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The Poincaré upper half-plane (with a boundary consisting of the real line
together with the point at infinity
) carries an action of the projective special linear group
via fractional linear transformations:
Here and in the rest of the post we will abuse notation by identifying elements of the special linear group
with their equivalence class
in
; this will occasionally create or remove a factor of two in our formulae, but otherwise has very little effect, though one has to check that various definitions and expressions (such as (1)) are unaffected if one replaces a matrix
by its negation
. In particular, we recommend that the reader ignore the signs
that appear from time to time in the discussion below.
As the action of on
is transitive, and any given point in
(e.g.
) has a stabiliser isomorphic to the projective rotation group
, we can view the Poincaré upper half-plane
as a homogeneous space for
, and more specifically the quotient space of
of a maximal compact subgroup
. In fact, we can make the half-plane a symmetric space for
, by endowing
with the Riemannian metric
(using Cartesian coordinates ), which is invariant with respect to the
action. Like any other Riemannian metric, the metric on
generates a number of other important geometric objects on
, such as the distance function
which can be computed to be given by the formula
the volume measure , which can be computed to be
and the Laplace-Beltrami operator, which can be computed to be (here we use the negative definite sign convention for
). As the metric
was
-invariant, all of these quantities arising from the metric are similarly
-invariant in the appropriate sense.
The Gauss curvature of the Poincaré half-plane can be computed to be the constant , thus
is a model for two-dimensional hyperbolic geometry, in much the same way that the unit sphere
in
is a model for two-dimensional spherical geometry (or
is a model for two-dimensional Euclidean geometry). (Indeed,
is isomorphic (via projection to a null hyperplane) to the upper unit hyperboloid
in the Minkowski spacetime
, which is the direct analogue of the unit sphere in Euclidean spacetime
or the plane
in Galilean spacetime
.)
One can inject arithmetic into this geometric structure by passing from the Lie group to the full modular group
or congruence subgroups such as
for natural number , or to the discrete stabiliser
of the point at infinity:
These are discrete subgroups of , nested by the subgroup inclusions
There are many further discrete subgroups of (known collectively as Fuchsian groups) that one could consider, but we will focus attention on these three groups in this post.
Any discrete subgroup of
generates a quotient space
, which in general will be a non-compact two-dimensional orbifold. One can understand such a quotient space by working with a fundamental domain
– a set consisting of a single representative of each of the orbits
of
in
. This fundamental domain is by no means uniquely defined, but if the fundamental domain is chosen with some reasonable amount of regularity, one can view
as the fundamental domain with the boundaries glued together in an appropriate sense. Among other things, fundamental domains can be used to induce a volume measure
on
from the volume measure
on
(restricted to a fundamental domain). By abuse of notation we will refer to both measures simply as
when there is no chance of confusion.
For instance, a fundamental domain for is given (up to null sets) by the strip
, with
identifiable with the cylinder formed by gluing together the two sides of the strip. A fundamental domain for
is famously given (again up to null sets) by an upper portion
, with the left and right sides again glued to each other, and the left and right halves of the circular boundary glued to itself. A fundamental domain for
can be formed by gluing together
copies of a fundamental domain for in a rather complicated but interesting fashion.
While fundamental domains can be a convenient choice of coordinates to work with for some computations (as well as for drawing appropriate pictures), it is geometrically more natural to avoid working explicitly on such domains, and instead work directly on the quotient spaces . In order to analyse functions
on such orbifolds, it is convenient to lift such functions back up to
and identify them with functions
which are
-automorphic in the sense that
for all
and
. Such functions will be referred to as
-automorphic forms, or automorphic forms for short (we always implicitly assume all such functions to be measurable). (Strictly speaking, these are the automorphic forms with trivial factor of automorphy; one can certainly consider other factors of automorphy, particularly when working with holomorphic modular forms, which corresponds to sections of a more non-trivial line bundle over
than the trivial bundle
that is implicitly present when analysing scalar functions
. However, we will not discuss this (important) more general situation here.)
An important way to create a -automorphic form is to start with a non-automorphic function
obeying suitable decay conditions (e.g. bounded with compact support will suffice) and form the Poincaré series
defined by
which is clearly -automorphic. (One could equivalently write
in place of
here; there are good argument for both conventions, but I have ultimately decided to use the
convention, which makes explicit computations a little neater at the cost of making the group actions work in the opposite order.) Thus we naturally see sums over
associated with
-automorphic forms. A little more generally, given a subgroup
of
and a
-automorphic function
of suitable decay, we can form a relative Poincaré series
by
where is any fundamental domain for
, that is to say a subset of
consisting of exactly one representative for each right coset of
. As
is
-automorphic, we see (if
has suitable decay) that
does not depend on the precise choice of fundamental domain, and is
-automorphic. These operations are all compatible with each other, for instance
. A key example of Poincaré series are the Eisenstein series, although there are of course many other Poincaré series one can consider by varying the test function
.
For future reference we record the basic but fundamental unfolding identities
for any function with sufficient decay, and any
-automorphic function
of reasonable growth (e.g.
bounded and compact support, and
bounded, will suffice). Note that
is viewed as a function on
on the left-hand side, and as a
-automorphic function on
on the right-hand side. More generally, one has
whenever are discrete subgroups of
,
is a
-automorphic function with sufficient decay on
, and
is a
-automorphic (and thus also
-automorphic) function of reasonable growth. These identities will allow us to move fairly freely between the three domains
,
, and
in our analysis.
When computing various statistics of a Poincaré series , such as its values
at special points
, or the
quantity
, expressions of interest to analytic number theory naturally emerge. We list three basic examples of this below, discussed somewhat informally in order to highlight the main ideas rather than the technical details.
The first example we will give concerns the problem of estimating the sum
where is the divisor function. This can be rewritten (by factoring
and
) as
which is basically a sum over the full modular group . At this point we will “cheat” a little by moving to the related, but different, sum
This sum is not exactly the same as (8), but will be a little easier to handle, and it is plausible that the methods used to handle this sum can be modified to handle (8). Observe from (2) and some calculation that the distance between and
is given by the formula
and so one can express the above sum as
(the factor of coming from the quotient by
in the projective special linear group); one can express this as
, where
and
is the indicator function of the ball
. Thus we see that expressions such as (7) are related to evaluations of Poincaré series. (In practice, it is much better to use smoothed out versions of indicator functions in order to obtain good control on sums such as (7) or (9), but we gloss over this technical detail here.)
The second example concerns the relative
of the sum (7). Note from multiplicativity that (7) can be written as , which is superficially very similar to (10), but with the key difference that the polynomial
is irreducible over the integers.
As with (7), we may expand (10) as
At first glance this does not look like a sum over a modular group, but one can manipulate this expression into such a form in one of two (closely related) ways. First, observe that any factorisation of
into Gaussian integers
gives rise (upon taking norms) to an identity of the form
, where
and
. Conversely, by using the unique factorisation of the Gaussian integers, every identity of the form
gives rise to a factorisation of the form
, essentially uniquely up to units. Now note that
is of the form
if and only if
, in which case
. Thus we can essentially write the above sum as something like
and one the modular group is now manifest. An equivalent way to see these manipulations is as follows. A triple
of natural numbers with
gives rise to a positive quadratic form
of normalised discriminant
equal to
with integer coefficients (it is natural here to allow
to take integer values rather than just natural number values by essentially doubling the sum). The group
acts on the space of such quadratic forms in a natural fashion (by composing the quadratic form with the inverse
of an element
of
). Because the discriminant
has class number one (this fact is equivalent to the unique factorisation of the gaussian integers, as discussed in this previous post), every form
in this space is equivalent (under the action of some element of
) with the standard quadratic form
. In other words, one has
which (up to a harmless sign) is exactly the representation ,
,
introduced earlier, and leads to the same reformulation of the sum (10) in terms of expressions like (11). Similar considerations also apply if the quadratic polynomial
is replaced by another quadratic, although one has to account for the fact that the class number may now exceed one (so that unique factorisation in the associated quadratic ring of integers breaks down), and in the positive discriminant case the fact that the group of units might be infinite presents another significant technical problem.
Note that has real part
and imaginary part
. Thus (11) is (up to a factor of two) the Poincaré series
as in the preceding example, except that
is now the indicator of the sector
.
Sums involving subgroups of the full modular group, such as , often arise when imposing congruence conditions on sums such as (10), for instance when trying to estimate the expression
when
and
are large. As before, one then soon arrives at the problem of evaluating a Poincaré series at one or more special points, where the series is now over
rather than
.
The third and final example concerns averages of Kloosterman sums
where and
is the inverse of
in the multiplicative group
. It turns out that the
norms of Poincaré series
or
are closely tied to such averages. Consider for instance the quantity
where is a natural number and
is a
-automorphic form that is of the form
for some integer and some test function
, which for sake of discussion we will take to be smooth and compactly supported. Using the unfolding formula (6), we may rewrite (13) as
To compute this, we use the double coset decomposition
where for each ,
are arbitrarily chosen integers such that
. To see this decomposition, observe that every element
in
outside of
can be assumed to have
by applying a sign
, and then using the row and column operations coming from left and right multiplication by
(that is, shifting the top row by an integer multiple of the bottom row, and shifting the right column by an integer multiple of the left column) one can place
in the interval
and
to be any specified integer pair with
. From this we see that
and so from further use of the unfolding formula (5) we may expand (13) as
The first integral is just . The second expression is more interesting. We have
so we can write
as
which on shifting by
simplifies a little to
and then on scaling by
simplifies a little further to
Note that as , we have
modulo
. Comparing the above calculations with (12), we can thus write (13) as
where
is a certain integral involving and a parameter
, but which does not depend explicitly on parameters such as
. Thus we have indeed expressed the
expression (13) in terms of Kloosterman sums. It is possible to invert this analysis and express varius weighted sums of Kloosterman sums in terms of
expressions (possibly involving inner products instead of norms) of Poincaré series, but we will not do so here; see Chapter 16 of Iwaniec and Kowalski for further details.
Traditionally, automorphic forms have been analysed using the spectral theory of the Laplace-Beltrami operator on spaces such as
or
, so that a Poincaré series such as
might be expanded out using inner products of
(or, by the unfolding identities,
) with various generalised eigenfunctions of
(such as cuspidal eigenforms, or Eisenstein series). With this approach, special functions, and specifically the modified Bessel functions
of the second kind, play a prominent role, basically because the
-automorphic functions
for and
non-zero are generalised eigenfunctions of
(with eigenvalue
), and are almost square-integrable on
(the
norm diverges only logarithmically at one end
of the cylinder
, while decaying exponentially fast at the other end
).
However, as discussed in this previous post, the spectral theory of an essentially self-adjoint operator such as is basically equivalent to the theory of various solution operators associated to partial differential equations involving that operator, such as the Helmholtz equation
, the heat equation
, the Schrödinger equation
, or the wave equation
. Thus, one can hope to rephrase many arguments that involve spectral data of
into arguments that instead involve resolvents
, heat kernels
, Schrödinger propagators
, or wave propagators
, or involve the PDE more directly (e.g. applying integration by parts and energy methods to solutions of such PDE). This is certainly done to some extent in the existing literature; resolvents and heat kernels, for instance, are often utilised. In this post, I would like to explore the possibility of reformulating spectral arguments instead using the inhomogeneous wave equation
Actually it will be a bit more convenient to normalise the Laplacian by , and look instead at the automorphic wave equation
This equation somewhat resembles a “Klein-Gordon” type equation, except that the mass is imaginary! This would lead to pathological behaviour were it not for the negative curvature, which in principle creates a spectral gap of that cancels out this factor.
The point is that the wave equation approach gives access to some nice PDE techniques, such as energy methods, Sobolev inequalities and finite speed of propagation, which are somewhat submerged in the spectral framework. The wave equation also interacts well with Poincaré series; if for instance and
are
-automorphic solutions to (15) obeying suitable decay conditions, then their Poincaré series
and
will be
-automorphic solutions to the same equation (15), basically because the Laplace-Beltrami operator commutes with translations. Because of these facts, it is possible to replicate several standard spectral theory arguments in the wave equation framework, without having to deal directly with things like the asymptotics of modified Bessel functions. The wave equation approach to automorphic theory was introduced by Faddeev and Pavlov (using the Lax-Phillips scattering theory), and developed further by by Lax and Phillips, to recover many spectral facts about the Laplacian on modular curves, such as the Weyl law and the Selberg trace formula. Here, I will illustrate this by deriving three basic applications of automorphic methods in a wave equation framework, namely
- Using the Weil bound on Kloosterman sums to derive Selberg’s 3/16 theorem on the least non-trivial eigenvalue for
on
(discussed previously here);
- Conversely, showing that Selberg’s eigenvalue conjecture (improving Selberg’s
bound to the optimal
) implies an optimal bound on (smoothed) sums of Kloosterman sums; and
- Using the same bound to obtain pointwise bounds on Poincaré series similar to the ones discussed above. (Actually, the argument here does not use the wave equation, instead it just uses the Sobolev inequality.)
This post originated from an attempt to finally learn this part of analytic number theory properly, and to see if I could use a PDE-based perspective to understand it better. Ultimately, this is not that dramatic a depature from the standard approach to this subject, but I found it useful to think of things in this fashion, probably due to my existing background in PDE.
I thank Bill Duke and Ben Green for helpful discussions. My primary reference for this theory was Chapters 15, 16, and 21 of Iwaniec and Kowalski.
The wave equation is usually expressed in the form
where is a function of both time
and space
, with
being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain
with some other manifold and replacing the Laplacian
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
. 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
which we can rewrite using integration by parts and the inner product
on
as
A key feature of the wave equation is finite speed of propagation: if, at time (say), the initial position
and initial velocity
are both supported in a ball
, then at any later time
, the position
and velocity
are supported in the larger ball
. This can be seen for instance (formally, at least) by inspecting the exterior energy
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 .
The wave equation is second order in time, but one can turn it into a first order system by working with the pair rather than just the single field
, where
is the velocity field. The system is then
and the conserved energy is now
Finite speed of propagation then tells us that if are both supported on
, then
are supported on
for all
. One also has time reversal symmetry: if
is a solution, then
is a solution also, thus for instance one can establish an analogue of finite speed of propagation for negative times
using this symmetry.
If one has an eigenfunction
of the Laplacian, then we have the explicit solutions
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 , the solution
is given via functional calculus by
and the propagator can be expressed as the average of half-wave operators:
One can view as a minor of the full wave propagator
which is unitary with respect to the energy form (1), and is the fundamental solution to the wave equation in the sense that
Viewing the contraction 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 now lives on the integers
rather than on
, and the spatial domain can be replaced by discrete domains also (such as graphs). Formally, the system is now of the form
where is now an integer,
take values in some Hilbert space (e.g.
functions on a graph
), and
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
where is a small parameter (representing the discretised time step),
now takes values in the integer multiples
of
, and
is the wave propagator operator
or the heat propagator
(the two operators are different, but agree to fourth order in
). One can then formally verify that the wave equation emerges from this rescaled system in the limit
. (Thus,
is not exactly the direct analogue of the Laplacian
, but can be viewed as something like
in the case of small
, or
if we are not rescaling to the small
case. The operator
is sometimes known as the diffusion operator)
Assuming is self-adjoint, solutions to the system (3) formally conserve the energy
This energy is positive semi-definite if is a contraction. We have the same time reversal symmetry as before: if
solves the system (3), then so does
. If one has an eigenfunction
to the operator , then one has an explicit solution
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 by one unit. Suppose we know that
has unit speed in the sense that whenever
is supported in a ball
, then
is supported in the ball
. Then an easy induction shows that if
are supported in
respectively, then
are supported in
.
The fundamental solution to the discretised wave equation (3), in the sense of (2), is given by the formula
where and
are the Chebyshev polynomials of the first and second kind, thus
and
In particular, is now a minor of
, and can also be viewed as an average of
with its inverse
:
As before, is unitary with respect to the energy form (4), so this is another instance of the dilation trick in action. The powers
and
are discrete analogues of the heat propagators
and wave propagators
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
be a (possibly infinite) regular graph, let
, and let
be vertices in
. Then the probability that the simple random walk at
lands at
at time
is at most
, where
is the graph distance.
This general inequality is quite sharp, as one can see using the standard Cayley graph on the integers . Very roughly speaking, it asserts that on a regular graph of reasonably controlled growth (e.g. polynomial growth), random walks of length
concentrate on the ball of radius
or so centred at the origin of the random walk.
Proof: Let be the graph Laplacian, thus
for any , where
is the degree of the regular graph and sum is over the
vertices
that are adjacent to
. This is a contraction of unit speed, and the probability that the random walk at
lands at
at time
is
where are the Dirac deltas at
. Using (5), we can rewrite this as
where we are now using the energy form (4). We can write
where is the simple random walk of length
on the integers, that is to say
where
are independent uniform Bernoulli signs. Thus we wish to show that
By finite speed of propagation, the inner product here vanishes if . For
we can use Cauchy-Schwarz and the unitary nature of
to bound the inner product by
. Thus the left-hand side may be upper bounded by
and the claim now follows from the Chernoff inequality.
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
, and let
be supported on compact sets
respectively. Then
where
is the Euclidean distance between
and
.
Proof: By Fourier inversion one has
for any real , and thus
By finite speed of propagation, the inner product vanishes when
; otherwise, we can use Cauchy-Schwarz and the contractive nature of
to bound this inner product by
. Thus
Bounding by
, we obtain the claim.
Observe that the argument is quite general and can be applied for instance to other Riemannian manifolds than .
Consider the free Schrödinger equation in spatial dimensions, which I will normalise as
where is the unknown field and
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
(if one wishes, one can restrict
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
solves (1), then the pseudoconformal solution
defined by
for can be seen after some computation to also solve (1). (If
has suitable decay at spatial infinity and one chooses a suitable branch cut for
, one can extend
continuously to the
spatial slice, whereupon it becomes essentially the spatial Fourier transform of
, but we will not need this fact for the current discussion.)
An analogous symmetry exists for the free wave equation in spatial dimensions, which I will write as
where is the unknown field. In analogy to pseudoconformal symmetry, we have conformal symmetry: if
solves (3), then the function
, defined in the interior
of the light cone by the formula
also solves (3).
There are also some direct links between the Schrödinger equation in dimensions and the wave equation in
dimensions. This can be easily seen on the spacetime Fourier side: solutions to (1) have spacetime Fourier transform (formally) supported on a
-dimensional hyperboloid, while solutions to (3) have spacetime Fourier transform formally supported on a
-dimensional cone. To link the two, one then observes that the
-dimensional hyperboloid can be viewed as a conic section (i.e. hyperplane slice) of the
-dimensional cone. In physical space, this link is manifested as follows: if
solves (1), then the function
defined by
solves (3). More generally, for any non-zero scaling parameter , the function
defined by
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 (or more generally
) to explicitly connect together the pseudoconformal transformation
and the conformal transformation
. It turns out that this connection is a little bit unusual, with the “obvious” guess (namely, that the embeddings
intertwine
and
) 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.
LLet be a self-adjoint operator on a finite-dimensional Hilbert space
. 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
of eigenvalues and an orthonormal basis
of eigenfunctions such that
for all
. In particular, given any function
on the spectrum
of
, one can then define the linear operator
by the formula
which then gives a functional calculus, in the sense that the map is a
-algebra isometric homomorphism from the algebra
of bounded continuous functions from
to
, to the algebra
of bounded linear operators on
. Thus, for instance, one can define heat operators
for
, Schrödinger operators
for
, resolvents
for
, and (if
is positive) wave operators
for
. 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
positive, they will be contractions). Among other things, this functional calculus can then be used to solve differential equations such as the heat equation
The functional calculus can also be associated to a spectral measure. Indeed, for any vectors , there is a complex measure
on
with the property that
indeed, one can set to be the discrete measure on
defined by the formula
One can also view this complex measure as a coefficient
of a projection-valued measure on
, defined by setting
Finally, one can view as unitarily equivalent to a multiplication operator
on
, where
is the real-valued function
, and the intertwining map
is given by
so that .
It is an important fact in analysis that many of these above assertions extend to operators on an infinite-dimensional Hilbert space , 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
. However, in the theory of partial differential equations, one often needs to apply the spectral theorem to unbounded, densely defined linear operators
, which (initially, at least), are only defined on a dense subspace
of the Hilbert space
. A very typical situation arises when
is the square-integrable functions on some domain or manifold
(which may have a boundary or be otherwise “incomplete”), and
are the smooth compactly supported functions on
, and
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
or to solve equations such as (1), (2), (3), (4).
In order to do this, some necessary conditions on the densely defined operator must be imposed. The most obvious is that of symmetry, which asserts that
for all . In some applications, one also wants to impose positive definiteness, which asserts that
for all . These hypotheses are sufficient in the case when
is bounded, and in particular when
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
are well-defined for any strictly complex
, which by duality implies that the image of
should be dense in
. However, this can fail if one just assumes symmetry, or symmetry and positive definiteness. A well-known example occurs when
is the Hilbert space
,
is the space of test functions, and
is the one-dimensional Laplacian
. Then
is symmetric and positive, but the operator
does not have dense image for any complex
, since
for all test functions , 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 ,
,
is the momentum operator
. Then the resolvent
can be uniquely defined for
in the upper half-plane, but not in the lower half-plane, due to the obstruction
for all test functions (note that the function
lies in
when
is in the lower half-plane). For related reasons, the translation operators
have a problem with either uniqueness or existence (depending on whether
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 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 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
, then
is self-adjoint. Examples of “expected consequences” include:
- Existence of resolvents
(or equivalently, dense image for
);
- Existence of a contractive heat propagator semigroup
(in the positive case);
- Existence of a unitary Schrödinger propagator group
;
- Existence of a unitary wave propagator group
(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 on a smooth Riemannian manifold
(using
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
and
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
(1)
where is the solution and
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
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 for all time.
However, this leaves open the question of the asymptotic behaviour of such solutions in the limit as . 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
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
, 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
(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,
(3)
where and
also have finite energy, so in particular norms such as
cannot decay to zero as
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 decay on the average:
Theorem 1. (Average
decay) If
is a finite energy solution to (1), then
tends to zero as
.
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 ). 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 in which the velocity
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
is a Lipschitz continuous function (with Lipschitz constant at most
).
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 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.
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