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The volumetric flow rate is the integral of axial velocity over the cross-section:The plane Poiseuille flow contribution isFor each correction eigenfunction,Uniform convergence permits termwise integration. Multiplying these two integrals by $A_n$ givesConsequentlyEvery summand in the subtracted quantity is positive. Thus the side walls reduce the flux below the infinite-width plane Poiseuille flow prediction. Its dimensions are volume per unit time, as required for a volumetric flow rate.

Substitute $W=f(z)e^{i(kx-\omega t)}$ into the transformed equation. It yieldsThe nonzero eigenfunctions of this Dirichlet boundary value problem are $f=A\sin (n\pi z/h)$, $n\ge 1$. Hence, with $N=\sqrt{\alpha g}$ and $b_n^2=(n\pi /h{)}^2+{\alpha }^2/4$,The physical vertical velocity includes the factor $e^{\alpha z/2}\sin (n\pi z/h)$. The zero horizontal wavenumber is the zero-frequency limiting case, not a propagating oscillation.

For $k>0$, the harmonic oscillator equation gives $X=C\sin (kx)+D\cos (kx)$. The condition at zero requires $D=0$, and the condition at $L$ requires $C\sin (kL)=0$. A nonzero normal mode therefore hasA multiplicative normalization has been absorbed into the temporal amplitude. The $k=0$ candidate is affine and the two zero endpoint values force it to vanish. A positive separation constant gives hyperbolic functions, whose two zero endpoint values likewise give only the trivial solution. Negative $n$ adds no new eigenfunctions because it only reverses their signs.

Suppose a Bloch stability operator has expansion $L(q)=L_0+iq{L}_1-q^2{L}_2+\cdots$, with a simple neutral eigenfunction $v_0$ and an adjoint neutral function $w$ normalized by $⟨w,v_0⟩=1$. Expand $v(q)=v_0+iq{v}_1+\cdots$ and $\lambda (q)=icq-D{q}^2+\cdots$. The Fredholm alternative gives $c=⟨w,L_1{v}_0⟩$ and the cell problem $L_0{v}_1=c{v}_0-L_1{v}_0$, $⟨w,v_1⟩=0$. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion coefficient. Multiple translation phases or a conserved mean require a coupled matrix problem rather than this scalar formula.

For an unstable Taylor–Goldstein equation mode in a finite channel, take $a=1$ in the power-transformed Taylor–Goldstein energy identity. Its imaginary part givesThus the real phase velocity lies strictly between the extremes of a nonconstant smooth shear profile. Equality would force the regular eigenfunction to vanish on an interval, hence everywhere by uniqueness for its ordinary differential equation.

A connected open subset of the Euclidean plane. For a bounded polygonal example, the Dirichlet Laplacian is the self-adjoint operator associated with the Dirichlet energy on $H_0^1(\Omega )$; its eigenfunctions are smooth in the interior but need not be smooth at reentrant boundary corners.

For a finite irreducible Markov chain with at least two states satisfying detailed balance, its Dirichlet form of a Markov chain obeysExpand $f-\pi (f)$ in an orthonormal eigenbasis. The denominator is ${\sum }_{j\ge 2}{a}_j^2$, and the numerator is ${\sum }_{j\ge 2}(1-{\lambda }_j)a_j^2$. Their ratio is at least $1-{\lambda }_2$ and equality holds for the second eigenfunction. Thus the optimal Poincare inequality for a reversible Markov chain constant is $1/\gamma$, for the ordinary rather than absolute spectral gap.

For nonnegative nonzero integrable initial data $f$ on $(0,L)$, the first coefficient in the normalized Fourier sine series is strictly positive, because the first sine eigenfunction is positive throughout the interval. For ${\rho }_t=D{\rho }_{xx}+\gamma \rho$ with zero Dirichlet boundary conditions, the first decay rate is ${\lambda }_1=D{\pi }^2/L^2-\gamma$. Every higher rate exceeds it by $D{\pi }^2(n^2-1)/L^2$, so smoothing for positive time and the spectral gap imply $e^{{\lambda }_{1}t}\rho (x,t)\to a_1\sqrt{2/L}\sin (\pi x/L)$ uniformly. Thus all interior points share the growth/decay threshold $\gamma =D{\pi }^2/L^2$; the endpoints remain zero. At equality the limiting profile is nonzero and stationary.

A principal component function is a normalized eigenfunction ${\varphi }_k$ of a covariance operator, ordered by decreasing eigenvalue ${\lambda }_k$.

Quantum mechanics describes a quantum system using a quantum state, with observables represented by self-adjoint operators. For a single particle in one dimension, a pure state can be described by a wavefunction $\psi (x,t)$: its squared modulus gives the probability density for position. The Schrödinger equation predicts how this probability amplitude evolves between interventions; measurement in quantum mechanics connects it to experimental outcomes through the Born rule. A wavefunction therefore predicts a distribution of possible positions rather than a single trajectory as in classical mechanics.

The infinite square well makes three central ideas visible with elementary sine functions. Confinement selects discrete energy eigenvalues; wavefunction normalization converts a shape into quantitative probabilities; and a quantum superposition can have a changing probability density even when each constituent energy eigenstate has a stationary one. This combination explains why quantum mechanics uses both linear algebra and differential equations: eigenfunctions describe the building blocks, while quantum phases determine how they interfere. The same tools help explain the discrete energies of chemical atoms and molecules, although their Quantum Hamiltonians require different potentials and often many particles.

In his first 1926 paper, Erwin Schrödinger formulated quantization as an eigenvalue problem for a hydrogen atom in nonrelativistic quantum mechanics. On its opening page he compared the emergence of integers with the number of nodes of a standing wave on a vibrating string. The infinite square well below realizes that appealing idea especially transparently: the allowed integers follow from the boundary conditions. This historical analogy concerns the origin of discrete energies; the Born rule supplies the statistical interpretation of the modern wavefunction. ^[ref].

For the radial Laplace operator $L=-r^{-2}(r^2{u}^{\prime }{)}^{\prime }$ on $0<r<R$, require a finite value at the origin and $u(R)=0$. The map $u\mapsto v=ru$ preserves the norm from $L^2((0,R),r^{2}dr)$ to $L^2(0,R)$ and sends $L$ to $-d^2/d{r}^2$. The regular endpoint condition becomes $v(0)=0$. The self-adjoint realization therefore corresponds to the ordinary Dirichlet boundary conditions for the one-dimensional second derivative. Its eigenvalues and eigenfunctions are displayed above, with $u_n(0)=n\pi /R$ understood by a limit. Their squared weighted norm is $R/2$. Completeness and orthogonality follow from the Fourier sine series on $(0,R)$.

For a nonzero trial function with $y(0)=y(L)=0$ and $y(L-x)=y(x)$, substituting $L-x$ for $x$ proves ${\int }_0^{L}x{y}^{2}dx=(L/2){\int }_0^L{y}^{2}dx$. This gives the displayed Rayleigh quotient. The Poincaré inequality on an interval bounds it below by $2{\pi }^2/L^3$, with equality for a nonzero multiple of $\sin (\pi x/L)$. Thus this sine is the best reflection-symmetric trial for $-y^{\prime \prime }=\lambda xy$, while the unrestricted lowest eigenfunction need not be reflection-symmetric. The polynomial $x(L-x)$ gives $20/L^3$, only a factor $10/{\pi }^2$ above the sine's value. Both are upper bounds on the unrestricted lowest eigenvalue by the Rayleigh-Ritz method.

For noninteger real $\gamma$, expand a smooth forcing as $F={\sum }_{n\ge 0}{F}_n{T}_n$ in Chebyshev polynomials. Because $(1-x^2)T_n^{\prime \prime }-x{T}_n^{\prime }=-n^2{T}_n$, division of each coefficient by ${\gamma }^2-n^2$ gives the regular solution. The coefficients use the weighted inner product with weight $(1-x^2{)}^{-1/2}$. Choosing the same endpoint domain as the regular eigenfunctions of the Sturm-Liouville problem is essential: finite values alone also allow homogeneous solutions $\cos (\gamma \arccos x)$ and $\sin (\gamma \arccos x)$, which generally have divergent endpoint derivatives. Setting $x=\cos \theta$ converts the regular endpoint condition into the Neumann boundary conditions $Z_{\theta }(0)=Z_{\theta }(\pi )=0$. The homogeneous equation then has no nonzero solution when $\gamma$ is noninteger, proving uniqueness in this domain.

For $u_t=u_{xx}-a(t)u$, the displayed substitution converts the equation exactly into the heat equation $v_t=v_{xx}$. The scalar prefactor preserves homogeneous Dirichlet boundary conditions. Consequently each spatial eigenfunction $X_n$ with $X_n^{\prime \prime }=-{\lambda }_n{X}_n$ acquires temporal factor $\exp [-{\lambda }_{n}t-{\int }_0^{t}a(s)ds]$. The same algebra works with an evolution variable named $y$; calling that variable spatial does not alter the separation of variables. For $a(t)=1+\sin t$, the prefactor normalized to one at zero is $e^{-t+\cos t-1}$.

For $L>0$, constant $\beta >0$ and $\mathrm{Re}p>0$, on $(0,L)$ put $b=\beta /2$, $\kappa =\sqrt{p+b^2}$ with the principal square root, andThese satisfy $(p-{\partial }_x^2-\beta {\partial }_x)u_p=(p-{\partial }_x^2-\beta {\partial }_x)v_p=0$ and the left and right homogeneous Neumann boundary conditions, respectively. The weighted Wronskian givesThe first derivative jumps by $-e^{-\beta y}$, so the resolvent integrates against $e^{\beta y}dy$. Its time Laplace transform interpretation is ${\mathcal{R}}_p={\int }_0^{\infty }{e}^{-pt}{K}_{\beta }(t)dt$. The factor $p$ records the zero eigenvalue and constant stationary eigenfunction.

These Robin boundary conditions give the spatial Sturm-Liouville operator $-d^2/d{x}^2$ one negative eigenvalue, $-1$, with eigenfunction $e^x$. The other eigenvalues are $n^2{\pi }^2$, with eigenfunctions $V_n=n\pi \cos (n\pi x)+\sin (n\pi x)$, $n\ge 1$; there is no zero eigenvalue. Hence the diffusion equation has a growing contribution $c_0{e}^{t+x}$ in addition to decaying modes.

There is no contradiction with ordinary diffusive smoothing. The outward normal derivative is $-u_x$ at the left endpoint and $u_x$ at the right endpoint, so imposing the same coordinate derivative at both ends does not impose dissipative outward-normal data at both ends. The energy identity is $\frac{1}{2}{\partial }_t\int u^2=[u{u}_x{]}_0^1-\int u_x^2$, whose boundary term is $u(1{)}^2-u(0{)}^2$ here and can supply energy. Orthogonal projection of initial data $f$ gives $c_0=2{\int }_0^{1}f{e}^{x}dx/(e^2-1)$; it vanishes precisely when the growing mode is absent.

The reflected boundary conditions $u_x(0,t)=-u(0,t)$ and $u_x(1,t)=-u(1,t)$ have the same spectrum, because reflection $x\mapsto 1-x$ reverses the coordinate derivative. Here the growing eigenfunction is $e^{-x}$ and the decaying functions are $W_n=n\pi \cos (n\pi x)-\sin (n\pi x)$, givingFor square-integrable initial data $f$, orthogonal projection givesThe norms are $(1-e^{-2})/2$ and $(n^2{\pi }^2+1)/2$. The spatial Sturm-Liouville problem has no zero eigenvalue and exactly one negative eigenvalue for $-D^2$, namely $-1$. The energy boundary term now reads $u(0{)}^2-u(1{)}^2$, so the left endpoint can supply energy. The growing mode disappears exactly when ${\int }_0^1{e}^{-x}f(x)dx=0$.

An ordinary scalar wavefunction assigns one complex amplitude to each physical configuration. In polar coordinates without an additional twisted boundary condition, the points $(r,\varphi )$ and $(r,\varphi +2\pi )$ are identical, so the angular factor must be periodic. An angular eigenfunction $e^{im\varphi }$ therefore requires integer $m$. This scalar orbital condition does not rule out half-integer intrinsic spin.

Adding $\frac{1}{2}\int \sigma (x)(h^{\prime }{)}^{2}dx$ to the quadratic bending energy of an elastic filament contributes $-(\sigma h^{\prime }{)}^{\prime }$ to its variational derivative. The endpoint term is $A{h}^{\prime \prime }{\eta }^{\prime }+(\sigma h^{\prime }-A{h}^{\prime \prime \prime })\eta$. If the real filament tension vanishes at both ends, the usual self-adjoint endpoint conditions for filament bending remain valid. A complete eigenfunction expansion diagonalizes the energy even when the eigenfunctions have no explicit formula. Thermal equipartition theorem arguments require positive eigenvalues after fixing any kernel; strong compression can violate this requirement.

Let $(w_m)$ be the orthonormal basis of Dirichlet eigenfunctions for a strictly positive Dirichlet realization of an elliptic operator, with eigenvalues ${\lambda }_m$. ThenFor integers $k$, these domains are the Sobolev domains of powers of an elliptic Dirichlet operator. At $k=0$ the condition is just Parseval identity and imposes no boundary condition. At $k=1,2$ it describes $H_0^1$ and $H^2\cap H_0^1$, respectively. Higher $k$ impose traces of powers of $L$, not only the trace of $u$. For $L=-d^2/d{x}^2$ on $(0,\pi )$, the smooth Dirichlet function $u=x(\pi -x)$ has sine coefficients proportional to $m^{-3}$ for odd $m$, so the weighted sum diverges at $k=3$ despite its ordinary $H^3$ regularity.

For the random adjacent transposition shuffle with $n\ge 2$, the generator $P-I$ is the interchange process on a path graph with each edge rate $1/n$. The Aldous spectral gap theorem reduces its spectral gap to that of the single-label random walk. The eigenfunctions $k\mapsto \cos (\ell \pi (k-1/2)/n)$, for $0\le \ell <n$, have generator eigenvalues $-2(1-\cos (\ell \pi /n))/n$; direct substitution verifies both the interior and endpoint equations. Consequently $\gamma \sim {\pi }^2/n^3$.