Pseudo-differential operator

Key takeaways

Pseudo-differential operator belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pseudo-differential operator to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pseudo-differential operator from memory before moving on to harder problems.

Reference excerpt

In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space.

History The study of pseudo-differential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza. They played an influential role in the second proof of the Atiyah–Singer index theorem via K-theory. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators.

Motivation

Linear differential operators with constant coefficients Consider a linear differential operator with constant coefficients,

P ( D ) := ∑ α a α D α {\displaystyle P(D):=\sum _{\alpha }a_{\alpha }\,D^{\alpha }}

which acts on smooth functions u {\displaystyle u} with compact support in Rn. This operator can be written as a composition of a Fourier transform, a simple multiplication by the polynomial function (called the symbol)

P ( ξ ) = ∑ α a α ξ α , {\displaystyle P(\xi )=\sum _{\alpha }a_{\alpha }\,\xi ^{\alpha },}

and an inverse Fourier transform, in the form:

Here, α = ( α 1 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})} is a multi-index, a α {\displaystyle a_{\alpha }} are complex numbers, and

D α = ( − i ∂ 1 ) α 1 ⋯ ( − i ∂ n ) α n {\displaystyle D^{\alpha }=(-i\partial _{1})^{\alpha _{1}}\cdots (-i\partial _{n})^{\alpha _{n}}}

is an iterated partial derivative, where ∂j means differentiation with respect to the j-th variable. We introduce the constants − i {\displaystyle -i} to facilitate the calculation of Fourier transforms.

Derivation of formula (1) The Fourier transform of a smooth function u, compactly supported in Rn, is

u ^ ( ξ ) := ∫ e − i y ξ u ( y ) d y {\displaystyle {\hat {u}}(\xi ):=\int e^{-iy\xi }u(y)\,dy}

and Fourier's inversion formula gives

u ( x ) = 1 ( 2 π ) n ∫ e i x ξ u ^ ( ξ ) d ξ = 1 ( 2 π ) n ∬ e i ( x − y ) ξ u ( y ) d y d ξ {\displaystyle u(x)={\frac {1}{(2\pi )^{n}}}\int e^{ix\xi }{\hat {u}}(\xi )d\xi ={\frac {1}{(2\pi )^{n}}}\iint e^{i(x-y)\xi }u(y)\,dy\,d\xi }

By applying P(D) to this representation of u and using

P ( D x ) e i ( x − y ) ξ = e i ( x − y ) ξ P ( ξ ) {\displaystyle P(D_{x})\,e^{i(x-y)\xi }=e^{i(x-y)\xi }\,P(\xi )}

one obtains formula (1).

Representation of solutions to partial differential equations To solve the partial differential equation

P ( D ) u = f {\displaystyle P(D)\,u=f}

we (formally) apply the Fourier transform on both sides and obtain the algebraic equation

P ( ξ ) u ^ ( ξ ) = f ^ ( ξ ) . {\displaystyle P(\xi )\,{\hat {u}}(\xi )={\hat {f}}(\xi ).}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Pseudo-differential operator

Start with the simplest possible case. Write down what Pseudo-differential operator claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pseudo-differential operator before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pseudo-differential operator ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pseudo-differential operator

In research: Pseudo-differential operator appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pseudo-differential operator in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pseudo-differential operator is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential operators, Functional analysis, Generalized functions, so understanding it makes those chapters shorter.
In everyday life: Look for Pseudo-differential operator outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Pseudo-differential operator in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pseudo-differential operator means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pseudo-differential operator out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pseudo-differential operator in simple terms?

In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-d…

Why does Pseudo-differential operator matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pseudo-differential operator?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pseudo-differential operator.