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Differential operator

A differential operator is a mathematical operator that applies differentiation to functions, typically represented by the symbol D = \frac{d}{dx} for the first derivative in one variable, and extended to higher-order derivatives via powers like D^n = \frac{d^n}{dx^n}.^[1] These operators are linear, satisfying D(af + bg) = aD(f) + bD(g) for constants a, b and functions f, g, and can be combined into polynomials such as L = a_n D^n + \cdots + a_1 D + a_0, where the coefficients a_k may be functions of the independent variable.^[2] In multiple variables, they generalize to partial differential operators, like the Laplacian \Delta = \sum \frac{\partial^2}{\partial x_i^2}, which measures the divergence of the gradient.^[1] Differential operators form the foundation of the theory of differential equations, where equations of the form L(u) = f describe how functions evolve under differentiation, enabling the modeling of dynamic systems.^[3] For instance, linear constant-coefficient operators facilitate the solution of ordinary differential equations by factoring into characteristic equations, revealing exponential solutions.^[2] In partial differential equations, they underpin fundamental laws in physics, such as the heat equation \frac{\partial u}{\partial t} = k \Delta u for diffusion processes and the wave equation \frac{\partial^2 u}{\partial t^2} = c^2 \Delta u for vibrations.^[4] Applications extend to engineering and mathematical physics, including electromagnetic fields via Maxwell's equations and fluid dynamics through the Navier-Stokes equations, where these operators capture spatial and temporal changes.^[5]^[6] Advanced concepts, such as pseudodifferential operators, arise in microlocal analysis to handle singularities and wavefronts in solutions.^[7]

Basic Concepts

Definition

A differential operator is a linear map D: C^\infty(\Omega) \to C^\infty(\Omega), where \Omega \subset \mathbb{R}^n is an open set and C^\infty(\Omega) denotes the vector space of smooth real-valued functions on \Omega, that satisfies a generalized Leibniz rule characterizing its finite order.^[8] Specifically, D is a differential operator of order at most k if, for every smooth function g \in C^\infty(\Omega), the commutator [D, m_g]: m \mapsto D(g m) - g D(m) (where m_g denotes multiplication by g) is a differential operator of order at most k-1, with order 0 operators being precisely the continuous linear maps (i.e., multiplication operators).^[8] The order of D is the minimal such nonnegative integer k for which the (k+1)-fold iterated commutator with multiplication operators vanishes identically.^[8] This inductive definition via commutators ensures that D locally behaves like a finite combination of partial derivatives, up to smooth multiplication factors. In local coordinates x = (x_1, \dots, x_n) on \Omega, any differential operator D of order at most k admits the explicit expression

D = \sum_{|\alpha| \leq k} a_\alpha(x) \partial^\alpha,

where \alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n is a multi-index with |\alpha| = \alpha_1 + \dots + \alpha_n, \partial^\alpha = \partial_{x_1}^{\alpha_1} \cdots \partial_{x_n}^{\alpha_n} denotes the corresponding partial derivative operator of order |\alpha|, and the coefficients a_\alpha: \Omega \to \mathbb{R} are smooth functions.^[9] The highest-order terms (with |\alpha| = k) determine the principal symbol of D, which plays a key role in analyzing its global properties. More generally, differential operators extend to smooth manifolds M of dimension n, acting as linear maps D: C^\infty(M) \to C^\infty(M) that are locally of the above form in any coordinate chart.^[9] On a manifold, the order k is independent of the choice of coordinates, preserving the commutator characterization relative to multiplication by smooth functions on M.^[8] This framework captures extensions of classical differentiation while ensuring consistency across overlapping charts via the partition of unity theorem.

Historical Development

The concept of differential operators traces its origins to the 18th century, amid the development of calculus, particularly in the calculus of variations. Leonhard Euler and Joseph-Louis Lagrange began treating higher-order derivatives as successive applications of a basic derivative operation acting on functions, facilitating the analysis of variational problems. Euler's foundational contributions in the 1740s, developed further through collaboration with Lagrange in the 1760s, marked an early recognition of derivatives as operator-like entities that could be composed and applied systematically to functionals.^[10] This intuitive approach gained formal structure in the early 19th century with Louis François Antoine Arbogast's introduction of the standalone differential operator notation D, which separated operational symbols from quantities and enabled algebraic manipulation of derivatives. Published in 1800 in Du calcul des dérivations et ses usages dans la théorie des suites et dans la géométrie, Arbogast's work represented a pivotal conceptual shift, allowing differential operators to be viewed independently of specific functions. Subsequent 19th-century advancements by Augustin-Louis Cauchy and Karl Weierstrass further embedded differential operators within the rigorous theory of partial differential equations (PDEs). Cauchy's 1827 analysis of the Cauchy-Riemann equations exemplified first-order differential operators in complex analysis, while his 1840 power series methods for nonlinear PDE initial value problems highlighted their role in solution existence and uniqueness. Weierstrass's mid-1870s emphasis on analytical rigor critiqued earlier informal approaches, influencing the study of elliptic operators and boundary value problems in PDEs.^[11]^[12] In the early 20th century, Élie Cartan's development of exterior differential systems from 1899 onward provided a geometric framework for higher-order differential operators, integrating them with Lie groups and moving frames for solving systems of PDEs. The mid-20th century brought abstract generalizations: Laurent Schwartz's 1950-1951 theory of distributions extended differential operators to act on generalized functions, enabling solutions to PDEs beyond classical smoothness. Pseudodifferential operators, building on singular integral techniques, were advanced by Alberto Calderón in 1959 to address Cauchy problems for broad classes of PDEs. Algebraically, the ring of differential operators on smooth manifolds was formalized by Alexander Grothendieck in the 1960s, revealing deep ties to sheaf theory and D-modules. During the 1940s-1950s, connections to Lie algebras emerged, with the ring of constant-coefficient differential operators identified as the universal enveloping algebra of the translation Lie algebra, influencing representation theory and quantization. Quantum mechanics profoundly shaped this evolution, as Erwin Schrödinger in 1926 introduced differential operators like the momentum operator -i\hbar \frac{d}{dx} to represent observables in wave mechanics, bridging classical analysis with operator algebras.^[12]^[13]^[14]

Examples and Notation

Examples

A fundamental example of a first-order differential operator in one variable is the differentiation operator D = \frac{d}{dx}, which maps a smooth function f(x) to its derivative Df = f'(x). This operator has order 1, as its principal symbol is the nonzero linear function \xi \mapsto i\xi in the cotangent variable \xi. It satisfies the Leibniz rule for derivations: for smooth functions f and g, D(fg) = f \, Dg + g \, Df = f g' + g f'.^[15] A typical second-order differential operator in one variable is P = \frac{d^2}{dx^2} + x \frac{d}{dx}, which arises in contexts such as the Airy differential equation. Its order is 2, determined by the highest-order term \frac{d^2}{dx^2}, with principal symbol \xi \mapsto -\xi^2. To verify it qualifies as a differential operator of order at most 2, consider its action under multiplication: for a smooth function f and variable h, compute P(f h) - f \, P h = f'' h + 2 f' h' + x f' h. This remainder equals f'' \cdot h + (2 f' + x f') \cdot h', which is a first-order differential operator in h (order reduced by 1), confirming the Leibniz condition recursively.^[15] In multiple variables, the divergence operator \operatorname{div} = \sum_{i=1}^n \frac{\partial}{\partial x_i} acts on a vector field V = (V_1, \dots, V_n) by \operatorname{div} V = \sum_{i=1}^n \frac{\partial V_i}{\partial x_i}. This is a first-order differential operator, with principal symbol \xi \mapsto i \sum_{i=1}^n \xi_i. It obeys the multivariable Leibniz rule: for a vector field V and scalar function f, \operatorname{div}(f V) = f \, \operatorname{div} V + \nabla f \cdot V = f \sum_{i=1}^n \frac{\partial V_i}{\partial x_i} + \sum_{i=1}^n \frac{\partial f}{\partial x_i} V_i. The Laplacian \Delta = \sum_{i=1}^n \frac{\partial^2}{\partial x_i^2} is a second-order operator on scalar functions, with principal symbol \xi \mapsto -\lVert \xi \rVert^2, and satisfies the corresponding higher-order Leibniz condition, such as \Delta(f h) - f \, \Delta h = 2 \sum_{i=1}^n \frac{\partial f}{\partial x_i} \frac{\partial h}{\partial x_i} + h \Delta f, where the remainder is order at most 1 in h.^[15] On Riemannian manifolds, the covariant derivative \nabla provides a first-order differential operator that generalizes partial differentiation to tensor fields while respecting the manifold's geometry. For a vector field Y along a curve with tangent X, \nabla_X Y measures the rate of change of Y parallel to the connection, satisfying \nabla_X (f Y) = f \nabla_X Y + (X f) Y as the Leibniz rule. Its order is 1, with the principal symbol determined by the metric and connection form.^[16] Similarly, the Dirac operator D on a spinor bundle over a Riemannian manifold is a first-order elliptic differential operator, locally expressed as D = \sum_{j=1}^n e_j \cdot \nabla_{e_j} for an orthonormal frame \{e_j\}, where \cdot denotes Clifford multiplication. As a first-order differential operator, it satisfies the Leibniz rule D(f s) = f D s + c(df) s for a smooth function f and spinor section s, where c denotes Clifford multiplication. Its principal symbol is \sigma_D(X) = i \sum_{j=1}^n X(e_j) e_j \cdot, which is invertible for X \neq 0.^[17] An example of a differential operator with variable coefficients and mixed derivatives is the heat equation operator L = \frac{\partial}{\partial t} - \Delta, acting on functions u(t, x) in \mathbb{R} \times \mathbb{R}^n. This has order 2, dominated by the second-order spatial Laplacian \Delta, with principal symbol i \tau + \lVert \xi \rVert^2 in variables (\tau, \xi). It satisfies the Leibniz condition for order 2; for instance, in one spatial dimension, L(f u) = \partial_t (f u) - \partial_{xx} (f u) = (\partial_t f) u + f \partial_t u - [f_{xx} u + 2 f_x u_x + f u_{xx}], and f L u = f (\partial_t u - u_{xx}), so the difference is (\partial_t f) u - f_{xx} u - 2 f_x u_x, which rearranges to multiplication by (\partial_t f - f_{xx}) times u plus multiplication by -2 f_x times u_x, a first-order operator in u.^[18]

Notations

In the univariate case, the differential operator corresponding to the first derivative is commonly denoted by D, representing \frac{d}{dx}, with higher powers D^k indicating the k-th derivative \frac{d^k}{dx^k}.^[19] In the multivariate setting, partial derivatives with respect to variables x_1, \dots, x_n are denoted by \partial_i = \frac{\partial}{\partial x_i} for i = 1, \dots, n.^[20] To compactly express higher-order partial derivatives, multi-index notation is standard: a multi-index \alpha = (\alpha_1, \dots, \alpha_n) is an n-tuple of nonnegative integers, with |\alpha| = \sum_{i=1}^n \alpha_i denoting its order, and \partial^\alpha = \prod_{i=1}^n \partial_i^{\alpha_i} representing the corresponding mixed partial derivative.^[20] This notation facilitates the summation over all partial derivatives of order at most k, as in \sum_{|\alpha| \leq k} a_\alpha(x) \partial^\alpha f(x).^[20] Polynomial differential operators are often symbolized as P(x, D), where D = (\partial_1, \dots, \partial_n) and P is a polynomial in the variables x and the formal symbols D_j, such as P(x, D) = \sum_{|\alpha| \leq m} a_\alpha(x) D^\alpha with D^\alpha = (-i)^{|\alpha|} \partial^\alpha in some conventions to align with Fourier analysis.^[21] On a smooth manifold M, the space of differential operators of order at most k acting on smooth sections of a vector bundle is denoted by \mathrm{Diff}^k(M), forming a filtered algebra under composition.^[22] Notational conventions vary between mathematics and physics: mathematicians typically use italicized or script \mathcal{D} for formal differential operators and emphasize formal adjoints, while physicists often employ boldface or upright \mathbf{D} and prioritize Hermitian adjoints in Hilbert space contexts.^[23] For instance, the Laplacian operator may appear as \Delta = \sum_i \partial_i^2 in mathematical texts but as -\hbar^2 \nabla^2 in quantum mechanics, highlighting domain-specific adjustments.^[19] Composition of differential operators can be left or right ordered, affecting the symbol in quantization schemes; in Weyl quantization, the symbol of the product Op(a) Op(b) corresponds to a symmetric (Weyl) ordering where multiplication operators act midway between left and right derivatives, given by the oscillatory integral formula for the composed symbol.^[23]

Fundamental Properties

General Properties

Differential operators are linear maps, meaning that for a differential operator P of order m, and functions u, v and scalars a, b, P(au + bv) = a P u + b P v.^[24] In appropriate function spaces, such as Sobolev spaces H^s(\mathbb{R}^n), these operators are continuous: a differential operator P(D) maps H^{s+m}_{\mathrm{loc}}(\Omega) continuously to H^s_{\mathrm{loc}}(\Omega) for all s \in \mathbb{R}.^[24] This continuity holds in the topology induced by the Sobolev norms, ensuring well-defined behavior on spaces of functions with controlled derivatives.^[25] The composition of two differential operators exhibits a Leibniz-type structure. If P has order k and Q has order m, then P \circ Q has order k + m, and the leading terms in the composition arise from the product of the leading coefficients via a generalized Leibniz rule.^[24] Specifically, for operators P = \sum_{|\alpha| \leq k} p_\alpha(x) \partial^\alpha and Q = \sum_{|\beta| \leq m} q_\beta(x) \partial^\beta, the highest-order part is simply the product of the principal parts.^[26] Commutators with multiplication operators further illustrate this: for a first-order operator D = \partial_j and smooth function f, [D, f] g = D(f g) - f D g = (\partial_j f) g, which is a zero-order operator, and in general, [D, f] reduces the order by at least one.^[24] Each differential operator P of order k has a principal symbol \sigma_k(P)(x, \xi) = \sum_{|\alpha| = k} a_\alpha(x) (i \xi)^\alpha, a homogeneous polynomial of degree k in the cotangent variable \xi.^[24] This symbol captures the highest-order behavior and is independent of lower-order terms. An operator is elliptic if its principal symbol satisfies |\sigma_k(P)(x, \xi)| \geq c |\xi|^k for some c > 0 and all \xi \neq 0, ensuring the operator is "invertible" in the high-frequency regime and leading to improved regularity properties for solutions.^[24] For systems, ellipticity requires the symbol matrix to be invertible for \xi \neq 0.^[25]

Fourier Interpretation

The Fourier transform provides a powerful interpretation of differential operators by transforming them into multiplication operators in the frequency domain. For a function u on \mathbb{R}^n, the Fourier transform \hat{u}(\xi) = \int_{\mathbb{R}^n} u(x) e^{-i x \cdot \xi} \, dx (up to normalization constants) converts partial derivatives into multiplications: the operator \partial_j acts as \widehat{\partial_j u}(\xi) = i \xi_j \hat{u}(\xi). More generally, for a multi-index \alpha, \partial^\alpha corresponds to (i \xi)^\alpha, so a linear differential operator P = \sum_{|\alpha| \leq m} a_\alpha(x) \partial^\alpha with smooth coefficients a_\alpha has symbol \sigma_P(x, \xi) = \sum_{|\alpha| \leq m} a_\alpha(x) (i \xi)^\alpha, a polynomial in \xi of degree at most m. For constant coefficients, \widehat{P u}(\xi) = \sigma_P(\xi) \hat{u}(\xi). For variable coefficients, P is a pseudodifferential operator whose action involves an oscillatory integral with this symbol, but \widehat{P u}(\xi) is not pointwise multiplication by \sigma_P(x, \xi) \hat{u}(\xi).^[27]^[28]^[29] A concrete example illustrates this correspondence: consider the first-order operator D = -i \frac{d}{dx} on \mathbb{R}, which is the momentum operator in quantum mechanics. Its Fourier transform yields \hat{D} f(\xi) = \xi \hat{f}(\xi), directly associating the operator with the frequency variable \xi as a multiplier. This multiplier property extends to higher-order operators, such as the Laplacian \Delta = \sum_{j=1}^n \partial_j^2, where \widehat{\Delta u}(\xi) = -|\xi|^2 \hat{u}(\xi), facilitating the solution of elliptic equations like \Delta u = f via \hat{u}(\xi) = -\hat{f}(\xi) / |\xi|^2 for \xi \neq 0. Such transformations underscore the role of differential operators as special cases of pseudodifferential operators, where the symbol is precisely a polynomial in \xi.^[27] Beyond basic multiplication, the Fourier interpretation connects to the propagation of singularities in solutions to partial differential equations. Singularities in the wavefront set of a distribution propagate along bicharacteristic curves defined by the Hamilton flow of the principal symbol \sigma_m(P)(x, \xi), the homogeneous leading term of degree m. This phenomenon is analyzed using Fourier integral operators, which generalize pseudodifferential operators to handle phase shifts and oscillatory integrals, ensuring that singularities neither appear nor disappear except along these flows for hyperbolic or properly supported operators. The propagation theorem, established through microlocal analysis, applies to solutions of P u = f, where the wavefront set of u is contained in that of f union the flow-out from characteristic sets.^[30] Finally, the Fourier framework links differential operators to symbol quantization in Weyl calculus, a symmetric quantization scheme where the operator associated to a symbol a(x, \xi) is \mathrm{Op}_w(a) f(x) = (2\pi)^{-n} \iint e^{i (x-y) \cdot \eta} a\left(\frac{x+y}{2}, \xi\right) f(y) \, dy \, d\eta with \eta = \xi adjusted via oscillatory integrals. For polynomial symbols of differential operators, Weyl quantization coincides with the standard left or right quantizations due to the exact polynomial structure, providing a bridge to semiclassical analysis and deformation quantization in phase space. This connection preserves the total symbol and enables precise control over operator composition via the Moyal product.^[31]^[29]

Adjoint Operators

Formal Adjoint in One Variable

In the context of linear differential operators acting on functions in one variable, the formal adjoint D^* of an operator D is defined such that for suitable test functions f and g with compact support, the integration by parts formula holds:

\int_{-\infty}^{\infty} (D f) g \, dx = \int_{-\infty}^{\infty} f (D^* g) \, dx,

where boundary terms vanish due to the compact support assumption.^[3] This definition ensures that the adjoint captures the duality between the operator and its action under the L^2 inner product, up to boundary contributions that are controlled in appropriate function spaces.^[3] For a polynomial differential operator D = \sum_{k=0}^m a_k(x) \frac{d^k}{dx^k} with smooth coefficients a_k(x), the explicit form of the formal adjoint is

D^* g = \sum_{k=0}^m (-1)^k \frac{d^k}{dx^k} \left( a_k(x) g \right).

This formula arises from applying the product rule (Leibniz rule) repeatedly during integration by parts to transfer all derivatives from f to g.^[3] The derivation begins with the first-order case and proceeds inductively: for the differentiation operator \frac{d}{dx}, integration by parts gives \int (f') g \, dx = -\int f g' \, dx, so \left( \frac{d}{dx} \right)^* = -\frac{d}{dx}.^[3] For higher orders, the Leibniz rule for adjoints follows: if D = P \frac{d^k}{dx^k} with multiplication by P(x), then (D)^* = (-1)^k \frac{d^k}{dx^k} (P \cdot), and the full operator sums these terms.^[3] A key example is the first-order operator, where D = \frac{d}{dx} yields D^* = -\frac{d}{dx}, as noted above. For a second-order operator D = \frac{d^2}{dx^2} + b(x) \frac{d}{dx} + c(x), repeated integration by parts produces

D^* = \frac{d^2}{dx^2} - b \frac{d}{dx} + (c - b'),

where b' denotes the derivative of b with respect to x.^[32] This reflects the sign flip for odd-order terms and the adjustment for variable coefficients via the product rule. An operator D is formally self-adjoint if D = D^*, a condition that simplifies the analysis of symmetric problems in partial differential equations and ensures real eigenvalues under suitable boundary conditions.^[3] For instance, the pure second derivative \frac{d^2}{dx^2} satisfies this property, as its adjoint is itself.^[3]

Formal Adjoint in Several Variables

In several variables, the formal adjoint of a linear partial differential operator extends the concept from the one-dimensional case by incorporating the multivariable integration by parts formula, often derived via the divergence theorem. For a domain \Omega \subset \mathbb{R}^n with smooth boundary, consider smooth functions f, g with appropriate support or boundary conditions such that boundary integrals vanish. The formal adjoint D^* of a linear partial differential operator D is defined by the relation

\int_\Omega (D f) g \, dx = \int_\Omega f (D^* g) \, dx,

where the integrals are taken with respect to the Lebesgue measure on \Omega. This definition ensures that D^* is the unique differential operator satisfying the bilinear pairing identity for test functions, ignoring boundary terms in the formal sense.^[3] For a general linear partial differential operator of order at most m,

D f = \sum_{|\alpha| \leq m} a_\alpha(x) \partial^\alpha f,

where \alpha = (\alpha_1, \dots, \alpha_n) is a multi-index with |\alpha| = \sum_{i=1}^n \alpha_i, \partial^\alpha = \prod_{i=1}^n \frac{\partial^{\alpha_i}}{\partial x_i^{\alpha_i}}, and the coefficients a_\alpha(x) are smooth functions on \Omega, the formal adjoint is given by

D^* g = \sum_{|\alpha| \leq m} (-1)^{|\alpha|} \partial^\alpha (a_\alpha(x) g).

This expression preserves the order m of the operator and transforms the leading terms accordingly.^[3] The formula for D^* arises from repeated applications of the multivariable integration by parts rule, which leverages the product rule for derivatives and the divergence theorem. For a single first-order partial derivative, integration by parts yields

\int_\Omega (\partial_i f) g \, dx = -\int_\Omega f (\partial_i g) \, dx + \int_{\partial \Omega} f g n_i \, dS,

where n_i is the i-th component of the outward unit normal, showing that the formal adjoint of \partial_i is -\partial_i when boundary terms are neglected. For a term of higher order \partial^\alpha f, integration by parts is applied successively to each factor \partial_i^{\alpha_i}, introducing a factor of (-1)^{\alpha_i} per variable and pulling the coefficient a_\alpha inside the derivatives via the Leibniz rule: \partial^\alpha (a_\alpha g) = \sum_{\beta \leq \alpha} \binom{\alpha}{\beta} (\partial^\beta a_\alpha) (\partial^{\alpha - \beta} g). The overall sign (-1)^{|\alpha|} accounts for the total number of integrations by parts across all variables.^[33] A fundamental illustration is the divergence operator \operatorname{div}: C^\infty(\Omega; \mathbb{R}^n) \to C^\infty(\Omega), defined by \operatorname{div} V = \sum_{i=1}^n \partial_i V_i for a vector field V = (V_1, \dots, V_n). Its formal adjoint is the negative gradient (\operatorname{div})^* \phi = -\nabla \phi = (-\partial_1 \phi, \dots, -\partial_n \phi), satisfying

\int_\Omega (\operatorname{div} V) \phi \, dx = -\sum_{i=1}^n \int_\Omega V_i (\partial_i \phi) \, dx + \int_{\partial \Omega} \phi (V \cdot n) \, dS.

This relation follows directly from applying integration by parts to each component, confirming the structure for vector-valued operators.^[34] The distinction between the formal adjoint and the L^2 adjoint lies in their settings: the formal adjoint D^* is a purely differential expression without specified domain, applicable to smooth functions, whereas the L^2 adjoint is the densely defined unbounded operator on the Hilbert space L^2(\Omega) whose graph ensures the pairing holds for functions in its maximal domain, typically requiring D^* g \in L^2(\Omega). Under assumptions that smooth compactly supported functions are dense in L^2(\Omega) and the coefficients a_\alpha are sufficiently regular (e.g., bounded and continuous), the L^2 adjoint coincides with the formal adjoint on this dense subspace, enabling extension by continuity.^[35]

Examples of Adjoints

A classic example in one variable is the Euler operator E = x \frac{d}{dx}, acting on smooth functions on (0, \infty). Its formal adjoint with respect to the L^2 inner product \langle f, g \rangle = \int_0^\infty f(x) g(x) \, dx is E^* = -x \frac{d}{dx} - 1. To verify this, consider \langle E f, g \rangle = \int_0^\infty g(x) \left( x f'(x) \right) dx. Integration by parts yields [g(x) x f(x)]_0^\infty - \int_0^\infty f(x) \frac{d}{dx} \left( x g(x) \right) dx = B - \int_0^\infty f(x) \left( x g'(x) + g(x) \right) dx, where B denotes boundary terms that vanish for suitable test functions with compact support. Thus, \langle E f, g \rangle = B - \langle f, x g' + g \rangle, so E^* g = - x g' - g = -x \frac{d}{dx} g - g.^[32] In several variables, the gradient operator \nabla: C_c^\infty(\mathbb{R}^n) \to C_c^\infty(\mathbb{R}^n, \mathbb{R}^n) has formal adjoint \nabla^* = -\operatorname{div}, where \operatorname{div} \mathbf{v} = \sum_{i=1}^n \frac{\partial v_i}{\partial x_i}. This follows from integration by parts: \langle \nabla u, \mathbf{v} \rangle = \int_{\mathbb{R}^n} \nabla u \cdot \mathbf{v} \, dx = -\int_{\mathbb{R}^n} u \operatorname{div} \mathbf{v} \, dx + B, with boundary terms B=0 for compactly supported functions. Consequently, the Laplacian \Delta = \operatorname{div} \circ \nabla is formally self-adjoint, \Delta^* = \Delta, as \langle \Delta u, v \rangle = \langle \nabla u, \nabla v \rangle = \langle u, \Delta v \rangle by applying the adjoint twice.^[34] In physics, the momentum operator p = -i \frac{d}{dx} on L^2(\mathbb{R}), restricted to the dense domain C_c^\infty(\mathbb{R}) of smooth compactly supported functions, is essentially self-adjoint. This means its closure is self-adjoint, ensuring a unique self-adjoint extension used in quantum mechanics for the free particle Hamiltonian. Essential self-adjointness follows from the fact that the deficiency indices are zero, verified via solutions to p^* \psi = \pm i \psi, which lie outside L^2(\mathbb{R}).^[36] For a variable coefficient operator in \mathbb{R}^2, consider P = \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}. The formal adjoint is P^* = -\frac{\partial}{\partial x} - x \frac{\partial}{\partial y} - 1. Verification proceeds by integration by parts in the inner product \langle P u, v \rangle = \int_{\mathbb{R}^2} v \left( u_x + x u_y \right) dx dy = B - \int_{\mathbb{R}^2} u \left( v_x + (x v)_y \right) dx dy = B - \int_{\mathbb{R}^2} u \left( v_x + x v_y + v \right) dx dy, where the extra -1 arises from \frac{\partial x}{\partial y} = 1. Thus, P^* v = - v_x - x v_y - v.^[32] The formal adjoint ignores boundary terms and is defined locally on smooth functions, but the actual adjoint in a Hilbert space like L^2(\Omega) depends on the domain, incorporating boundary conditions to ensure \langle L u, v \rangle = \langle u, [L^*](/page/Adjoint) v \rangle without boundary contributions. For instance, on a bounded interval [a,b], the operator \frac{d}{dx} requires boundary conditions (e.g., Dirichlet u(a)=u(b)=0) for self-adjointness, altering the domain of the adjoint relative to the formal version.^[32]

Algebraic Structures

Ring of Univariate Polynomial Differential Operators

The ring of univariate polynomial differential operators, often denoted \operatorname{Diff}(R) for a commutative ring R, is the associative algebra generated by the polynomials R and the differentiation operator \frac{d}{dx}, where elements are finite sums \sum_{i=0}^n f_i \left(\frac{d}{dx}\right)^i with f_i \in R.^[37] The multiplication in \operatorname{Diff}(R) is defined via operator composition, incorporating the Leibniz rule: for f, g \in R, (f \frac{d}{dx})(g) = f \frac{d g}{dx} + f' g, where f' denotes the formal derivative of f with respect to x.^[37] This structure ensures that \operatorname{Diff}(R) acts naturally on R as a ring of endomorphisms. \operatorname{Diff}(R) admits an Ore extension presentation: \operatorname{Diff}(R) \cong R[\partial; \delta], where \delta is the standard derivation on R given by \delta(x) = 1 and extended R-linearly, with the multiplication rule \partial \cdot a = \sigma(a) \partial + \delta(a) for a \in R and \sigma = \mathrm{id}.^[38] This construction highlights the non-commutative nature of the ring, arising from the commutation relation [\frac{d}{dx}, x] = \frac{d}{dx} \cdot x - x \cdot \frac{d}{dx} = 1.^[37] The algebra \operatorname{Diff}(R) is the first Weyl algebra when R is a field of characteristic zero, generated by x and \partial (with \partial corresponding to \frac{d}{dx}) subject to the key relation \partial x - x \partial = 1.^[37] This relation generates the entire non-commutative structure, distinguishing the Weyl algebra from commutative polynomial rings. For R = \mathbb{C}, the Weyl algebra is simple, possessing no nontrivial two-sided ideals.^[39]

Ring of Multivariate Polynomial Differential Operators

The ring of multivariate polynomial differential operators, often denoted as \operatorname{Diff}(\mathbb{R}^n) or D(\mathbb{R}^n), is the noncommutative associative algebra over \mathbb{R} (or \mathbb{C}) generated by the coordinate multiplication operators x_1, \dots, x_n and the partial differentiation operators \partial_1 = \frac{\partial}{\partial x_1}, \dots, \partial_n = \frac{\partial}{\partial x_n}, subject to the commutation relations [\partial_i, x_j] = \delta_{ij}, [\partial_i, \partial_j] = 0, and [x_i, x_j] = 0 for all i, j = 1, \dots, n, where \delta_{ij} is the Kronecker delta.^[40] These relations ensure that the algebra faithfully represents the action of linear partial differential operators with polynomial coefficients on the space of smooth functions on \mathbb{R}^n.^[40] This algebra is known as the n-th Weyl algebra, denoted A_n, and can be formally constructed as the quotient

A_n = \mathbb{C}\langle x_1, \dots, x_n, \partial_1, \dots, \partial_n \rangle / I,

where I is the two-sided ideal generated by the specified commutators.^[40] The multivariate structure generalizes the univariate case, introducing additional generators and relations that capture interactions across multiple variables.^[40] The Weyl algebra A_n carries a natural filtration by operator order, where the j-th filtered component F^j A_n consists of elements of total degree at most j in the \partial_i's (with x_i's having order 0).^[40] The associated graded ring \operatorname{gr} A_n = \bigoplus_j F^j A_n / F^{j-1} A_n is isomorphic to the commutative polynomial ring \mathbb{C}[x_1, \dots, x_n, \partial_1, \dots, \partial_n] in $2n variables, reflecting the commutative approximation of the noncommutative structure.^[40] In the global setting, A_n acts on the entire space \mathbb{R}^n or \mathbb{C}^n, whereas local versions arise as stalks of the sheaf of differential operators on algebraic varieties, such as the sheaf \mathcal{D}_X over a smooth variety X.^[40] Over an algebraically closed field of characteristic zero, such as \mathbb{C}, the Weyl algebra A_n(\mathbb{C}) is simple, possessing no nontrivial two-sided ideals.^[40] A key application arises in quantum mechanics, where the commutation relations [\partial_i, x_j] = \delta_{ij} (up to scaling by i\hbar) encode the canonical commutation relations of the Heisenberg algebra, governing the position and momentum operators in n-dimensional phase space.^[41]

Coordinate-Independent Description

In the coordinate-independent setting, differential operators on a smooth manifold M act between smooth sections of vector bundles E \to M and F \to M. The space \mathrm{Diff}^k(E, F) consists of all linear maps P: C^\infty(M, E) \to C^\infty(M, F) of order at most k, defined such that for any point x \in M, there exists a neighborhood U of x where the iterated commutator [\cdots [P, m_{s_1}], m_{s_2}] \cdots, m_{s_{k+1}}] vanishes for all smooth sections s_1, \dots, s_{k+1} of E, with m_s denoting pointwise multiplication by the section s.^[42] This characterization ensures the operators satisfy a generalized Leibniz rule, extending the product rule to higher orders via tensor products of sections: for an order-k operator, P(f \cdot s) = \sum_{j=0}^k \binom{k}{j} (P_j f) \cdot \nabla^j s, where \nabla is a connection and P_j are lower-order terms, though the precise form depends on the bundle structure.^[42] Locally, in a trivialization of E and F over a chart (U, \phi) on M, such operators reduce to the classical coordinate form \sum_{|\alpha| \leq k} a_\alpha(x) \partial^\alpha, where a_\alpha are smooth coefficient sections and \partial^\alpha are partial derivatives.^[35] Globally, however, the coordinate-free description employs jet bundles J^k(E), which parametrize the k-th order jets of sections of E—equivalence classes of sections agreeing up to k-th order derivatives at a point—allowing differential operators to be viewed as morphisms between jet bundles and F.^[43] Covariant derivatives induced by a connection on E provide prototypical order-one differential operators in \mathrm{Diff}^1(E, E \otimes T^*M), mapping sections to covector-valued sections while preserving the Leibniz rule \nabla_X (f s) = (\nabla_X f) s + f \nabla_X s for vector fields X.^[44] Higher-order operators arise naturally from compositions of such covariant derivatives, generating the full space \mathrm{Diff}^*(E, F) in a manner independent of local coordinates.^[43] The collection of all differential operators \bigcup_k \mathrm{Diff}^k(E, F) forms a filtered ring under composition, with the filtration \mathrm{Diff}^k(E, F) \subseteq \mathrm{Diff}^{k+1}(E, F) preserved such that the product of order-k and order-l operators has order at most k+l.^[42] The associated graded ring is isomorphic to the ring of symbols via the principal symbol map \sigma_k: \mathrm{Diff}^k(E, F)/\mathrm{Diff}^{k-1}(E, F) \to \Gamma(S^k(T^*M) \otimes \mathrm{Hom}(E, F)), where S^k(T^*M) denotes symmetric k-th powers of the cotangent bundle, rendering the symbol sequence exact and facilitating algebraic analysis.^[45] A canonical example is the de Rham complex on M, a chain complex of differential forms \Omega^\bullet(M) where the exterior derivative d: \Omega^p(M) \to \Omega^{p+1}(M) serves as a first-order differential operator satisfying d^2 = 0 and the graded Leibniz rule d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\deg \alpha} \alpha \wedge d\beta.^[35]

Advanced Variants

Differential Operators of Infinite Order

Differential operators of infinite order generalize finite-order differential operators by allowing formal power series expansions in the differentiation operator, typically expressed in exponential form to ensure convergence on suitable function spaces. Specifically, such an operator D on functions f is defined as

Df(x) = \sum_{k=0}^\infty \frac{a_k}{k!} \partial_x^k f(x),

where the coefficients \{a_k\} form a sequence with positive radius of convergence, making the series converge for analytic functions or in appropriate topologies. This form arises naturally from composing the operator with the exponential generating function \phi(z) = \sum_{k=0}^\infty a_k \frac{z^k}{k!}, which is entire, ensuring the operator \phi(\partial_x) acts continuously on spaces beyond smooth functions.^[46]^[47] The operator satisfies a Leibniz rule derived from that of the derivative:

D(fg)(x) = \sum_{k=0}^\infty \frac{a_k}{k!} \sum_{j=0}^k \binom{k}{j} \partial_x^j f(x) \cdot \partial_x^{k-j} g(x).

Prominent examples include the translation operator e^{h \partial_x}, which shifts functions via e^{h \partial_x} f(x) = f(x + h) for h \in \mathbb{R}, and the heat semigroup operator e^{t \Delta} for t > 0, where \Delta is the Laplacian, generating solutions to the heat equation \partial_t u = \Delta u. These operators extend the finite-order case, where the series truncates, and are well-defined on entire functions of exponential type, preserving properties like reality of zeros under iteration.^[46]^[48] In terms of topology, infinite-order differential operators are continuous when acting on Gevrey classes \mathcal{G}^s, spaces of functions where higher derivatives satisfy |\partial^k f(x)| \leq C^{k+1} (k!)^s for some C > 0 and s \geq 1, with analytic functions corresponding to s=1. This continuity holds for symbols in Gevrey classes, ensuring boundedness in norms adapted to the growth of derivatives. The symbol of such an operator, obtained via Fourier transform, is an entire function in the frequency variable \xi, extending the principal symbol concept from finite-order operators to \phi(i\xi), where \phi is of exponential type.^[49]^[50]^[48] Applications of infinite-order differential operators appear prominently in solving partial differential equations (PDEs) through formal power series methods, where they facilitate the construction of fundamental solutions or semigroups for evolution equations. For instance, operators like e^{t \Delta} directly yield the Green's function for the heat equation, while more general \phi(\Delta_{\theta,\omega}) solve Cauchy problems for second-order PDEs with variable coefficients, converging in spaces of entire functions to provide asymptotic behaviors or zero distributions of solutions.^[46]^[48]

Bidifferential Operators

A bidifferential operator is a bilinear map B: C^\infty(M) \times C^\infty(M) \to C^\infty(M) that acts as a differential operator in each argument separately, generalizing the notion of bilinear forms to incorporate differentiation. Locally, on an open set in \mathbb{R}^n, a bidifferential operator of total order at most k takes the form

B(f,g) = \sum_{|\alpha| + |\beta| \leq k} a_{\alpha\beta}(x) \, \partial^\alpha f(x) \, \partial^\beta g(x),

where the coefficients a_{\alpha\beta} are smooth functions and \alpha, \beta are multi-indices.^[51] This expression ensures bilinearity and the property that, for fixed g, B(\cdot, g) is a differential operator of order at most k in the first variable, and analogously for fixed f.^[52] The total order of B is defined as the maximum of |\alpha| + |\beta| over all nonzero terms, measuring the highest combined degree of differentiation. Composition of bidifferential operators preserves this structure: if B has order k and C has order l, then C \circ B has order at most k + l, as derivatives compose additively.^[53] The operators satisfy the standard Leibniz rule in each argument separately, arising from the product rule for derivatives.^[52] Representative examples include the zeroth-order product operator B(f,g) = f g, which is simply multiplication, and the first-order operator B(f,g) = f \partial_x g in one dimension, combining multiplication in the first argument with differentiation in the second. In number theory, Rankin–Cohen brackets provide higher-order examples, defined for modular forms f_1, f_2 of weights \lambda', \lambda'' and order \ell by

R_{\lambda',\lambda''}^{\lambda' + \lambda'' + \ell}(f_1, f_2)(z) = \sum_{j=0}^\ell \binom{\lambda' + \lambda'' + \ell - 1 - j}{\ell - j} \frac{f_1^{(\ell - j)}(z) f_2^{(j)}(z)}{(2i\pi)^{\ell}},

a bidifferential operator of order \ell that is invariant under the modular group.^[54] In quantum field theory, Wick products, such as the normal-ordered bilinear form on fields \partial \phi \cdot \partial \psi, function as bidifferential operators by subtracting vacuum expectations to ensure proper renormalization. The formal adjoint B^* of a bidifferential operator B is defined such that integration by parts yields \int B(f,g) \, h = \int f \, B^*(g,h) (up to boundary terms), resulting in B^*(f,g) = \sum (-1)^{|\beta|} a_{\alpha\beta} \partial^\alpha g \, \partial^\beta f after transposing arguments and applying signs from the adjoint of each derivative (\partial^* = -\partial). This introduces sign changes depending on the order in the second argument.^[55] Bidifferential operators are closely related to tensor products of differential operators: the space of such operators of order at most k corresponds to elements in the tensor product \mathrm{Diff}^k(M) \otimes \mathrm{Diff}^k(M), where \mathrm{Diff}^k(M) is the module of differential operators of order \leq k on M, acting diagonally on pairs of functions via (P \otimes Q)(f \otimes g) = P(f) Q(g). This structure underlies their role in deformation quantization and representation theory.^[55]

Microdifferential Operators

Microdifferential operators arise in microlocal analysis as a refinement of differential operators, enabling precise localization of their action on the cotangent bundle T^*M of a smooth manifold M. These operators act on Lagrangian distributions, which are distributions associated to Lagrangian submanifolds of T^*(M \times M) and generalize smooth functions and their singularities in phase space. Formally, a microdifferential operator P of order m is defined via an oscillatory integral representation:

(Pu)(x) = \int e^{i\phi(x,y,\theta)} a(x,y,\theta) u(y) \, dy \, d\theta,

where \phi is a non-degenerate phase function whose graph is a Lagrangian submanifold \Lambda \subset T^*(M \times M), and the amplitude a belongs to the symbol class S^m(T^*M \times T^*M), consisting of smooth functions satisfying estimates |\partial^\alpha_x \partial^\beta_y \partial^\gamma_\theta a| \leq C (1+|\theta|)^{m - |\gamma|} for multi-indices \alpha, \beta, \gamma. This structure allows microdifferential operators to capture propagation phenomena in phase space, extending the classical Leibniz rule to infinite-order formal series while preserving algebraic properties like composition.^[56] A central concept in the theory is microlocal ellipticity, which occurs when the principal symbol p_m(x,y,\theta) \in S^m(T^*M \times T^*M) is invertible on \Lambda away from the characteristic variety \mathrm{Char}(P) = \{(x,y,\theta) \in \Lambda \mid p_m(x,y,\theta) = 0\}. Elliptic microdifferential operators propagate singularities along bicharacteristic strips in the cotangent bundle, ensuring that singularities of solutions to equations Pu = f follow the Hamiltonian flow of the principal symbol. This propagation of singularities is crucial for analyzing hyperbolic and elliptic partial differential equations, where the operator dictates how wavefronts evolve microlocally. For instance, in wave equations, microlocal ellipticity guarantees finite propagation speed, with singularities confined to conical neighborhoods of the light cone in phase space. Pseudodifferential operators provide a key example of microdifferential operators of order zero, where the phase function is the standard bilinear form \phi(x,y,\theta) = (x-y) \cdot \theta, and the Lagrangian \Lambda is the conormal bundle to the diagonal in M \times M. In this case, the full symbol admits an asymptotic expansion a(x,\theta) \sim \sum_{j=0}^\infty a_{m-j}(x,\theta) with a_{m-j} homogeneous of degree m-j in \theta, enabling the operator to be expressed as P = \sum_{k=0}^\infty \frac{(-i)^k}{k!} \partial_\theta^k a(x,\theta) D_x^k in local coordinates. This representation highlights their role in smoothing or amplifying singularities based on the symbol's behavior.^[57] The relation to wavefront sets underscores the microlocal nature of these operators: the wavefront set \mathrm{WF}(u) \subset T^*M \setminus 0 measures the singular directions of a distribution u, and a microdifferential operator P smooths singularities outside its characteristic set, meaning that if (x,\xi) \notin \mathrm{Char}(P), then u is smooth microlocally near (x,\xi) implies Pu is smooth there. More precisely, \mathrm{WF}(Pu) \cap (T^*M \setminus \mathrm{Char}(P)) \subset \mathrm{WF}(u) \cap (T^*M \setminus \mathrm{Char}(P)), ensuring that P does not introduce new singularities away from the characteristic variety. This property is foundational for proving local solvability and hypoellipticity in PDE theory. As an advanced generalization, Fourier integral operators extend microdifferential operators by allowing arbitrary clean canonical relations—Lagrangian immersions \Lambda \subset T^*M \times T^*N \setminus 0—rather than restricting to graph-like structures over the diagonal. These operators, also realized via oscillatory integrals with symbols in appropriate classes, model changes of variables and scattering processes, preserving microlocal ellipticity and wavefront set propagation in broader geometric settings.^[58]

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### Summary of arXiv:math/0311213
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Differential operators of infinite order with symbols that are infinitely differentiable functions (of Gevrey class) in some domain in are considered.
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### Summary of Bi-Differential Operators in Deformation Quantization from arXiv:math/0111094
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