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The wave equation is usually expressed in the form

$\displaystyle \partial_{tt} u - \Delta u = 0$

where ${u \colon {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ is a function of both time ${t \in {\bf R}}$ and space ${x \in {\bf R}^d}$, with ${\Delta}$ being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain ${{\bf R}^d}$ with some other manifold and replacing the Laplacian ${\Delta}$ with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on ${{\bf R}^d}$. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy

$\displaystyle \int_{{\bf R}^d} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx$

which we can rewrite using integration by parts and the ${L^2}$ inner product ${\langle, \rangle}$ on ${{\bf R}^d}$ as

$\displaystyle \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle \partial_t u(t), \partial_t u(t) \rangle.$

A key feature of the wave equation is finite speed of propagation: if, at time ${t=0}$ (say), the initial position ${u(0)}$ and initial velocity ${\partial_t u(0)}$ are both supported in a ball ${B(x_0,R) := \{ x \in {\bf R}^d: |x-x_0| \leq R \}}$, then at any later time ${t>0}$, the position ${u(t)}$ and velocity ${\partial_t u(t)}$ are supported in the larger ball ${B(x_0,R+t)}$. This can be seen for instance (formally, at least) by inspecting the exterior energy

$\displaystyle \int_{|x-x_0| > R+t} \frac{1}{2} |\nabla u(t,x)|^2 + \frac{1}{2} |\partial_t u(t,x)|^2\ dx$

and observing (after some integration by parts and differentiation under the integral sign) that it is non-increasing in time, non-negative, and vanishing at time ${t=0}$.

The wave equation is second order in time, but one can turn it into a first order system by working with the pair ${(u(t),v(t))}$ rather than just the single field ${u(t)}$, where ${v(t) := \partial_t u(t)}$ is the velocity field. The system is then

$\displaystyle \partial_t u(t) = v(t)$

$\displaystyle \partial_t v(t) = \Delta u(t)$

and the conserved energy is now

$\displaystyle \frac{1}{2} \langle -\Delta u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (1)$

Finite speed of propagation then tells us that if ${u(0),v(0)}$ are both supported on ${B(x_0,R)}$, then ${u(t),v(t)}$ are supported on ${B(x_0,R+t)}$ for all ${t>0}$. One also has time reversal symmetry: if ${t \mapsto (u(t),v(t))}$ is a solution, then ${t \mapsto (u(-t), -v(-t))}$ is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times ${t<0}$ using this symmetry.

If one has an eigenfunction

$\displaystyle -\Delta \phi = \lambda^2 \phi$

of the Laplacian, then we have the explicit solutions

$\displaystyle u(t) = e^{\pm it \lambda} \phi$

$\displaystyle v(t) = \pm i \lambda e^{\pm it \lambda} \phi$

of the wave equation, which formally can be used to construct all other solutions via the principle of superposition.

When one has vanishing initial velocity ${v(0)=0}$, the solution ${u(t)}$ is given via functional calculus by

$\displaystyle u(t) = \cos(t \sqrt{-\Delta}) u(0)$

and the propagator ${\cos(t \sqrt{-\Delta})}$ can be expressed as the average of half-wave operators:

$\displaystyle \cos(t \sqrt{-\Delta}) = \frac{1}{2} ( e^{it\sqrt{-\Delta}} + e^{-it\sqrt{-\Delta}} ).$

One can view ${\cos(t \sqrt{-\Delta} )}$ as a minor of the full wave propagator

$\displaystyle U(t) := \exp \begin{pmatrix} 0 & t \\ t\Delta & 0 \end{pmatrix}$

$\displaystyle = \begin{pmatrix} \cos(t \sqrt{-\Delta}) & \frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}} \\ \sin(t\sqrt{-\Delta}) \sqrt{-\Delta} & \cos(t \sqrt{-\Delta} ) \end{pmatrix}$

which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that

$\displaystyle \begin{pmatrix} u(t) \\ v(t) \end{pmatrix} = U(t) \begin{pmatrix} u(0) \\ v(0) \end{pmatrix}. \ \ \ \ \ (2)$

Viewing the contraction ${\cos(t\sqrt{-\Delta})}$ as a minor of a unitary operator is an instance of the “dilation trick“.

It turns out (as I learned from Yuval Peres) that there is a useful discrete analogue of the wave equation (and of all of the above facts), in which the time variable ${t}$ now lives on the integers ${{\bf Z}}$ rather than on ${{\bf R}}$, and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form

$\displaystyle u(t+1) = P u(t) + v(t) \ \ \ \ \ (3)$

$\displaystyle v(t+1) = P v(t) - (1-P^2) u(t)$

where ${t}$ is now an integer, ${u(t), v(t)}$ take values in some Hilbert space (e.g. ${\ell^2}$ functions on a graph ${G}$), and ${P}$ is some operator on that Hilbert space (which in applications will usually be a self-adjoint contraction). To connect this with the classical wave equation, let us first consider a rescaling of this system

$\displaystyle u(t+\varepsilon) = P_\varepsilon u(t) + \varepsilon v(t)$

$\displaystyle v(t+\varepsilon) = P_\varepsilon v(t) - \frac{1}{\varepsilon} (1-P_\varepsilon^2) u(t)$

where ${\varepsilon>0}$ is a small parameter (representing the discretised time step), ${t}$ now takes values in the integer multiples ${\varepsilon {\bf Z}}$ of ${\varepsilon}$, and ${P_\varepsilon}$ is the wave propagator operator ${P_\varepsilon := \cos( \varepsilon \sqrt{-\Delta} )}$ or the heat propagator ${P_\varepsilon := \exp( - \varepsilon^2 \Delta/2 )}$ (the two operators are different, but agree to fourth order in ${\varepsilon}$). One can then formally verify that the wave equation emerges from this rescaled system in the limit ${\varepsilon \rightarrow 0}$. (Thus, ${P}$ is not exactly the direct analogue of the Laplacian ${\Delta}$, but can be viewed as something like ${P_\varepsilon = 1 - \frac{\varepsilon^2}{2} \Delta + O( \varepsilon^4 )}$ in the case of small ${\varepsilon}$, or ${P = 1 - \frac{1}{2}\Delta + O(\Delta^2)}$ if we are not rescaling to the small ${\varepsilon}$ case. The operator ${P}$ is sometimes known as the diffusion operator)

Assuming ${P}$ is self-adjoint, solutions to the system (3) formally conserve the energy

$\displaystyle \frac{1}{2} \langle (1-P^2) u(t), u(t) \rangle + \frac{1}{2} \langle v(t), v(t) \rangle. \ \ \ \ \ (4)$

This energy is positive semi-definite if ${P}$ is a contraction. We have the same time reversal symmetry as before: if ${t \mapsto (u(t),v(t))}$ solves the system (3), then so does ${t \mapsto (u(-t), -v(-t))}$. If one has an eigenfunction

$\displaystyle P \phi = \cos(\lambda) \phi$

to the operator ${P}$, then one has an explicit solution

$\displaystyle u(t) = e^{\pm it \lambda} \phi$

$\displaystyle v(t) = \pm i \sin(\lambda) e^{\pm it \lambda} \phi$

to (3), and (in principle at least) this generates all other solutions via the principle of superposition.

Finite speed of propagation is a lot easier in the discrete setting, though one has to offset the support of the “velocity” field ${v}$ by one unit. Suppose we know that ${P}$ has unit speed in the sense that whenever ${f}$ is supported in a ball ${B(x,R)}$, then ${Pf}$ is supported in the ball ${B(x,R+1)}$. Then an easy induction shows that if ${u(0), v(0)}$ are supported in ${B(x_0,R), B(x_0,R+1)}$ respectively, then ${u(t), v(t)}$ are supported in ${B(x_0,R+t), B(x_0, R+t+1)}$.

The fundamental solution ${U(t) = U^t}$ to the discretised wave equation (3), in the sense of (2), is given by the formula

$\displaystyle U(t) = U^t = \begin{pmatrix} P & 1 \\ P^2-1 & P \end{pmatrix}^t$

$\displaystyle = \begin{pmatrix} T_t(P) & U_{t-1}(P) \\ (P^2-1) U_{t-1}(P) & T_t(P) \end{pmatrix}$

where ${T_t}$ and ${U_t}$ are the Chebyshev polynomials of the first and second kind, thus

$\displaystyle T_t( \cos \theta ) = \cos(t\theta)$

and

$\displaystyle U_t( \cos \theta ) = \frac{\sin((t+1)\theta)}{\sin \theta}.$

In particular, ${P}$ is now a minor of ${U(1) = U}$, and can also be viewed as an average of ${U}$ with its inverse ${U^{-1}}$:

$\displaystyle \begin{pmatrix} P & 0 \\ 0 & P \end{pmatrix} = \frac{1}{2} (U + U^{-1}). \ \ \ \ \ (5)$

As before, ${U}$ is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers ${P^n}$ and ${U^n}$ are discrete analogues of the heat propagators ${e^{t\Delta/2}}$ and wave propagators ${U(t)}$ respectively.

One nice application of all this formalism, which I learned from Yuval Peres, is the Varopoulos-Carne inequality:

Theorem 1 (Varopoulos-Carne inequality) Let ${G}$ be a (possibly infinite) regular graph, let ${n \geq 1}$, and let ${x, y}$ be vertices in ${G}$. Then the probability that the simple random walk at ${x}$ lands at ${y}$ at time ${n}$ is at most ${2 \exp( - d(x,y)^2 / 2n )}$, where ${d}$ is the graph distance.

This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers ${{\bf Z}}$. Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length ${n}$ concentrate on the ball of radius ${O(\sqrt{n})}$ or so centred at the origin of the random walk.

Proof: Let ${P \colon \ell^2(G) \rightarrow \ell^2(G)}$ be the graph Laplacian, thus

$\displaystyle Pf(x) = \frac{1}{D} \sum_{y \sim x} f(y)$

for any ${f \in \ell^2(G)}$, where ${D}$ is the degree of the regular graph and sum is over the ${D}$ vertices ${y}$ that are adjacent to ${x}$. This is a contraction of unit speed, and the probability that the random walk at ${x}$ lands at ${y}$ at time ${n}$ is

$\displaystyle \langle P^n \delta_x, \delta_y \rangle$

where ${\delta_x, \delta_y}$ are the Dirac deltas at ${x,y}$. Using (5), we can rewrite this as

$\displaystyle \langle (\frac{1}{2} (U + U^{-1}))^n \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle$

where we are now using the energy form (4). We can write

$\displaystyle (\frac{1}{2} (U + U^{-1}))^n = {\bf E} U^{S_n}$

where ${S_n}$ is the simple random walk of length ${n}$ on the integers, that is to say ${S_n = \xi_1 + \dots + \xi_n}$ where ${\xi_1,\dots,\xi_n = \pm 1}$ are independent uniform Bernoulli signs. Thus we wish to show that

$\displaystyle {\bf E} \langle U^{S_n} \begin{pmatrix} 0 \\ \delta_x\end{pmatrix}, \begin{pmatrix} 0 \\ \delta_y\end{pmatrix} \rangle \leq 2 \exp(-d(x,y)^2 / 2n ).$

By finite speed of propagation, the inner product here vanishes if ${|S_n| < d(x,y)}$. For ${|S_n| \geq d(x,y)}$ we can use Cauchy-Schwarz and the unitary nature of ${U}$ to bound the inner product by ${1}$. Thus the left-hand side may be upper bounded by

$\displaystyle {\bf P}( |S_n| \geq d(x,y) )$

and the claim now follows from the Chernoff inequality. $\Box$

This inequality has many applications, particularly with regards to relating the entropy, mixing time, and concentration of random walks with volume growth of balls; see this text of Lyons and Peres for some examples.

For sake of comparison, here is a continuous counterpart to the Varopoulos-Carne inequality:

Theorem 2 (Continuous Varopoulos-Carne inequality) Let ${t > 0}$, and let ${f,g \in L^2({\bf R}^d)}$ be supported on compact sets ${F,G}$ respectively. Then

$\displaystyle |\langle e^{t\Delta/2} f, g \rangle| \leq \sqrt{\frac{2t}{\pi d(F,G)^2}} \exp( - d(F,G)^2 / 2t ) \|f\|_{L^2} \|g\|_{L^2}$

where ${d(F,G)}$ is the Euclidean distance between ${F}$ and ${G}$.

Proof: By Fourier inversion one has

$\displaystyle e^{-t\xi^2/2} = \frac{1}{\sqrt{2\pi t}} \int_{\bf R} e^{-s^2/2t} e^{is\xi}\ ds$

$\displaystyle = \sqrt{\frac{2}{\pi t}} \int_0^\infty e^{-s^2/2t} \cos(s \xi )\ ds$

for any real ${\xi}$, and thus

$\displaystyle \langle e^{t\Delta/2} f, g\rangle = \sqrt{\frac{2}{\pi}} \int_0^\infty e^{-s^2/2t} \langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds.$

By finite speed of propagation, the inner product ${\langle \cos(s \sqrt{-\Delta} ) f, g \rangle\ ds}$ vanishes when ${s < d(F,G)}$; otherwise, we can use Cauchy-Schwarz and the contractive nature of ${\cos(s \sqrt{-\Delta} )}$ to bound this inner product by ${\|f\|_{L^2} \|g\|_{L^2}}$. Thus

$\displaystyle |\langle e^{t\Delta/2} f, g\rangle| \leq \sqrt{\frac{2}{\pi t}} \|f\|_{L^2} \|g\|_{L^2} \int_{d(F,G)}^\infty e^{-s^2/2t}\ ds.$

Bounding ${e^{-s^2/2t}}$ by ${e^{-d(F,G)^2/2t} e^{-d(F,G) (s-d(F,G))/t}}$, we obtain the claim. $\Box$

Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than ${{\bf R}^d}$.

Consider the free Schrödinger equation in ${d}$ spatial dimensions, which I will normalise as

$\displaystyle i u_t + \frac{1}{2} \Delta_{{\bf R}^d} u = 0 \ \ \ \ \ (1)$

where ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ is the unknown field and ${\Delta_{{\bf R}^{d+1}} = \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}}$ is the spatial Laplacian. To avoid irrelevant technical issues I will restrict attention to smooth (classical) solutions to this equation, and will work locally in spacetime avoiding issues of decay at infinity (or at other singularities); I will also avoid issues involving branch cuts of functions such as ${t^{d/2}}$ (if one wishes, one can restrict ${d}$ to be even in order to safely ignore all branch cut issues). The space of solutions to (1) enjoys a number of symmetries. A particularly non-obvious symmetry is the pseudoconformal symmetry: if ${u}$ solves (1), then the pseudoconformal solution ${pc(u): {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ defined by

$\displaystyle pc(u)(t,x) := \frac{1}{(it)^{d/2}} \overline{u(\frac{1}{t}, \frac{x}{t})} e^{i|x|^2/2t} \ \ \ \ \ (2)$

for ${t \neq 0}$ can be seen after some computation to also solve (1). (If ${u}$ has suitable decay at spatial infinity and one chooses a suitable branch cut for ${(it)^{d/2}}$, one can extend ${pc(u)}$ continuously to the ${t=0}$ spatial slice, whereupon it becomes essentially the spatial Fourier transform of ${u(0,\cdot)}$, but we will not need this fact for the current discussion.)

An analogous symmetry exists for the free wave equation in ${d+1}$ spatial dimensions, which I will write as

$\displaystyle u_{tt} - \Delta_{{\bf R}^{d+1}} u = 0 \ \ \ \ \ (3)$

where ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if ${u: {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ solves (3), then the function ${conf(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$, defined in the interior ${\{ (t,x): |x| < |t| \}}$ of the light cone by the formula

$\displaystyle conf(u)(t,x) := (t^2-|x|^2)^{-d/2} u( \frac{t}{t^2-|x|^2}, \frac{x}{t^2-|x|^2} ), \ \ \ \ \ (4)$

also solves (3).

There are also some direct links between the Schrödinger equation in ${d}$ dimensions and the wave equation in ${d+1}$ dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a ${d}$-dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a ${d+1}$-dimensional cone. To link the two, one then observes that the ${d}$-dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the ${d+1}$-dimensional cone. In physical space, this link is manifested as follows: if ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}}$ solves (1), then the function ${\iota_{1}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by

$\displaystyle \iota_{1}(u)(t,x_1,\ldots,x_{d+1}) := e^{-i(t+x_{d+1})} u( \frac{t-x_{d+1}}{2}, x_1,\ldots,x_d)$

solves (3). More generally, for any non-zero scaling parameter ${\lambda}$, the function ${\iota_{\lambda}(u): {\bf R} \times {\bf R}^{d+1} \rightarrow {\bf C}}$ defined by

$\displaystyle \iota_{\lambda}(u)(t,x_1,\ldots,x_{d+1}) :=$

$\displaystyle \lambda^{d/2} e^{-i\lambda(t+x_{d+1})} u( \lambda \frac{t-x_{d+1}}{2}, \lambda x_1,\ldots,\lambda x_d) \ \ \ \ \ (5)$

solves (3).

As an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings ${\iota_1}$ (or more generally ${\iota_\lambda}$) to explicitly connect together the pseudoconformal transformation ${pc}$ and the conformal transformation ${conf}$. It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings ${\iota_\lambda}$ intertwine ${pc}$ and ${conf}$) being incorrect, and as such this particular task was perhaps too difficult even for a challenge question. I’ve been asked a couple times to provide the connection more explicitly, so I will do so below the fold.

Let ${L: H \rightarrow H}$ be a self-adjoint operator on a finite-dimensional Hilbert space ${H}$. The behaviour of this operator can be completely described by the spectral theorem for finite-dimensional self-adjoint operators (i.e. Hermitian matrices, when viewed in coordinates), which provides a sequence ${\lambda_1,\ldots,\lambda_n \in {\bf R}}$ of eigenvalues and an orthonormal basis ${e_1,\ldots,e_n}$ of eigenfunctions such that ${L e_i = \lambda_i e_i}$ for all ${i=1,\ldots,n}$. In particular, given any function ${m: \sigma(L) \rightarrow {\bf C}}$ on the spectrum ${\sigma(L) := \{ \lambda_1,\ldots,\lambda_n\}}$ of ${L}$, one can then define the linear operator ${m(L): H \rightarrow H}$ by the formula

$\displaystyle m(L) e_i := m(\lambda_i) e_i,$

which then gives a functional calculus, in the sense that the map ${m \mapsto m(L)}$ is a ${C^*}$-algebra isometric homomorphism from the algebra ${BC(\sigma(L) \rightarrow {\bf C})}$ of bounded continuous functions from ${\sigma(L)}$ to ${{\bf C}}$, to the algebra ${B(H \rightarrow H)}$ of bounded linear operators on ${H}$. Thus, for instance, one can define heat operators ${e^{-tL}}$ for ${t>0}$, Schrödinger operators ${e^{itL}}$ for ${t \in {\bf R}}$, resolvents ${\frac{1}{L-z}}$ for ${z \not \in \sigma(L)}$, and (if ${L}$ is positive) wave operators ${e^{it\sqrt{L}}}$ for ${t \in {\bf R}}$. These will be bounded operators (and, in the case of the Schrödinger and wave operators, unitary operators, and in the case of the heat operators with ${L}$ positive, they will be contractions). Among other things, this functional calculus can then be used to solve differential equations such as the heat equation

$\displaystyle u_t + Lu = 0; \quad u(0) = f \ \ \ \ \ (1)$

the Schrödinger equation

$\displaystyle u_t + iLu = 0; \quad u(0) = f \ \ \ \ \ (2)$

the wave equation

$\displaystyle u_{tt} + Lu = 0; \quad u(0) = f; \quad u_t(0) = g \ \ \ \ \ (3)$

or the Helmholtz equation

$\displaystyle (L-z) u = f. \ \ \ \ \ (4)$

The functional calculus can also be associated to a spectral measure. Indeed, for any vectors ${f, g \in H}$, there is a complex measure ${\mu_{f,g}}$ on ${\sigma(L)}$ with the property that

$\displaystyle \langle m(L) f, g \rangle_H = \int_{\sigma(L)} m(x) d\mu_{f,g}(x);$

indeed, one can set ${\mu_{f,g}}$ to be the discrete measure on ${\sigma(L)}$ defined by the formula

$\displaystyle \mu_{f,g}(E) := \sum_{i: \lambda_i \in E} \langle f, e_i \rangle_H \langle e_i, g \rangle_H.$

One can also view this complex measure as a coefficient

$\displaystyle \mu_{f,g} = \langle \mu f, g \rangle_H$

of a projection-valued measure ${\mu}$ on ${\sigma(L)}$, defined by setting

$\displaystyle \mu(E) f := \sum_{i: \lambda_i \in E} \langle f, e_i \rangle_H e_i.$

Finally, one can view ${L}$ as unitarily equivalent to a multiplication operator ${M: f \mapsto g f}$ on ${\ell^2(\{1,\ldots,n\})}$, where ${g}$ is the real-valued function ${g(i) := \lambda_i}$, and the intertwining map ${U: \ell^2(\{1,\ldots,n\}) \rightarrow H}$ is given by

$\displaystyle U ( (c_i)_{i=1}^n ) := \sum_{i=1}^n c_i e_i,$

so that ${L = U M U^{-1}}$.

It is an important fact in analysis that many of these above assertions extend to operators on an infinite-dimensional Hilbert space ${H}$, so long as one one is careful about what “self-adjoint operator” means; these facts are collectively referred to as the spectral theorem. For instance, it turns out that most of the above claims have analogues for bounded self-adjoint operators ${L: H \rightarrow H}$. However, in the theory of partial differential equations, one often needs to apply the spectral theorem to unbounded, densely defined linear operators ${L: D \rightarrow H}$, which (initially, at least), are only defined on a dense subspace ${D}$ of the Hilbert space ${H}$. A very typical situation arises when ${H = L^2(\Omega)}$ is the square-integrable functions on some domain or manifold ${\Omega}$ (which may have a boundary or be otherwise “incomplete”), and ${D = C^\infty_c(\Omega)}$ are the smooth compactly supported functions on ${\Omega}$, and ${L}$ is some linear differential operator. It is then of interest to obtain the spectral theorem for such operators, so that one build operators such as ${e^{-tL}, e^{itL}, \frac{1}{L-z}, e^{it\sqrt{L}}}$ or to solve equations such as (1), (2), (3), (4).

In order to do this, some necessary conditions on the densely defined operator ${L: D \rightarrow H}$ must be imposed. The most obvious is that of symmetry, which asserts that

$\displaystyle \langle Lf, g \rangle_H = \langle f, Lg \rangle_H \ \ \ \ \ (5)$

for all ${f, g \in D}$. In some applications, one also wants to impose positive definiteness, which asserts that

$\displaystyle \langle Lf, f \rangle_H \geq 0 \ \ \ \ \ (6)$

for all ${f \in D}$. These hypotheses are sufficient in the case when ${L}$ is bounded, and in particular when ${H}$ is finite dimensional. However, as it turns out, for unbounded operators these conditions are not, by themselves, enough to obtain a good spectral theory. For instance, one consequence of the spectral theorem should be that the resolvents ${(L-z)^{-1}}$ are well-defined for any strictly complex ${z}$, which by duality implies that the image of ${L-z}$ should be dense in ${H}$. However, this can fail if one just assumes symmetry, or symmetry and positive definiteness. A well-known example occurs when ${H}$ is the Hilbert space ${H := L^2((0,1))}$, ${D := C^\infty_c((0,1))}$ is the space of test functions, and ${L}$ is the one-dimensional Laplacian ${L := -\frac{d^2}{dx^2}}$. Then ${L}$ is symmetric and positive, but the operator ${L-k^2}$ does not have dense image for any complex ${k}$, since

$\displaystyle \langle (L-\overline{k}^2) f, e^{\overline{k}x} \rangle_H = 0$

for all test functions ${f \in C^\infty_c((0,1))}$, as can be seen from a routine integration by parts. As such, the resolvent map is not everywhere uniquely defined. There is also a lack of uniqueness for the wave, heat, and Schrödinger equations for this operator (note that there are no spatial boundary conditions specified in these equations).

Another example occurs when ${H := L^2((0,+\infty))}$, ${D := C^\infty_c((0,+\infty))}$, ${L}$ is the momentum operator ${L := i \frac{d}{dx}}$. Then the resolvent ${(L-z)^{-1}}$ can be uniquely defined for ${z}$ in the upper half-plane, but not in the lower half-plane, due to the obstruction

$\displaystyle \langle (L-z) f, e^{i \bar{z} x} \rangle_H = 0$

for all test functions ${f}$ (note that the function ${e^{i\bar{z} x}}$ lies in ${L^2((0,+\infty))}$ when ${z}$ is in the lower half-plane). For related reasons, the translation operators ${e^{itL}}$ have a problem with either uniqueness or existence (depending on whether ${t}$ is positive or negative), due to the unspecified boundary behaviour at the origin.

The key property that lets one avoid this bad behaviour is that of essential self-adjointness. Once ${L}$ is essentially self-adjoint, then spectral theorem becomes applicable again, leading to all the expected behaviour (e.g. existence and uniqueness for the various PDE given above).

Unfortunately, the concept of essential self-adjointness is defined rather abstractly, and is difficult to verify directly; unlike the symmetry condition (5) or the positive condition (6), it is not a “local” condition that can be easily verified just by testing ${L}$ on various inputs, but is instead a more “global” condition. In practice, to verify this property, one needs to invoke one of a number of a partial converses to the spectral theorem, which roughly speaking asserts that if at least one of the expected consequences of the spectral theorem is true for some symmetric densely defined operator ${L}$, then ${L}$ is self-adjoint. Examples of “expected consequences” include:

• Existence of resolvents ${(L-z)^{-1}}$ (or equivalently, dense image for ${L-z}$);
• Existence of a contractive heat propagator semigroup ${e^{tL}}$ (in the positive case);
• Existence of a unitary Schrödinger propagator group ${e^{itL}}$;
• Existence of a unitary wave propagator group ${e^{it\sqrt{L}}}$ (in the positive case);
• Existence of a “reasonable” functional calculus.
• Unitary equivalence with a multiplication operator.

Thus, to actually verify essential self-adjointness of a differential operator, one typically has to first solve a PDE (such as the wave, Schrödinger, heat, or Helmholtz equation) by some non-spectral method (e.g. by a contraction mapping argument, or a perturbation argument based on an operator already known to be essentially self-adjoint). Once one can solve one of the PDEs, then one can apply one of the known converse spectral theorems to obtain essential self-adjointness, and then by the forward spectral theorem one can then solve all the other PDEs as well. But there is no getting out of that first step, which requires some input (typically of an ODE, PDE, or geometric nature) that is external to what abstract spectral theory can provide. For instance, if one wants to establish essential self-adjointness of the Laplace-Beltrami operator ${L = -\Delta_g}$ on a smooth Riemannian manifold ${(M,g)}$ (using ${C^\infty_c(M)}$ as the domain space), it turns out (under reasonable regularity hypotheses) that essential self-adjointness is equivalent to geodesic completeness of the manifold, which is a global ODE condition rather than a local one: one needs geodesics to continue indefinitely in order to be able to (unitarily) solve PDEs such as the wave equation, which in turn leads to essential self-adjointness. (Note that the domains ${(0,1)}$ and ${(0,+\infty)}$ in the previous examples were not geodesically complete.) For this reason, essential self-adjointness of a differential operator is sometimes referred to as quantum completeness (with the completeness of the associated Hamilton-Jacobi flow then being the analogous classical completeness).

In these notes, I wanted to record (mostly for my own benefit) the forward and converse spectral theorems, and to verify essential self-adjointness of the Laplace-Beltrami operator on geodesically complete manifolds. This is extremely standard analysis (covered, for instance, in the texts of Reed and Simon), but I wanted to write it down myself to make sure that I really understood this foundational material properly.

Hans Lindblad and I have just uploaded to the arXiv our joint paper “Asymptotic decay for a one-dimensional nonlinear wave equation“, submitted to Analysis & PDE.  This paper, to our knowledge, is the first paper to analyse the asymptotic behaviour of the one-dimensional defocusing nonlinear wave equation

${}-u_{tt}+u_{xx} = |u|^{p-1} u$ (1)

where $u: {\bf R} \times {\bf R} \to {\bf R}$ is the solution and $p>1$ is a fixed exponent.  Nowadays, this type of equation is considered a very simple example of a non-linear wave equation (there is only one spatial dimension, the equation is semilinear, the conserved energy is positive definite and coercive, and there are no derivatives in the nonlinear term), and indeed it is not difficult to show that any solution whose conserved energy

$E[u] := \int_{{\bf R}} \frac{1}{2} |u_t|^2 + \frac{1}{2} |u_x|^2 + \frac{1}{p+1} |u|^{p+1}\ dx$

is finite, will exist globally for all time (and remain finite energy, of course).  In particular, from the one-dimensional Gagliardo-Nirenberg inequality (a variant of the Sobolev embedding theorem), such solutions will remain uniformly bounded in $L^\infty_x({\bf R})$ for all time.

However, this leaves open the question of the asymptotic behaviour of such solutions in the limit as $t \to \infty$.  In higher dimensions, there are a variety of scattering and asymptotic completeness results which show that solutions to nonlinear wave equations such as (1) decay asymptotically in various senses, at least if one is in the perturbative regime in which the solution is assumed small in some sense (e.g. small energy).  For instance, a typical result might be that spatial norms such as $\|u(t)\|_{L^q({\bf R})}$ might go to zero (in an average sense, at least).   In general, such results for nonlinear wave equations are ultimately based on the fact that the linear wave equation in higher dimensions also enjoys an analogous decay as $t \to +\infty$, as linear waves in higher dimensions spread out and disperse over time.  (This can be formalised by decay estimates on the fundamental solution of the linear wave equation, or by basic estimates such as the (long-time) Strichartz estimates and their relatives.)  The idea is then to view the nonlinear wave equation as a perturbation of the linear one.

On the other hand, the solution to the linear one-dimensional wave equation

$-u_{tt} + u_{xx} = 0$ (2)

does not exhibit any decay in time; as one learns in an undergraduate PDE class, the general (finite energy) solution to such an equation is given by the superposition of two travelling waves,

$u(t,x) = f(x+t) + g(x-t)$ (3)

where $f$ and $g$ also have finite energy, so in particular norms such as $\|u(t)\|_{L^\infty_x({\bf R})}$ cannot decay to zero as $t \to \infty$ unless the solution is completely trivial.

Nevertheless, we were able to establish a nonlinear decay effect for equation (1), caused more by the nonlinear right-hand side of (1) than by the linear left-hand side, to obtain $L^\infty_x({\bf R})$ decay on the average:

Theorem 1. (Average $L^\infty_x$ decay) If $u$ is a finite energy solution to (1), then $\frac{1}{2T} \int_{-T}^T \|u(t)\|_{L^\infty_x({\bf R})}$ tends to zero as $T \to \infty$.

Actually we prove a slightly stronger statement than Theorem 1, in that the decay is uniform among all solutions with a given energy bound, but I will stick to the above formulation of the main result for simplicity.

Informally, the reason for the nonlinear decay is as follows.  The linear evolution tries to force waves to move at constant velocity (indeed, from (3) we see that linear waves move at the speed of light $c=1$).  But the defocusing nature of the nonlinearity will spread out any wave that is propagating along a constant velocity worldline.  This intuition can be formalised by a Morawetz-type energy estimate that shows that the nonlinear potential energy must decay along any rectangular slab of spacetime (that represents the neighbourhood of a constant velocity worldline).

Now, just because the linear wave equation propagates along constant velocity worldlines, this does not mean that the nonlinear wave equation does too; one could imagine that a wave packet could propagate along a more complicated trajectory $t \mapsto x(t)$ in which the velocity $x'(t)$ is not constant.  However, energy methods still force the solution of the nonlinear wave equation to obey finite speed of propagation, which in the wave packet context means (roughly speaking) that the nonlinear trajectory $t \mapsto x(t)$ is a Lipschitz continuous function (with Lipschitz constant at most $1$).

And now we deploy a trick which appears to be new to the field of nonlinear wave equations: we invoke the Rademacher differentiation theorem (or Lebesgue differentiation theorem), which asserts that Lipschitz continuous functions are almost everywhere differentiable.  (By coincidence, I am teaching this theorem in my current course, both in one dimension (which is the case of interest here) and in higher dimensions.)  A compactness argument allows one to extract a quantitative estimate from this theorem (cf. this earlier blog post of mine) which, roughly speaking, tells us that there are large portions of the trajectory $t \mapsto x(t)$ which behave approximately linearly at an appropriate scale.  This turns out to be a good enough control on the trajectory that one can apply the Morawetz inequality and rule out the existence of persistent wave packets over long periods of time, which is what leads to Theorem 1.

There is still scope for further work to be done on the asymptotics.  In particular, we still do not have a good understanding of what the asymptotic profile of the solution should be, even in the perturbative regime; standard nonlinear geometric optics methods do not appear to work very well due to the extremely weak decay.