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In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting {d=1} for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain {{\bf R}} as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.

In previous notes we have often performed various localisations in either physical space or Fourier space {{\bf R}}, for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions {{\mathcal S}({\bf R})}, the position operator {X: {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})} defined by

\displaystyle  (Xf)(x) := x f(x)

and the momentum operator {D: {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})}, defined by

\displaystyle  (Df)(x) := \frac{1}{2\pi i} \frac{d}{dx} f(x). \ \ \ \ \ (1)

(The terminology comes from quantum mechanics, where it is customary to also insert a small constant {h} on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit {h \rightarrow 0}, but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity

\displaystyle  \widehat{Df}(\xi) = \xi \hat f(\xi)

for any {\xi \in {\bf R}} and {f \in {\mathcal S}({\bf R})}. We observe that both operators {X,D} are formally self-adjoint in the sense that

\displaystyle  \langle Xf, g \rangle = \langle f, Xg \rangle; \quad \langle Df, g \rangle = \langle f, Dg \rangle

for all {f,g \in {\mathcal S}({\bf R})}, where we use the {L^2({\bf R})} Hermitian inner product

\displaystyle  \langle f, g\rangle := \int_{\bf R} f(x) \overline{g(x)}\ dx.

Clearly, for any polynomial {P(x)} of one real variable {x} (with complex coefficients), the operator {P(X): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})} is given by the spatial multiplier operator

\displaystyle  (P(X) f)(x) = P(x) f(x)

and similarly the operator {P(D): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})} is given by the Fourier multiplier operator

\displaystyle  \widehat{P(D) f}(\xi) = P(\xi) \hat f(\xi).

Inspired by this, if {m: {\bf R} \rightarrow {\bf C}} is any smooth function that obeys the derivative bounds

\displaystyle  \frac{d^j}{dx^j} m(x) \lesssim_{m,j} \langle x \rangle^{O_{m,j}(1)} \ \ \ \ \ (2)

for all {j \geq 0} and {x \in {\bf R}} (that is to say, all derivatives of {m} grow at most polynomially), then we can define the spatial multiplier operator {m(X): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})} by the formula

\displaystyle  (m(X) f)(x) := m(x) f(x);

one can easily verify from several applications of the Leibniz rule that {m(X)} maps Schwartz functions to Schwartz functions. We refer to {m(x)} as the symbol of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator {m(D)} associated to the symbol {m(\xi)} by the formula

\displaystyle  \widehat{m(D) f}(\xi) := m(\xi) \hat f(\xi).

For instance, any constant coefficient linear differential operators {\sum_{k=0}^n c_k \frac{d^k}{dx^k}} can be written in this notation as

\displaystyle \sum_{k=0}^n c_k \frac{d^k}{dx^k} =\sum_{k=0}^n c_k (2\pi i D)^k;

however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators {\langle D \rangle^s = (1- \frac{1}{4\pi^2} \frac{d^2}{dx^2})^{s/2}} for non-integer values of {s}, which is a Fourier multiplier operator with symbol {\langle \xi \rangle^s}. It is also very common to use spatial cutoffs {\psi(X)} and Fourier cutoffs {\psi(D)} for various bump functions {\psi} to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting {d>1}).

We observe that the maps {m \mapsto m(X)} and {m \mapsto m(D)} are ring homomorphisms, thus for instance

\displaystyle  (m_1 + m_2)(D) = m_1(D) + m_2(D)


\displaystyle  (m_1 m_2)(D) = m_1(D) m_2(D)

for any {m_1,m_2} obeying the derivative bounds (2); also {m(D)} is formally adjoint to {\overline{m(D)}} in the sense that

\displaystyle  \langle m(D) f, g \rangle = \langle f, \overline{m}(D) g \rangle

for {f,g \in {\mathcal S}({\bf R})}, and similarly for {m(X)} and {\overline{m}(X)}. One can interpret these facts as part of the functional calculus of the operators {X,D}, which can be interpreted as densely defined self-adjoint operators on {L^2({\bf R})}. However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.

In the field of PDE and ODE, it is also very common to study variable coefficient linear differential operators

\displaystyle  \sum_{k=0}^n c_k(x) \frac{d^k}{dx^k} \ \ \ \ \ (3)

where the {c_0,\dots,c_n} are now functions of the spatial variable {x} obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian {-\frac{d^2}{dx^2} + x^2}. One can rewrite this operator in our notation as

\displaystyle  \sum_{k=0}^n c_k(X) (2\pi i D)^k

and so it is natural to interpret this operator as a combination {a(X,D)} of both the position operator {X} and the momentum operator {D}, where the symbol {a: {\bf R} \times {\bf R} \rightarrow {\bf C}} this operator is the function

\displaystyle  a(x,\xi) := \sum_{k=0}^n c_k(x) (2\pi i \xi)^k. \ \ \ \ \ (4)

Indeed, from the Fourier inversion formula

\displaystyle  f(x) = \int_{\bf R} \hat f(\xi) e^{2\pi i x \xi}\ d\xi

for any {f \in {\mathcal S}({\bf R})} we have

\displaystyle  (2\pi i D)^k f(x) = \int_{\bf R} (2\pi i \xi)^k \hat f(\xi) e^{2\pi i x \xi}\ d\xi

and hence on multiplying by {c_k(x)} and summing we have

\displaystyle (\sum_{k=0}^n c_k(X) (2\pi i D)^k) f(x) = \int_{\bf R} a(x,\xi) \hat f(\xi) e^{2\pi i x \xi}\ d\xi.

Inspired by this, we can introduce the Kohn-Nirenberg quantisation by defining the operator {a(X,D) = a_{KN}(X,D): {\mathcal S}({\bf R}) \rightarrow {\mathcal S}({\bf R})} by the formula

\displaystyle  a(X,D) f(x) = \int_{\bf R} a(x,\xi) \hat f(\xi) e^{2\pi i x \xi}\ d\xi \ \ \ \ \ (5)

whenever {f \in {\mathcal S}({\bf R})} and {a: {\bf R} \times {\bf R} \rightarrow {\bf C}} is any smooth function obeying the derivative bounds

\displaystyle  \frac{\partial^j}{\partial x^j} \frac{\partial^l}{\partial \xi^l} a(x,\xi) \lesssim_{a,j,l} \langle x \rangle^{O_{a,j}(1)} \langle \xi \rangle^{O_{a,j,l}(1)} \ \ \ \ \ (6)

for all {j,l \geq 0} and {x \in {\bf R}} (note carefully that the exponent in {x} on the right-hand side is required to be uniform in {l}). This quantisation clearly generalises both the spatial multiplier operators {m(X)} and the Fourier multiplier operators {m(D)} defined earlier, which correspond to the cases when the symbol {a(x,\xi)} is a function of {x} only or {\xi} only respectively. Thus we have combined the physical space {{\bf R} = \{ x: x \in {\bf R}\}} and the frequency space {{\bf R} = \{ \xi: \xi \in {\bf R}\}} into a single domain, known as phase space {{\bf R} \times {\bf R} = \{ (x,\xi): x,\xi \in {\bf R} \}}. The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.

Exercise 1

  • (i) Show that for {a} obeying (6), that {a(X,D)} does indeed map {{\mathcal S}({\bf R})} to {{\mathcal S}({\bf R})}.
  • (ii) Show that the symbol {a} is uniquely determined by the operator {a(X,D)}. That is to say, if {a,b} are two functions obeying (6) with {a(X,D) f = b(X,D) f} for all {f \in {\mathcal S}({\bf R})}, then {a=b}. (Hint: apply {a(X,D)-b(X,D)} to a suitable truncation of a plane wave {x \mapsto e^{2\pi i x \xi}} and then take limits.)

In principle, the quantisations {a(X,D)} are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols {a} to operators {a(X,D)} is now no longer a ring homomorphism, in particular

\displaystyle  (a_1 a_2)(X,D) \neq a_1(X,D) a_2(X,D) \ \ \ \ \ (7)

in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as {X} and {D} does not necessarily commute. This lack of commutativity can be measured by introducing the commutator

\displaystyle  [A,B] := AB - BA

of two operators {A,B}, and noting from the product rule that

\displaystyle  [X,D] = -\frac{1}{2\pi i} \neq 0.

(In the language of Lie groups and Lie algebras, this tells us that {X,D} are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:

Exercise 2 (Heisenberg uncertainty principle) For any {x_0, \xi_0 \in {\bf R}} and {f \in \mathcal{S}({\bf R})}, show that

\displaystyle  \| (X-x_0) f \|_{L^2({\bf R})} \| (D-\xi_0) f\|_{L^2({\bf R})} \geq \frac{1}{4\pi} \|f\|_{L^2({\bf R})}^2.

(Hint: evaluate the expression {\langle [X-x_0, D - \xi_0] f, f \rangle} in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty {\Delta x} and the frequency uncertainty {\Delta \xi} of a function obey the Heisenberg uncertainty relation {\Delta x \Delta \xi \gtrsim 1}.

Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators {X,D} (and the various operators {a(X,D)} constructed from them) commute up to “lower order” errors. This can be formalised using the pseudodifferential calculus, which we give below the fold, in which we restrict the symbol {a} to certain “symbol classes” of various orders (which then restricts {a(X,D)} to be pseudodifferential operators of various orders), and obtains approximate identities such as

\displaystyle  (a_1 a_2)(X,D) \approx a_1(X,D) a_2(X,D)

where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions {f \in {\mathcal S}({\bf R})} as having some sort of “phase space portrait{\tilde f(x,\xi)} which somehow combines the physical space representation {x \mapsto f(x)} with its Fourier representation {\xi \mapsto f(\xi)}, and pseudodifferential operators {a(X,D)} behave approximately like “phase space multiplier operators” in this representation in the sense that

\displaystyle  \widetilde{a(X,D) f}(x,\xi) \approx a(x,\xi) \tilde f(x,\xi).

Unfortunately the uncertainty principle (or the non-commutativity of {X} and {D}) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait {\tilde f} of a function {f} precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.

To complement the pseudodifferential calculus we have the basic Calderón-Vaillancourt theorem, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on {L^p({\bf R})} for {1 < p < \infty}. The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of almost orthogonality; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.

Pseudodifferential operators (especially when generalised to higher dimensions {d \geq 1}) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait {\tilde f(x,\xi)} of a function by some multiplier {a(x,\xi)}, but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.

This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.

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The Schrödinger equation

\displaystyle  i \hbar \partial_t |\psi \rangle = H |\psi\rangle

is the fundamental equation of motion for (non-relativistic) quantum mechanics, modeling both one-particle systems and {N}-particle systems for {N>1}. Remarkably, despite being a linear equation, solutions {|\psi\rangle} to this equation can be governed by a non-linear equation in the large particle limit {N \rightarrow \infty}. In particular, when modeling a Bose-Einstein condensate with a suitably scaled interaction potential {V} in the large particle limit, the solution can be governed by the cubic nonlinear Schrödinger equation

\displaystyle  i \partial_t \phi = \Delta \phi + \lambda |\phi|^2 \phi. \ \ \ \ \ (1)

I recently attended a talk by Natasa Pavlovic on the rigorous derivation of this type of limiting behaviour, which was initiated by the pioneering work of Hepp and Spohn, and has now attracted a vast recent literature. The rigorous details here are rather sophisticated; but the heuristic explanation of the phenomenon is fairly simple, and actually rather pretty in my opinion, involving the foundational quantum mechanics of {N}-particle systems. I am recording this heuristic derivation here, partly for my own benefit, but perhaps it will be of interest to some readers.

This discussion will be purely formal, in the sense that (important) analytic issues such as differentiability, existence and uniqueness, etc. will be largely ignored.

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My penultimate article for my PCM series is a very short one, on “Hamiltonians“. The PCM has a number of short articles to define terms which occur frequently in the longer articles, but are not substantive enough topics by themselves to warrant a full-length treatment. One of these is the term “Hamiltonian”, which is used in all the standard types of physical mechanics (classical or quantum, microscopic or statistical) to describe the total energy of a system. It is a remarkable feature of the laws of physics that this single object (which is a scalar-valued function in classical physics, and a self-adjoint operator in quantum mechanics) suffices to describe the entire dynamics of a system, although from a mathematical perspective it is not always easy to read off all the analytic aspects of this dynamics just from the form of the Hamiltonian.

In mathematics, Hamiltonians of course arise in the equations of mathematical physics (such as Hamilton’s equations of motion, or Schrödinger’s equations of motion), but also show up in symplectic geometry (as a special case of a moment map) and in microlocal analysis.

For this post, I would also like to highlight an article of my good friend Andrew Granville on one of my own favorite topics, “Analytic number theory“, focusing in particular on the classical problem of understanding the distribution of the primes, via such analytic tools as zeta functions and L-functions, sieve theory, and the circle method.

This post is derived from an interesting conversation I had several years ago with my friend Jason Newquist on trying to find some intuitive analogies for the non-classical nature of quantum mechanics. It occurred to me that this type of informal, rambling discussion might actually be rather suited to the blog medium, so here goes nothing…

Quantum mechanics has a number of weird consequences, but here we are focusing on three (inter-related) ones:

  1. Objects can behave both like particles (with definite position and a continuum of states) and waves (with indefinite position and (in confined situations) quantised states);
  2. The equations that govern quantum mechanics are deterministic, but the standard interpretation of the solutions of these equations is probabilistic; and
  3. If instead one applies the laws of quantum mechanics literally at the macroscopic scale, then the universe itself must split into the superposition of many distinct “worlds”.

In trying to come up with a classical conceptual model in which to capture these non-classical phenomena, we eventually hit upon using the idea of using computer games as an analogy. The exact choice of game is not terribly important, but let us pick Tomb Raider – a popular game from about ten years ago (back when I had the leisure to play these things), in which the heroine, Lara Croft, explores various tombs and dungeons, solving puzzles and dodging traps, in order to achieve some objective. It is quite common for Lara to die in the game, for instance by failing to evade one of the traps. (I should warn that this analogy will be rather violent on certain computer-generated characters.)

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