Differential Equations and Green Operators

This chapter connects three decisions that are often collapsed into the word “solve.” First identify the differential equation and its principal type. Then specify the spaces, domain, and initial or boundary data that make a well-posed realization. Only after that should one select an inverse by boundary, support, decay, spectral, or pole conditions. The right route therefore begins with the reader’s QFT need, not with a requirement to read all seven pages in sidebar order.

There are two entry routes: mode equations and PDE and Green operators. Whichever route is chosen, the core rules are the same: name the operator, spaces, domain, and boundary data before writing an inverse, and distinguish elliptic, hyperbolic, and parabolic questions.

The chapter covers linear ODE evolution, regular Sturm–Liouville systems, principal symbols and PDE type, a bounded introduction to weak solutions and well-posedness, distributional fundamental solutions and Green operators, hyperbolic causal inverses, and elliptic boundary realizations with heat kernels. Developed physical applications remain in Foundations, Renormalization and EFT, and Curved Spacetime. General theorem-first operator theory, global Lorentzian geometry, nonlinear PDE, rough-domain theory, and numerical production workflows also remain with their specialist destinations.

Parent volume: Mathematical Methods

Enter this chapter

Use the following checks to find the smallest useful entry point.

Readiness check	Ready	If unsure	Repair and return
Can you rewrite a scalar linear ODE as a first-order system and state what data determine a solution?	Enter Linear ODEs, Evolution Operators, and Wronskians.	Check whether you can distinguish an initial condition from a two-endpoint boundary condition.	Start with the ODE page itself; it has no hard prerequisite.
With the site’s positive forward Fourier phase, can you translate $\partial_\mu$ into its momentum multiplier?	Enter Symbols, Characteristics, and PDE Type.	Derive the multiplier rather than recalling it from another convention.	Review Fourier Series, Fourier Transforms, and Plancherel Theory, then return to the symbol page.
Can you interpret $PE=\delta_0$ as a distributional identity and name the test space?	Enter Fundamental Solutions and Green Operators.	Pair both sides with a compactly supported smooth test function and move derivatives by duality.	Review Test-Function Spaces, Distributions, Support, and Convergence or the Learn remediation route linked below.
Can you name the trial, test, and data spaces in a variational equation and state the norm used for stability?	Enter Weak Solutions, Sobolev Spaces, and Well-Posedness.	Ask where derivatives live, how boundary traces are represented, and which estimate proves continuous dependence.	Review Test-Function Spaces, Distributions, Support, and Convergence and Lp Spaces, Inequalities, and Weak Convergence.
Can you distinguish formal symmetry from a self-adjoint operator domain and test for a zero mode?	Take Linear ODEs before the regular Sturm–Liouville route; take Symbols plus Fundamental Solutions before the elliptic route.	Vanishing of the boundary form proves symmetry only. Also check that the proposed domain equals the adjoint domain; for a regular Sturm–Liouville problem, the boundary subspace must be maximal isotropic. Then inspect the kernel.	Review Bilinear and Hermitian Forms, Adjoints, and Isometries; use Spectra, Resolvents, and Functional Calculus for the general spectral language.
Can you say whether an inverse is selected by cone support, a boundary domain, decay, or an $i0$ boundary value?	Complete both Symbols and Fundamental Solutions, then choose the hyperbolic or elliptic route.	Check whether the requested object is a response, a homogeneous propagator, a Euclidean inverse, or a time-ordered kernel.	Repair either hard input before continuing; for pole placement also use Contour Deformation, Pinches, and Causal Prescriptions.

For a focused review of Fourier methods, distributions, and Green functions, use Fourier, Distributions, and Green Functions Repair.

Choose a route

An arrow below means a hard dependency in the stated route. A plus sign means that both branches are required before the final page. “Recommended” material improves fluency but is not promoted to a hard prerequisite.

Reader goal	Minimum coherent route	Useful addition	Capability at the end
Evolve and match mode functions	Linear ODEs	None required	Construct an evolution operator, use variation of constants, and reduce boundary or matching data to a finite linear system
Expand cavity, compact-direction, or regular radial modes	Linear ODEs → Sturm–Liouville Problems	Forms and Adjoints	State the regular self-adjoint hypotheses, normalize modes with the weight, and recognize zero-mode compatibility
Classify a PDE or candidate data surface	Symbols, Characteristics, and PDE Type	Fourier Transforms	Compute the principal symbol and distinguish elliptic, hyperbolic, and weighted-parabolic questions without claiming an inverse
Formulate a weak problem and test stability	Test-Function Spaces and Distributions + Lp Spaces and Inequalities → Weak Solutions, Sobolev Spaces, and Well-Posedness	Symbols when PDE type matters	State a typed variational or energy problem and separate existence, uniqueness, stability, and extra regularity
Construct a Green kernel	Test-Function Spaces and Distributions → Fundamental Solutions and Green Operators	Symbols and Tempered Fourier Calculus	Distinguish a free-space fundamental solution from a Green operator with declared spaces, measure, domain, and selection condition
Select causal response or a Feynman inverse	Symbols + Fundamental Solutions → Hyperbolic Equations and Causal Propagators	Contour and Causal Prescriptions	Separate retarded, advanced, causal, and Feynman objects by support and pole data
Solve a Euclidean boundary problem or use heat time	Symbols + Fundamental Solutions → Elliptic Boundary Problems and Heat Kernels	Sturm–Liouville Problems and Spectral Calculus are recommended	Specify an elliptic realization, diagnose zero modes, and choose between spectral, heat-kernel, and proper-time representations

The dependency graph is branched:

• linear ODEs are required for the regular Sturm–Liouville theorem;
• distributions and $L^p$ spaces are required for the weak-solution route;
• distributions are required for fundamental solutions, while symbols are recommended there;
• symbols and fundamental solutions are both required for the hyperbolic and elliptic routes; and
• Sturm–Liouville theory and spectral calculus are recommended, not required, before the elliptic heat-kernel page.

The chapter’s objects are related but not interchangeable

Object	Defining relation	What selects it	What it is not
Fundamental matrix	$F'=AF$ with $\det F\neq0$	A basis of homogeneous solutions	The basis-independent transport between two times
Evolution operator	$\mathsf U(t,s)=F(t)F(s)^{-1}$, so $\mathsf U(s,s)=I$	A homogeneous ODE system and two times	A distributional inverse of a PDE
Free-space fundamental solution	$P E_{\mathrm{fund}}=\delta_0$ for a nonzero constant-coefficient $P$; this equation need not select a unique solution	A declared free-space distribution setting; a separately proved support, growth, or radiation condition may select one solution	Automatically a boundary-value Green kernel
Green operator and kernel	$P_B G_B=I$ on a declared compatible data space	Spaces, operator domain, measure, and boundary, support, decay, or asymptotic data	Unique from the differential expression alone
Retarded or advanced Green operator	$P G_{\mathrm{ret/adv}}=I$	Future or past support relative to the source	A time-ordered inverse
Causal propagator	$E_{\mathrm{caus}}=G_{\mathrm{ret}}-G_{\mathrm{adv}}$, so $P E_{\mathrm{caus}}=0$	Difference of the two support-selected inverses	An inverse of $P$
Feynman inverse	$P G_F=I$; in the page’s mathematical normalization, $\widetilde G_F=-1/(p^2-m^2+i0)$	Pole prescription and QFT normalization	Future- or past-cone supported
Heat kernel	Kernel of $e^{-tL_B}$ for $t>0$	A lower-bounded self-adjoint, strongly elliptic realization in the smooth compact setting, plus heat initial data	An inverse until proper-time integration and spectral qualifications are justified

Thus a fundamental matrix supplies a homogeneous solution basis, while the evolution operator transports Cauchy data independently of that basis. A fundamental solution responds to a delta source in free space; a Green operator is a typed selected inverse; causal and Feynman constructions make different selections; and a heat kernel represents a semigroup. The defining equation is only the first line of the specification.

Shared conventions

These are the recurrent chapter-level choices. Leaf-specific hypotheses still belong on the leaf that uses them.

Issue	Convention	Invariant check
Lorentzian metric	Signature $(+---)$, so $p^2=(p^0)^2-\lvert\mathbf p\rvert^2$ and $\Box=\partial_t^2-\nabla^2$	A Klein–Gordon plane wave obeys $p^2=m^2$
Fourier transform	Forward phase $e^{+ip\cdot x}$ and inverse phase $e^{-ip\cdot x}$	$\partial_\mu\mapsto-ip_\mu$
Spatial Fourier phase	At fixed time, the forward spatial phase is $e^{-i\mathbf k\cdot\mathbf x}$ because $p\cdot x=p^0t-\mathbf p\cdot\mathbf x$	It is consistent with the full spacetime convention, not a sign change
Euclidean operator	$L=-\Delta_g+V$, with $-\Delta_g\geq0$ before potential and boundary contributions	Heat evolution is $e^{-tL}$ and carries no implicit $i0$
Pairings	Hilbert inner products are conjugate-linear in the first slot; distributional and Sobolev duality pairings are different objects	Conjugation is never imported into a duality formula without declaration
Delta normalization	The delta is defined relative to the integration measure	The weighted Sturm–Liouville identity kernel is $\delta(x-x')/w(x')$
Mathematical versus QFT kernels	A mathematical Green kernel satisfies $PG=\delta$; the site uses $\widetilde\Delta_F=i/(p^2-m^2+i0)$ and $P\Delta_F=-i\delta$	The delta-normalized Feynman Green kernel is $G_F=i\Delta_F$
Causal names	Retarded means future-supported response; advanced means past-supported response; $E_{\mathrm{caus}}=G_{\mathrm{ret}}-G_{\mathrm{adv}}$	Check support before translating a source’s terminology
Boundary normal	The elliptic page uses the outward normal and Robin data $(\partial_n+\beta)u=0$	Green’s identity fixes the sign of the boundary form
Zero modes and convergence	On the ambient Hilbert space, $L G^\perp=I-\Pi_0$; on $(\ker L)^\perp$, $G^\perp$ is a genuine inverse	For a lower-bounded realization, zero and negative modes do not obstruct finite heat time; zero modes prevent large-time decay and negative modes grow exponentially, so either makes the unmodified proper-time inverse diverge

Fourier, mode, and kernel formulas are operator- or distribution-valued unless pointwise convergence is proved separately. Likewise, a short-time heat expansion is asymptotic local information and cannot determine the lowest spectrum or large-time behavior.

Topics in this chapter

Linear ODEs, Evolution Operators, and Wronskians

The ODE page answers the exact question: How do fundamental matrices, Wronskians, and boundary data organize linear mode equations? It constructs $\mathsf U(t,s)$, proves its composition and basis-change properties, derives Abel–Liouville evolution of the Wronskian, and uses variation of constants for a source. Boundary and matching data become a finite-dimensional solvability problem.

There is no hard prerequisite. Continue to the Sturm–Liouville page for a regular spectral boundary problem, or to the symbol and fundamental-solution pages when the mode equation came from a PDE.

Sturm–Liouville Problems and Eigenfunction Expansions

The Sturm–Liouville page answers the exact question: When does a boundary-value problem yield orthogonal modes and a useful completeness relation? On a finite interval it makes the weight, maximal domain, boundary form, and self-adjoint boundary conditions explicit before stating real discrete spectrum and weighted completeness. It then compares spectral and ODE constructions of a resolvent kernel and diagnoses a zero mode.

Linear ODEs are hard preparation; forms and adjoints are recommended. This theorem-focused route is useful for cavity, compact-direction, and regular radial modes. A singular radial endpoint needs the separate limit-point/limit-circle or extension analysis marked as outside this leaf. Continue to spectral analysis for general operator theory or to the elliptic page for heat evolution.

Symbols, Characteristics, and PDE Type

The symbol page answers the exact question: How do principal symbols and characteristics distinguish elliptic, hyperbolic, and parabolic behavior? It computes principal symbols in the site’s Fourier convention, identifies characteristic covectors and hypersurfaces, and separates ordinary homogeneous classification from the weighted scaling used by parabolic evolution.

There is no hard prerequisite; Fourier transforms are recommended. Its output is a PDE classification and a test for candidate data surfaces—not an existence theorem, stability estimate, or choice of Green operator. Continue to weak solutions for functional-analytic well-posedness, or combine it with the fundamental-solution page before the hyperbolic or elliptic method pages.

Weak Solutions, Sobolev Spaces, and Well-Posedness

The weak-solution page answers the exact question: What counts as a solution when derivatives are weak, and what existence, uniqueness, and stability should mean? It distinguishes distributional from $L^p$-represented weak derivatives, introduces Sobolev and trace spaces, formulates a coercive elliptic problem through Lax–Milgram, and gives an energy-estimate route for a linear wave equation.

Test-function distributions and $L^p$ spaces are hard preparation. This advanced treatment does not claim a general nonlinear or rough-coefficient theory, and it does not turn weak existence into classical regularity. Continue to the fundamental-solution page when a kernel representation is wanted, or to the elliptic page when boundary realizations and spectra are the next question.

Evans 2010, Chapters 5–7 supplies the Sobolev, weak-formulation, and energy-method framework summarized by this branch; the leaf states the narrower hypotheses actually used.

Fundamental Solutions and Green Operators

The Green-operator page answers the exact question: How does a Green operator invert a differential operator once function spaces and boundary data are fixed? It separates the free-space identity $PE_{\mathrm{fund}}=\delta_0$ from a typed right inverse, gives the kernel and convolution viewpoints, tracks the integration measure in the delta normalization, and treats homogeneous freedom, compatibility, and reduced inverses.

Test-function distributions are hard preparation; symbols are recommended, and tempered Fourier calculus is needed for Fourier division. The method is shared by the two final branches. Continue to hyperbolic equations for support and pole selection or to elliptic problems for boundary spectra and heat evolution.

Hyperbolic Equations and Causal Propagators

The hyperbolic page answers the exact question: How do hyperbolicity and support conditions distinguish retarded, advanced, causal, and Feynman-type inverses? The causal difference itself is homogeneous rather than an inverse. The page states the well-posedness and finite-propagation hypotheses, constructs support-selected Green operators, and contrasts support selection with a momentum-space pole boundary value.

Symbols and fundamental solutions are both hard preparation; contour and causal prescriptions are recommended. Foundations supplies state dependence and physical propagator interpretation; Curved Spacetime supplies the developed global causal and Hadamard theory.

Elliptic Boundary Problems and Heat Kernels

The elliptic page answers the exact question: How do ellipticity, boundary conditions, and heat evolution control inverses, spectra, and short-time behavior? It distinguishes a differential expression from its Dirichlet, Neumann, or Robin realization; derives the boundary form; treats compact resolvent and the resulting discrete finite-multiplicity spectrum, compatibility, and reduced inverses; and relates the heat semigroup, heat kernel, proper-time inverse, and local short-time expansion.

Grubb 1996, Chapter I, §§1.4–1.7, PDF supports the boundary-realization and strong-ellipticity framework, while Grieser 2004, Theorem 1.1 and §3, PDF supports the smooth compact short-time expansion and boundary layer.

Symbols and fundamental solutions are hard preparation. Sturm–Liouville modes and spectral calculus are recommended. Continue to heat-kernel zeta methods for spectral determinants, or to Renormalization and EFT for the physical heavy-field expansion.

One scalar operator across the chapter

A free scalar supplies a useful thread, but each page changes the mathematical question.

1. Mode evolution. Spatial decomposition reduces the Klein–Gordon equation to oscillator ODEs. The ODE page transports Cauchy data and handles matching.
2. Mode spectrum. A finite interval, cavity, compact direction, or regular radial reduction supplies a Sturm–Liouville weight and boundary domain. The spectral theorem licenses the mode sum; a singular radial endpoint needs additional domain analysis.
3. PDE type. The Lorentzian principal symbol identifies a characteristic cone. After Euclidean continuation, the Laplace-type principal symbol is elliptic. Lower-order mass terms do not change either principal classification.
4. Weak formulation. Integration by parts moves derivatives to test functions and exposes boundary terms. Existence and stability then depend on the chosen topology and estimate.
5. Inverse kernel. A source equation requires a fundamental solution or typed Green operator, with the delta normalized to the integration measure.
6. Lorentzian selection. Future support gives the retarded response; past support gives the advanced response; their difference is homogeneous. The Feynman inverse instead uses a pole boundary value.
7. Euclidean evolution. A positive self-adjoint, strongly elliptic realization in the smooth compact setting generates $e^{-tL}$; justified proper-time integration recovers an inverse, while short heat time isolates local high-frequency information.

What stays fixed is the obligation to name the operator, spaces, data, and normalization. What changes is the selection principle. The thread stops before state-dependent physical interpretation, renormalized loop coefficients, curved-spacetime Hadamard states, gauge constraints, or nonlinear dynamics.

Chapter synthesis

The reusable starting point is a typed realization

$$P_B: \mathcal D_B(P)\subset X \longrightarrow Y_0,$$

where $B$ records boundary, initial, support, decay, or asymptotic data and $Y_0$ is the compatible data space. A Green operator, when it exists, is a map in the opposite direction satisfying

$$P_B G_B=I_{Y_0}.$$

The chapter’s central conclusions are:

• the principal symbol classifies high-frequency PDE behavior but does not choose a domain, prove well-posedness, or select an inverse;
• initial and boundary conditions are part of the operator problem, not decorations attached after solving the differential expression;
• well-posedness means existence, uniqueness, and continuous dependence in declared spaces and norms;
• a fundamental solution is a distributional free-space seed, whereas a Green operator is a typed inverse with additional selection data;
• self-adjointness is a statement about an operator and its domain, not merely a formally symmetric differential expression;
• zero modes impose compatibility and nonuniqueness and require a reduced inverse or another explicit prescription;
• retarded, advanced, Feynman, and Euclidean inverses answer different boundary or support questions; and
• heat evolution is defined at finite positive time even in the presence of zero modes, while an unmodified proper-time inverse is not.

The ODE and regular Sturm–Liouville structure is developed systematically in Teschl 2012, Chapters 3 and 5, PDF. The distributional symbol and fundamental-solution framework follows Dyatlov 2022, Chapters 7, 9–10, and 12–14, PDF. Dyatlov uses the negative Fourier phase $e^{-ix\cdot\xi}$ and $D=-i\partial$; setting $\xi=-p$ translates his multiplier formulas to the site’s positive forward phase and $\partial\mapsto-ip$. The support-selected hyperbolic theory is stated at greater generality by Bär, Ginoux, and Pfäffle 2007, §§3.2–3.4, PDF, and the form, realization, and semigroup bridge used by the elliptic branch is developed in Arendt et al. 2015, Chapters 5–7, PDF. Hunter 2014, §§2.5–2.7, 4.8–4.10, 5.1–5.4, and 7.1–7.5, PDF provides a complementary route through Green identities, free-space kernels, compact-resolvent spectral theory, heat evolution, and weak hyperbolic solutions.

Review the chapter

Object retrieval and comparison. Given the phrases “fundamental matrix,” “fundamental solution,” “Green kernel,” “causal propagator,” and “heat kernel,” state one defining equation and one selection datum for each. Success distinguishes the homogeneous ODE object, the delta-normalized distribution, the typed inverse, the homogeneous causal difference, and the finite-time semigroup kernel. Repair a conflation with Fundamental Solutions and Green Operators and the object table above.

Derivation and representation change. Starting from the positive forward Fourier phase, derive $\partial_\mu\mapsto-ip_\mu$, compute the principal symbol of $\Box+m^2$, and explain why the mass does not affect PDE type. Then translate the site’s $\widetilde\Delta_F=i/(p^2-m^2+i0)$ into the delta-normalized mathematical kernel. Success obtains $P\Delta_F=-i\delta$ and $G_F=i\Delta_F$. Repair a sign error with Symbols, Characteristics, and PDE Type and Hyperbolic Equations and Causal Propagators.

Theorem-hypothesis check. For a regular Sturm–Liouville expression on a finite interval, list the coefficient, weight, and boundary-domain hypotheses needed before claiming a real complete mode basis. Success states $p,w>0$, regular real coefficients, the weighted Hilbert space, and a self-adjoint boundary domain; it also checks a possible zero mode before writing a resolvent. Repair with Sturm–Liouville Problems and Eigenfunction Expansions.

Failure diagnosis. Diagnose the claim “$P$ is elliptic, so $P^{-1}$ exists and is unique.” Success names the missing spaces, domain and boundary condition, compatibility and kernel test, and selection of a data subspace. It also distinguishes interior ellipticity from the boundary condition needed for the standard heat expansion. Repair with the elliptic page.

Transfer to a weak problem. Given $-\nabla\!\cdot(a\nabla u)+cu=f$ on a bounded domain, state trial, test, and data spaces, identify essential versus natural boundary data, and name the estimate that would prove stability. The data supplied do not determine a unique realization, so success first refuses to choose spaces until boundary conditions and coefficient hypotheses are declared. One valid answer takes homogeneous Dirichlet data, trial and test space $H_0^1$, data in $H^{-1}$, bounded uniformly positive $a$, and a coercive form; the Lax–Milgram estimate then gives existence, uniqueness, and continuous dependence. A Neumann choice instead uses $H^1$ and must test compatibility and zero modes; the pure massless Neumann realization has the constant obstruction, whereas a positive zeroth-order term can remove it. Neither answer infers classical regularity from weak existence. Repair with Weak Solutions, Sobolev Spaces, and Well-Posedness.

Chapter-scale synthesis. A scalar source is switched on in Lorentzian spacetime and the same quadratic operator is later used in a Euclidean one-loop calculation. Give the minimum route for each task and state what extra datum changes the inverse. Success uses symbols plus fundamental solutions for both branches, future support for retarded response, a pole boundary value for the Feynman inverse, and a positive self-adjoint, strongly elliptic realization with zero-mode treatment for proper time. Repair the Lorentzian branch with Hyperbolic Equations and Causal Propagators and the Euclidean branch with Elliptic Boundary Problems and Heat Kernels.

Connections to QFT and further theory

The chapter stops when physical interpretation or specialist global theory becomes the main question.

• The Klein–Gordon Field and Its Modes develops the physical meaning of scalar mode functions and their normalization.
• Boundaries, Variations, and Well-Posed Actions treats boundary terms as part of a physical variational principle.
• Scalar Propagators, Ordered Correlators, and Sources develops time ordering, states, and the physical propagator taxonomy.
• Retarded, Advanced, and Spectral Correlators treats causal response and spectral relations in QFT.
• Green Operators, Causal Propagators, and State-Dependent Two-Point Functions develops global curved-spacetime causal theory and state-dependent kernels.
• Heat Kernels and the Schwinger–DeWitt Expansion develops the curved-spacetime asymptotic expansion and its effective-action use.
• Integrating Out Heavy Fields uses heat-kernel asymptotics in the physical local heavy-scale expansion.

For general closed and self-adjoint operators, spectral measures, and functional calculus, continue to Functional and Spectral Analysis. For zeta continuation and spectral determinants, continue to Heat Kernels, Zeta Functions, and Spectral Determinants. To choose another mathematical route, return to Mathematical Methods.

References

• Wolfgang Arendt, Ralph Chill, Christian Seifert, Hendrik Vogt, and Jürgen Voigt (2015), Form Methods for Evolution Equations, and Applications, PDF, Chapters 5–7. This is the structural source for form-defined realizations, self-adjointness, compact resolvent, Neumann and Robin conditions, semigroups, and zero modes. Its first-slot-linear inner product is translated to the site’s first-slot-conjugate-linear convention.
• Christian Bär, Nicolas Ginoux, and Frank Pfäffle (2007), Wave Equations on Lorentzian Manifolds and Quantization, PDF, §§3.2–3.4. These sections establish Cauchy well-posedness, finite propagation, support-selected Green operators, and the causal difference. Its signature and advanced/retarded naming are translated by support on the hyperbolic leaf.
• Semyon Dyatlov (2022), Lecture Notes for 18.155: Distributions, Elliptic Regularity, and Applications to PDEs, PDF, Chapters 7, 9–10, and 12–14. These chapters develop distributional kernels, fundamental solutions, wave propagation, symbols, characteristics, and ellipticity. Its negative Fourier phase and $D=-i\partial$ convention are translated by $\xi=-p$ to the site’s positive forward phase.
• Lawrence C. Evans (2010), Partial Differential Equations, second edition, Chapters 2, 5–7. This is the structural reference for model elliptic, hyperbolic, and evolution equations, Sobolev spaces, weak formulations, energy methods, and well-posedness.
• Daniel Grieser (2004), Notes on Heat Kernel Asymptotics, PDF, Theorem 1.1 and §3. This is the structural source for the smooth compact short-time expansion, the boundary layer, and boundary half-powers.
• Gerd Grubb (1996), Functional Calculus of Pseudodifferential Boundary Problems, Chapter I, PDF, §§1.4–1.7. These sections establish boundary realizations, the Shapiro–Lopatinski condition, parameter-ellipticity, adjoints, and semiboundedness on compact manifolds with boundary.
• John K. Hunter (2014), Notes on Partial Differential Equations, PDF, §§2.5–2.7, 4.8–4.10, 5.1–5.4, and 7.1–7.5. This is the teaching source for Green identities, fundamental solutions, compact-resolvent spectral theory, heat evolution, and weak hyperbolic solutions.
• Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, PDF, Chapters 3 and 5, American Mathematical Society, 2012. This is the teaching and structural source for evolution matrices, variation of constants, Wronskians, and regular Sturm–Liouville theory.