velocity or displacement magnitude.

In a complex normed vector space, a nonzero complex scalar $c$ instead gives $\widehat{cv}=(c/|c|)\hat{v}$. The remaining complex argument is not removed by normalization. For a pure quantum state, vectors differing only by this overall global phase represent the same physical state; vector normalization alone does not identify them.

The zero vector cannot be normalized this way. The construction also applies in a function space: an eigenfunction of a Sturm-Liouville problem can be scaled to unit weighted norm byThe positive weight $w$ specifies the weighted inner product. Multiplication by this nonzero constant preserves the eigenvalue equation and its homogeneous boundary conditions.

For a positive weight $w$, the weighted inner product of functions is $⟨f,g{⟩}_w=\int \overline{f}gw$. It is the usual inner product in the L2 space for the measure $w(x)dx$. A Sturm-Liouville eigenfunction expansion uses this weight to make its eigenfunctions orthogonal.

The weight specifies how different locations contribute to a squared error: $\Vert f-g{\Vert }_w^2=\int |f-g{|}^{2}w$. More weight makes an error in that region more costly. The weighted Euclidean inner product makes the same choice for a finite list of coordinates.

For an explicit real-function example on $[0,1]$, let $w(x)=1+x$ and approximate $f(x)=x$ by a constant function. The orthogonal projection onto a finite-dimensional subspace spanned by the function $1$ givesThe best unweighted constant is $1/2$. Giving greater weight to larger $x$ moves the best constant upward, so a weighting choice changes the approximation even though the allowed family of functions is unchanged.