Analysis and Differential Equations

The Schrödinger equation is a differential equation, an operator equation, an initial- or boundary-value problem, and often a spectral problem at the same time. Solving it responsibly requires more than manipulating derivatives: one must specify the function space, domain, data, convergence mode, and approximation regime.

This chapter organizes the analysis used in wave mechanics, exact canonical systems, scattering, variational methods, and semiclassical approximations. It does not collect every theorem of analysis. Its purpose is to identify which mathematical problem has been posed, which tool addresses it, and which hypotheses make the resulting operations legitimate.

The anatomy of a differential problem

A well-posed statement normally includes five pieces.

Piece	Question to ask	Quantum example
Differential expression	What derivatives and coefficients appear?	$-\hbar^2\nabla^2/(2m)+V$
Function space and domain	Which regularity and integrability conditions are required?	a Sobolev subspace of $L^2$
Initial or boundary data	What selects one solution or operator realization?	initial wavefunction, Dirichlet data, or scattering condition
Spectral or source data	Is the equation homogeneous, inhomogeneous, or parameter dependent?	$H\psi=E\psi$ or $(H-z)u=f$
Convergence and error notion	In what sense is a series, limit, or approximation valid?	norm, pointwise, distributional, or asymptotic control

The same differential formula can describe different mathematical and physical problems when any of these pieces changes.

Identify the problem type first

Four common forms should not be conflated.

1. Initial-value problem: data are specified at one time or point, as in $\dot y=F(t,y)$ with $y(t_0)=y_0$ or the time-dependent Schrödinger equation with $\psi(0)$ given.
2. Boundary-value problem: conditions are imposed at spatial endpoints or boundaries, as in $Lu=f$ with $Bu=0$.
3. Eigenvalue problem: a spectral parameter is part of the unknown, as in $Lu=\lambda wu$ together with boundary conditions.
4. Inhomogeneous or resolvent problem: one solves $(L-z)u=f$, often through an integral kernel or Green function.

Existence, uniqueness, and admissible data differ among these forms. An initial-value solver does not automatically solve a two-point boundary problem, and a local general solution of an ODE does not determine its allowed eigenvalues.

Convergence controls legal operations

The phrase “the series converges” is incomplete until the convergence mode is named.

Mode	What converges	What it does not guarantee by itself
Pointwise	$f_n(x)\to f(x)$ for each fixed $x$	uniform error control, continuity of $f$, or interchange of limits and integrals
Uniform	$\sup_x\lvert f_n(x)-f(x)\rvert\to0$	convergence of derivatives without additional hypotheses
Norm	$\lVert f_n-f\rVert_X\to0$ in a specified function space	pointwise convergence everywhere
Distributional	pairings against test functions converge	ordinary pointwise values or square integrability
Asymptotic	a remainder is ordered in a limiting regime	convergence of the resulting infinite series

Uniform limits of continuous functions are continuous. $L^2$ convergence controls inner products against fixed $L^2$ test vectors by Cauchy–Schwarz, but it can ignore changes on sets of measure zero. Distributional convergence legitimizes delta functions and jump conditions. Termwise differentiation generally needs stronger hypotheses than termwise integration.

Read Sequences, Series, and Convergence before manipulating infinite basis expansions. Real Analysis Essentials supplies the continuity, differentiability, integrability, and function-space context behind those statements.

ODEs, PDEs, and separation

An ordinary differential equation involves one independent variable; a partial differential equation involves several. The time-dependent Schrödinger equation,

$$i\hbar\frac{\partial\Psi}{\partial t} =\left[-\frac{\hbar^2}{2m}\nabla^2+V\right]\Psi,$$

is first order in time and usually second order in space. Initial data govern time evolution, while spatial boundary conditions help define the Hamiltonian.

When the geometry, coefficients, and boundary data are compatible, a product ansatz can separate variables. For a time-independent Hamiltonian,

$$\Psi(\mathbf x,t)=\psi(\mathbf x)T(t)$$

leads to

$$i\hbar\frac{T'}{T}=E, \qquad H\psi=E\psi.$$

The separation constant becomes a spectral value because both factors must satisfy their own equations and data. Separation is a structural property of the operator and domain, not a guarantee for every PDE or coordinate system. Use Ordinary Differential Equations, Partial Differential Equations, and Separation of Variables as a sequence.

Boundary conditions and eigenvalue problems

Dirichlet, Neumann, Robin, periodic, and mixed conditions can encode physical constraints and operator-domain choices. They determine which integrations by parts lose their boundary terms and therefore whether a differential realization is symmetric or self-adjoint.

A regular Sturm–Liouville problem has the form

$$-\frac{d}{dx} \left(p(x)\frac{dy}{dx}\right) +q(x)y =\lambda w(x)y,$$

on a finite interval, with $p(x)\gt0$, $w(x)\gt0$, suitable coefficient regularity, and self-adjoint boundary conditions. Under the standard regular hypotheses, eigenvalues are real and discrete, and eigenfunctions belonging to distinct eigenvalues are orthogonal in the weighted inner product

$$\langle y_m,y_n\rangle_w =\int_a^b y_m(x)^*y_n(x)w(x)\,dx.$$

Completeness is a theorem with hypotheses, not a consequence of writing a second-order equation. Singular endpoints, unbounded intervals, and continuous spectral sectors require extensions of the regular theory. The route is Boundary Conditions $\to$ Eigenvalue Problems $\to$ Sturm–Liouville Theory.

Green functions and resolvents

For a linear operator $L$ with specified boundary conditions, a Green function satisfies

$$L_xG(x,\xi)=\delta(x-\xi),$$

with the same homogeneous boundary conditions in the $x$ variable. When the inverse exists, the solution of $Lu=f$ is

$$u(x)=\int G(x,\xi)f(\xi)\,d\xi.$$

The differential equation determines the behavior away from $x=\xi$; the delta source determines a jump condition; the boundary conditions select the kernel. Changing the boundary condition changes the Green function even when the differential expression is unchanged.

For $G(z)=(L-z)^{-1}$, spectral values appear as obstructions to inversion. In discrete settings they produce poles of matrix elements of the resolvent; continuous spectra require boundary values, branch cuts, or distributional prescriptions. Green Functions owns the kernel construction. Propagators, causal prescriptions, and scattering applications live in their respective physics volumes.

Complex analysis in quantum problems

Analytic continuation organizes resolvents, scattering amplitudes, inverse transforms, and asymptotic integrals. Poles can encode bound states or resonances depending on the function and sheet; branch points and cuts signal multivalued structure or continuum thresholds. A contour may be deformed only while accounting for crossed singularities and endpoint contributions.

Complex Analysis Essentials gives the physics-facing survey. Canonical derivations of Analytic Functions, Contour Integration, the Residue Theorem, and Branch Cuts remain in the complex-functions chapter.

Variational and functional methods

A functional assigns a scalar to a function. For

$$J[y]=\int_a^b \mathcal L(x,y,y')\,dx,$$

the first variation is

$$\delta J =\left[ \frac{\partial\mathcal L}{\partial y'}\delta y \right]_a^b +\int_a^b \left( \frac{\partial\mathcal L}{\partial y} -\frac{d}{dx} \frac{\partial\mathcal L}{\partial y'} \right) \delta y\,dx.$$

Fixed endpoint variations remove the boundary term and yield the Euler–Lagrange equation. Other boundary conditions produce natural boundary terms that must be treated explicitly.

In quantum mechanics, the Rayleigh quotient

$$R[\psi] =\frac{\langle\psi,H\psi\rangle} {\langle\psi,\psi\rangle}$$

connects functional stationarity to eigenvalue problems and variational bounds. Calculus of Variations owns finite-dimensional and function-space variations; Functional Derivatives develops relations such as

$$\delta F[\phi] =\int \frac{\delta F}{\delta\phi(x)} \delta\phi(x)\,dx.$$

Physical trial-state strategies are canonical in the variational-methods chapter, and path-integral applications remain in Dynamics and Formulations.

Asymptotic reasoning

An asymptotic expansion describes an ordered approximation as a parameter approaches a limit:

$$f(\epsilon) \sim\sum_{n=0}^{\infty}a_n\epsilon^n, \qquad \epsilon\to0.$$

It means that truncating after finitely many terms leaves a remainder smaller than the last retained order under a stated definition. The infinite series need not converge, and adding more terms can eventually worsen a fixed-$\epsilon$ approximation. A trustworthy asymptotic claim identifies the dimensionless small parameter, limiting regime, retained order, and remainder or validation strategy.

Asymptotic Analysis introduces big-$O$ notation, asymptotic series, stationary phase, and saddle points. WKB, semiclassical propagators, and matched physical approximations remain in Approximation and Semiclassical Methods.

Page map

Page	Use it to answer…
Sequences, Series, and Convergence	In what sense does a limit or expansion converge?
Real Analysis Essentials	Which continuity, differentiability, and integrability assumptions are being used?
Complex Analysis Essentials	How do analyticity, poles, contours, and branch cuts enter quantum calculations?
Ordinary Differential Equations	How are one-variable initial and boundary problems classified?
Partial Differential Equations	How do initial data, spatial boundaries, and several variables interact?
Boundary Conditions	Which endpoint data complete the operator problem?
Eigenvalue Problems	How are admissible spectral values selected?
Sturm–Liouville Theory	Why do many separated equations yield weighted orthogonal eigenfunctions?
Separation of Variables	When can one PDE become several ODE eigenproblems?
Green Functions	How does an inverse differential operator become an integral kernel?
Calculus of Variations	How does stationarity produce differential equations and boundary terms?
Functional Derivatives	How are functionals differentiated with respect to fields or wavefunctions?
Asymptotic Analysis	What does a controlled limiting expansion claim, and what does it not claim?

A practical workflow

1. State the function space, differential action, domain, and initial or boundary data.
2. Classify the task as an initial-value, boundary-value, eigenvalue, or inhomogeneous problem.
3. Test for symmetry, conservation laws, separability, and self-adjoint structure before solving components.
4. Choose an exact representation: basis expansion, special function, transform, Green kernel, or variational equation.
5. Name the convergence or approximation mode used by every infinite operation.
6. Check dimensions, boundary data, normalization, residuals, and limiting cases.
7. If no controlled analytic route is available, move to Numerical Mathematics with a benchmark and convergence plan.

Common mistakes

Mistake	Correction
Solving a differential expression without imposing data	the general local solution is not yet the physical solution or operator spectrum
Interchanging a limit, derivative, sum, or integral formally	identify a theorem and verify its hypotheses
Treating pointwise, uniform, and norm convergence as equivalent	state the topology or norm being used
Assuming separation of variables always works	check operator, geometry, and boundary compatibility
Calling every second-order eigenproblem Sturm–Liouville	verify the self-adjoint form, weight, interval, and endpoint hypotheses
Reusing a Green function after changing boundary conditions	rebuild the kernel for the new operator domain
Dropping variational boundary terms silently	state which variations or natural conditions make them vanish
Treating an asymptotic series as a convergent power series	specify the limiting regime and truncation error

Exercises

1. Pointwise but not uniform convergence

On $[0,1]$, let $f_n(x)=x^n$. Find the pointwise limit and show that convergence is not uniform.

Solution

For $0\leq x\lt1$, $x^n\to0$, while $f_n(1)=1$. Thus the pointwise limit is zero on $[0,1)$ and one at $x=1$. The limit is discontinuous even though every $f_n$ is continuous, so convergence cannot be uniform. Equivalently, values arbitrarily close to one make $\lvert f_n(x)-f(x)\rvert$ arbitrarily close to one for every fixed $n$.

2. Sturm–Liouville orthogonality

Let $Ly_n=\lambda_nwy_n$ for

$$L=-\frac{d}{dx}p(x)\frac{d}{dx}+q(x),$$

with self-adjoint boundary conditions. Show that eigenfunctions with $\lambda_m\ne\lambda_n$ are orthogonal in the weight $w$.

Solution

Multiply the $n$ equation by $y_m^*$, multiply the complex conjugate of the $m$ equation by $y_n$, subtract, and integrate. Integration by parts gives

$$(\lambda_n-\lambda_m) \int_a^b y_m^*y_nw\,dx =\left[ p\left(y_n y_m'^*-y_m^*y_n'\right) \right]_a^b.$$

The self-adjoint boundary conditions make the boundary form vanish. Since the eigenvalues differ, the weighted inner product must be zero.

3. Green-function jump condition

Suppose $G$ satisfies $-\partial_x^2G(x,\xi)=\delta(x-\xi)$. Integrate across a small interval around $\xi$ and find the jump in $\partial_xG$.

Solution

Integrating from $\xi-\varepsilon$ to $\xi+\varepsilon$ gives

$$-\left[ \partial_xG(x,\xi) \right]_{\xi-\varepsilon}^{\xi+\varepsilon} =1.$$

Taking $\varepsilon\to0$ yields

$$\partial_xG(\xi^+,\xi) -\partial_xG(\xi^-,\xi) =-1.$$

Away from $x=\xi$, the Green function satisfies the homogeneous equation; the delta source is encoded by this derivative jump.

4. A first variation

For fixed endpoint values, find the stationary equation of

$$J[y]=\int_a^b \left[ \frac12(y')^2+V(y) \right]dx.$$

Solution

The first variation is

$$\delta J =\int_a^b \left(y'\delta y'+V'(y)\delta y\right)dx.$$

Integrating the first term by parts and using $\delta y(a)=\delta y(b)=0$ gives

$$\delta J =\int_a^b \left[-y''+V'(y)\right]\delta y\,dx.$$

Arbitrary interior variations therefore imply $-y''+V'(y)=0$.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer, 1999.
• R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume I, Wiley, 1989.
• L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
• E. Kreyszig, Introductory Functional Analysis with Applications, Wiley, 1978.
• F. W. J. Olver, Asymptotics and Special Functions, A K Peters, 1997.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• W. A. Strauss, Partial Differential Equations: An Introduction, 2nd ed., Wiley, 2008.