Quantum ergodicity in mixed and KAM Hamiltonian systems

Declaration

This thesis is an account of research undertaken between August 2012 and May 2017 at the Mathematical Sciences Institute, The Australian National University, Canberra, Australia.

Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university.

The bulk of the original work in this thesis is contained in Chapters 3 and 6. Chapters 1 and 2 are a summary of known background material and Chapters 4 and 5 closely follow the work in [36] and [37].

 

Seán P Gomes

May, 2016

Acknowledgements

First and foremeost I would like to express my immense appreciation and gratitude to my thesis adviser, Professor Andrew Hassell, you have been a fantastic mentor and role model in these early stages of academic life. Your advice and support on matters both related to mathematics and career progress has been of immeasurable value to me.

I would also like to thank my committee members, Professors Ben Andrews and Xu-Jia Wang and the the HDR convenor Associate Professor Scott Morrison for making my thesis defence an enjoyable experience, and for providing useful feedback.

Thanks also go to Professor Georgi Popov for several enlightening email exchanges elaborating on aspects of his work in the KAM setting, and to Assistant Professor Semyon Dyatlov for a fruitful discussion that motivated a weakening of the “slow torus” condition in the the results of Chapter 6.

Thanks to the Australian federal government, whose funding via the Australian postgraduate award and research training program stipend have made this research possible.

Thanks to all of my friends and colleagues, for always encouraging me to strive for my goals.

A special thanks to Frank, Ronette, Karen and Tanya. The unconditional love, support, and encouragement that a family like ours provides is something that cannot be overstated.

Last but certainly not least, I must acknowledge my partner, Adeline. You have been a boundless source of personal support at the times when it was needed most. Being a thesis-widow is no easy burden, yet you have done everything in your power to support me on this journey and have rode the highs and lows alongside me, making even the most challenging obstacles seem surmountable.

Abstract

In this thesis, we investigate quantum ergodicity for two classes of Hamiltonian systems satisfying intermediate dynamical hypotheses between the well understood extremes of ergodic flow and quantum completely integrable flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian systems.

Hamiltonian systems with mixed phase space decompose into finitely many invariant subsets, only some of which are of ergodic character. It has been conjectured by Percival that the eigenfunctions of the quantisation of this system decompose into associated families of analogous character. The first project in this thesis proves a weak form of this conjecture for a class of dynamical billiards, namely the mushroom billiards of Bunimovich for a full measure subset of a shape parameter .

KAM Hamiltonian systems arise as perturbations of completely integrable Hamiltonian systems. The dynamics of these systems are well understood and have near-integrable character. The classical-quantum correspondence suggests that the quantisation of KAM systems will not have quantum ergodic character. The second project in this thesis proves an initial negative quantum ergodicity result for a class of positive Gevrey perturbations of a Gevrey Hamiltonian that satisfy a mild slow torus condition.

Chapter 1 Introduction to Quantum Ergodicity

The central objective in quantum chaos is to understand how chaotic dynamical assumptions about a classical mechanical system manifest themselves in the behaviour of its quantum mechanical analogue.

A natural setting for studying this correspondence is that of Hamiltonian flow on a compact Riemannian manifold , and this is the setting of the original work in this thesis. In this setting, our dynamical assumption is based on the measure-theoretic concept of ergodicity.

We shall begin in Section 1.1 by summarising the aspects of the Hamiltonian formalism relevant to our work. A more comprehensive treatment can be found in the book [3]. In particular, we shall highlight the opposing concepts of ergodicity and complete integrability.

In Section 1.2 we introduce Schrödinger’s equation, the quantum mechanical counterpart to Hamilton’s equations. We shall then discuss the semiclassical formalism and its relevance to studying the classical-quantum correspondence.

In Section 1.3 we define the quantum mechanical analogue to ergodicity of Hamiltonian flow and survey the major results and conjectures in this field.

In Section 1.4 we discuss the quantisations of Hamiltonian systems that are either completely integrable, or are small perturbations of completely integrable systems. As the Hamiltonian flow in these settings is far from ergodic, intuition suggests that the eigenfunctions for such a system will be far from equidistributed.

1.1 Hamiltonian flow

Suppose that we have a smooth -dimensional compact Riemannian manifold (possibly with boundary). Given a smooth Hamiltonian function which we interpret as an energy, we obtain the Hamiltonian flow generated by Hamilton’s equations

(1.1.1)

with coordinates corresponding to the cotangent vector . In this work we shall assume that our Hamiltonians are such that the flow does not blow up in finite time. We denote the Hamiltonian vector field given by (1.1.1) as .

The primary Hamiltonians of interest in this thesis will be Schrödinger Hamiltonians of the form

(1.1.2)

however the results of Chapter 6 could be generalised to symbols of more general classes of self-adjoint pseudodifferential operators. In Chapter 5, the function is a compactly supported symbol in the Gevrey class from Definition 2.2.5 with self-adjoint quantisation. In the special case of as in Chapter 3, this Hamiltonian system is referred to as billiards on . The trajectories of billiard flow can be identified with the geodesics of under the canonical isomorphism between the tangent and co-tangent bundles of Riemannian manifolds.

A major advantage of the Hamiltonian formulation of mechanics over the Newtonian and Lagrangian formulations is the duality of the variables and , best highlighted through the lens of symplectic geometry.

Definition 1.1.1 (Symplectic Form).

A symplectic form on a smooth manifold is a closed non-degenerate differential -form.

Definition 1.1.2 (Symplectomorphism).

A symplectomorphism between two symplectic manifolds and is a diffeomorphism such that .

Definition 1.1.3.

An exact symplectic form on a smooth manifold is an exact non-degenerate differential -form.

Definition 1.1.4 (Exact Symplectomorphism).

A symplectomorphism between two exact symplectic manifolds and is a diffeomorphism such that is an exact -form.

Given a smooth function on a smooth manifold equipped with a symplectic form , we can obtain a Hamiltonian vector field on defined implicitly by

(1.1.3)

In the case there is a canonical choice of symplectic form

(1.1.4)

and the vector field generates the flow given by Hamilton’s equations (1.1.1).

In fact for a general symplectic manifold , Darboux’s theorem asserts that local coordinates can be chosen such that (1.1.4) holds. Such coordinates are said to be canonical coordinates.

Writing in a canonical coordinate system on allows us to write

(1.1.5)

where

(1.1.6)

Indeed we say that a matrix is symplectic if

(1.1.7)

and a diffeomorphism on a symplectic manifolds is a symplectomorphism if and only if its Jacobian with respect to canonical coordinates is a symplectic matrix.

Symplectomorphisms are the natural class of coordinate transformations of a Hamiltonian system to work with as they preserve Hamilton’s equations.

Proposition 1.1.5.

If is a symplectomorphism, and is a smooth Hamiltonian then the transformed Hamiltonian generates a Hamiltonian flow in the coordinates given by

(1.1.8)

The Hamiltonian vector field on is the pullback of the Hamiltonian vector field on .

Hamiltonian flows give rise to symplectomorphisms in a natural way.

Proposition 1.1.6.

If is a Hamiltonian on the symplectic manifold , then the Hamiltonian flow on is a one-parameter family of symplectomorphisms.

Another useful method of constructing symplectomorphisms is through the use of a generating function.

Proposition 1.1.7.

If is such that the Hessian is non-singular, then a solution to the implicit equation

(1.1.9)

is symplectic on its domain.

For the particularly simple symplectic manifold , with , we can make this construction global, provided that is -periodic in . The resulting symplectomorphism is then an exact symplectomorphism.

Provided that all are regular values for the Hamiltonian , the canonical symplectic form on determines a family of measures on each of the energy hypersurfaces

(1.1.10)

defined implicitly by

(1.1.11)

for . In the special case of of and , upon normalisation we obtain the Liouville measure on .

The measures allow us to study the ergodic properties of the Hamiltonian flow .

Definition 1.1.8.

If denotes a measure preserving flow on a finite measure space , we say that is ergodic if

(1.1.12)

for -almost all .

That is to say, a flow is ergodic if and only if almost all trajectories equidistribute in the measure space. An equivalent characterisation can be made in terms of flow invariant subsets.

Proposition 1.1.9.

A measure preserving flow on a finite measure space is ergodic if and only if the only -invariant measurable sets are of full measure or null measure.

In particular, we say that the Hamiltonian flow generated by is ergodic on the energy surface if satisfies 1.1.8 on with respect to the measure

Two particularly famous examples of ergodic Hamiltonian systems are the Bunimovich stadium billiard [6] and the Sinai billiard [42].

Figure 1.1: An ergodic trajectory on the Bunimovich stadium billiard. Image from [45].

From Definition 1.1.8, we can see that for ergodic flows, the time average of a smooth classical observable tends to its space average. That is, we have

(1.1.13)

for -almost all .

A strictly stronger property of a flow is the mixing property which asserts that for smooth classical observables we have

(1.1.14)

as . The strong assumption of Anosov flow leads to (1.1.14) with an exponential rate of convergence. Thus, manifolds with non-positive sectional curvature also give rise to ergodic billiards. In this thesis we shall not discuss the stronger property of (1.1.14), and restrict ourself to the study of ergodicity.

In order to study the Hamiltonian evolution of functions on our phase space , we define the Poisson bracket.

Definition 1.1.10 (Poisson Bracket).

If , we define

(1.1.15)

An immediate consequence of the chain rule is that if is a trajectory of the Hamiltonian flow, then for a smooth function , we have

(1.1.16)

Motivated by this calculation we can define invariants of our flow.

Definition 1.1.11.

An invariant or first integral of the Hamiltonian flow is a smooth function such that .

The Hamiltonian itself is of course an invariant of the flow that it generates. Hence Hamiltonian flow is constrained to energy shells . Often we can find additional invariants that are mathematical manifestations of conservation laws from physics, such as that of angular momentum. Symmetries in a system lead to an abundance of such flow invariants, as follows from Noether’s theorem (See Chapter 4, Section 20 of [3]).

We say that a collection of flow invariants are in involution if their pairwise Poisson brackets vanish and we say that they are independent if their differentials are linearly independent.

Definition 1.1.12.

If an invariant subset of Hamiltonian flow admits independent invariants that are in involution, we say that the corresponding Hamiltonian system is completely integrable.

In a completely integrable Hamiltonian system, trajectories are constrained to -dimensional submanifolds and are thus far from equidistributed on the -dimensional energy surfaces. The Liouville–Arnold theorem from classical mechanics asserts that for completely integrable systems, we can find a neighbourhood of an arbitrary invariant manifold and a symplectomorphism for some such that the transformed Hamiltonian is independent of . Thus the invariant manifolds are diffeomorphic to -dimensional tori. Moreover, the Hamiltonian flow is quasiperiodic, with trajectories given by

(1.1.17)

in the -coordinates, referred to as action-angle variables. A construction of these coordinates can be found in Section 50 of [3].

We note that the invariant tori of a completely integrable Hamiltonian system are Lagrangian, that is the restriction of the symplectic form to any of the invariant tori vanishes.

An example of a completely integrable Hamiltonian system is geodesic billiards on an ellipsoid, pictured in Figure 1.2.

Figure 1.2: Geodesic flow on an ellipsoid. Image generated using GeographicLib.

We can extend the above discussion to manifolds with boundary by extending Hamiltonian flow by reflection at non-tangential boundary collisions. We shall postpone this somewhat technical discussion until we require it in Chapter 3.

1.2 Quantum dynamics

The quantum mechanical analogue of the system (3.1.1) is the evolution of a wave-function governed by Schrödinger’s equation.

(1.2.1)

where

(1.2.2)

is Planck’s constant and

(1.2.3)

is the Laplace–Beltrami operator with the positive sign convention.

The Bohr correspondence principle asserts that a classical Hamiltonian system is in a vague sense the macroscopic or high-energy limit of the corresponding quantum dynamical system. By scaling the units in (1.2.1), we may instead consider to be a small parameter in our problem. This is known as the semiclassical formalism. In the semiclassical formalism we replace differential operators with semiclassical differential operators to account for the scaling factor. By the Bohr correspondence principle, we then expect chaotic behaviour of the classical system to be manifest in the quantum system in the limit .

In solving (1.2.1), we can expand in terms of the basis of comprised by eigenfunctions of , and so localisation of quantum dynamics can be understood by the study of the localisation of these eigenfunctions.

For Hamiltonian systems that are more general than billiards, such as the Schrödinger type Hamiltonians in (1.1.2), we consider eigenfunction of the semiclassical Schrödinger operator

(1.2.4)

obtained by formally applying the Hamiltonian function to the operators and .

One can then ask how the localisation properties of the Hamiltonian flow on are reflected by the spectral theory of the associated Schrodinger operator in the limit .

Typically it is not possible to find exact, or even approximate expressions for chaotic eigenfunctions. Nevertheless, the machinery of microlocal analysis allows us to rigorously prove state and prove phase space equidistribution properties. The key tools here are pseudodifferential operators and Fourier integral operators, which correspond to quantisations of classical observables and symplectomorphisms respectively.

1.3 Quantum ergodicity

Under the assumption of ergodic Hamiltonian flow, one can make a remarkable statement of phase space equidistribution of the steady-state solutions to the corresponding Schrödinger’s equation (3.1.5).

A primitive version of this theorem asserts that the sequence of probability measures on must have a full density subsequence which tends to the uniform measure.

The stronger statement of phase space equidistribution requires some additional machinery to state, such as the pseudodifferential calculus which we introduce in Chapter 2.

Theorem 1.3.1 (Quantum Ergodicity).

If the Hamiltonian on the smooth compact boundaryless Riemannian manifold generates ergodic flow on the regular energy band for a smooth real potential that is bounded below, then there exists a family of subsets of eigenvalues of such that

(1.3.1)

and

(1.3.2)

uniformly for for any zero-th order semiclassical pseudodifferential operator with the property that

(1.3.3)

is independent of .

A semiclassical proof of this theorem can be found in [50], whilst the initial result goes back to [43],[48],[9]. For billiards on manifolds with boundary, there are additional technical considerations even from the purely dynamical perspective. Nevertheless, the quantum ergodicity theorem generalises to this setting [49],[17].

We can also define a notion of quantum ergodicity localised to an individual energy surface that is motivated by the results of [23]. We will make use of this definition in the proof of the negative quantum ergodicity result Theorem 6.1.3, in Chapter 6.

Suppose the semiclassical pseudodifferential operator on the smooth manifold has principal symbol and has purely point spectrum, with eigenpairs in increasing order.

If is a regular value of with nonempty preimage, then we can define quantum ergodicity localised to the energy surface as follows.

Definition 1.3.2.

We say that is quantum ergodic at energy if for each sufficiently small , there exists a family such that

(1.3.4)

and

(1.3.5)

uniformly for for any zero-th order semiclassical pseudodifferential operator .

Remark 1.3.3.

We could replace the width energy windows in Definition 1.3.2 with for . The size energy windows in Definition 1.3.2 are the smallest windows in which quantum ergodicity results are possible because the Weyl asymptotics break down in smaller energy windows.

An alternate formulation of quantum ergodicity can be made in the language of semiclassical measures. We shall state this version of quantum ergodicity in the special case , as we only make use of it in this setting of billiards in Chapter 3.

To each subsequence of , we can associate at least one non-negative Radon measure on which provides a notion of phase space concentration in the semiclassical limit. We say that the eigenfunction subsequence has unique semiclassical measure if

(1.3.6)

for each semiclassical pseudodifferential operator with principal symbol compactly supported supported away from the boundary of . In Chapter 5 of [50], the existence and basic properties of semiclassical measures are established using the calculus of semiclassical pseudodifferential operators (see also [17]).

A billiard can then be said to be quantum ergodic if there is a full density subsequence of eigenfunctions such that the the Liouville measure on is the unique semiclassical measure associated to the sequence . This statement can be interpreted as saying that the sequence of eigenfunctions equidistributes in phase space with the possible exception of a sparse subsequence.

Under the stronger assumption of Anosov flow, it is conjectured that the full sequence of eigenfunctions equidistributes in the sense of Theorem 1.3.1. This is known as the quantum unique ergodicity conjecture.

The prize-winning work of Lindenstrauss [29] verified this conjecture in certain arithmetic cases where we work with the Hecke joint eigenfunctions. The study of quantum ergodicity where we have this additional arithmetic structure is known as arithmetic QE. Sarnak [40] has written a survey on the recent developments in this field.

On the other hand, it is known that quantum ergodicity is strictly weaker than quantum unique ergodicity. Indeed, Hassell [22] showed that on the Bunimovich stadium there exist semiclassical measures that have positive mass on the union of the bouncing ball trajectories in phase space.

1.4 Negative results

In the extreme case of quantum unique ergodicity, there is a unique semiclassical measure, which is the Liouville measure. It is natural to ask what we can say about the semiclassical measures associated with sequences of eigenfunctions of Hamiltonian systems that are not quantum uniquely ergodic.

For the Bunimovich stadium, the quantum ergodicity theorem implies that any non-uniform limit can only arise from a density-zero subsequence.

Whilst Burq–Zworski [8] showed that concentration in a strict subrectangle is not possible, numerical evidence suggests that there could well be a sparse sequence of eigenfunctions with semiclassical limit supported in the rectangle itself. Rigorous proof of this phenomenon remains an open problem, with the most notable progress being Hassell’s proof that a semiclassical measure exists with positive mass on the union of bouncing ball trajectories [22].

Figure 1.3: An apparent “bouncing ball” eigenfunction in the quarter stadium corresponding to the eigenvalue . Image courtesy of Dr. Barnett.

On the other hand, if a Hamiltonian system is assumed to be completely integrable, any trajectory is constrained to a single invariant torus corresponding to the intersection of the level sets of the conserved quantities. The intuition stemming from the classical-quantum correspondence suggests that this extreme concentration of trajectories should manifest itself in a statement about the concentration of eigenfunctions onto the Lagrangian tori.

Such a result is proven in [46] for systems satisfying a stronger notion of quantum integrability and tori satisfying a certain nonresonance condition, however rigorous results in the general setting of complete integrability seem to be elusive.

At this point we introduce the notion of approximate eigenfunctions, or quasimodes.

Definition 1.4.1.

Given a semiclassical pseudodifferential operator , a quasimode is a family of functions such that

(1.4.1)

for some and some real , referred to as the quasi-eigenvalue.

Remark 1.4.2.

As a consequence of the semiclassical rescaling, it should be noted that for the semiclassical Laplacian correspond to quasimodes of the Laplacian .

We can of course replace the with a stronger bound in this definition. The uses of quasimodes are plentiful. Most results about eigenfunctions apply just as well to quasimodes, and it is easier to construct quasimodes than exact eigenfunctions. In fact we can often construct quasimodes with desirable localisation properties, as they are better behaved with respect to taking cutoffs than exact eigenfunctions which are generally destroyed. The

In [10], Colin de Verdière established that for completely integrable Hamiltonian systems, there exist quasimodes with exponentially small error term that localise onto the certain individual invariant Langrangian tori. This result relies on the construction of a quantum Birkhoff normal form.

1.5 Mixed and KAM systems

Between the extremes of completely integrable Hamiltonian dynamics and ergodic Hamiltonian dynamics, results are rather sparse. The original work in this thesis explores two intermediate classes of Hamiltonian dynamical systems for which questions of quantum ergodicity are tractable.

The two main original results in this thesis, Theorem 3.5.4 and Theorem 6.1.3, both make use of known quasimode constructions and perturbation arguments in order to prove eigenfunction localisation statements in the cases of mixed billiards and KAM system respectively. We now outline these two classes of Hamiltonian systems.

Mixed billards

If a dynamical billiard can be separated into multiple invariant subsets, only some of which are ergodic, it is said that the billiard is mixed. In this case, it is conjectured that we can divide the sequence of eigenfunctions into corresponding families, with the eigenfunctions corresponding to an ergodic invariant subset satisfying a suitable equidistribution property. For the sake of simplicity, we shall state the conjecture in the case of billiards with exactly two invariant subsets, one ergodic and one completely integrable.

Conjecture 1.5.1 (Percival’s Conjecture).

For every compact Riemannian manifold such that is the disjoint union of two invariant subsets , with ergodic and completely integrable, we can find two subsets such that

  1. has density ;

  2. equidistributes in the ergodic region ;

  3. Each semiclassical measure associated to the subset is supported in the completely integrable region ;

  4. The density of is equal to .

Numerical evidence [4] strongly supports this conjecture, but no rigorous proofs have been discovered, even for concrete examples.

A weaker version of Percival’s conjecture is formulated by slightly relaxing the density requirements of the subsets and .

Conjecture 1.5.2 (Weak Percival’s Conjecture).

For every compact Riemannian manifold such that is the disjoint union of two invariant subsets , with ergodic and completely integrable, we can find two subsets such that

  1. has upper density

  2. equidistributes in the ergodic region

  3. Each semiclassical measure associated to the subset is supported in the completely integrable region

  4. The upper densities of and are equal to and respectively.

In Chapter 3, I provide the first verification of the weak Percival’s conjecture for a family of “mushroom” billiards, defined in Section 3.1. The main result is

Theorem 1.5.3.

Conjecture 1.5.2 holds for the mushroom billiard for any fixed inner and outer radii, and almost all “stalk lengths” .

KAM Hamiltonian systems

A particularly interesting class of Hamiltonian systems arise if we apply a small perturbation to a completely integrable real analytic Hamiltonian in action-angle coordinates.

(1.5.1)

Motivated by the geometry of completely integrable systems, where our phase space is foliated by the Lagrangian tori

(1.5.2)

where , it is natural to ask whether there are any such invariant Lagrangian tori that survive the perturbation. This problem is one of real physical significane, as one application is the study of celestial stability by viewing the dynamics of the solar system as a small perturbation of the completely integrable system that results from neglecting forces between pairs of planets. This perturbation is of course “small” because of the considerably greater mass of the sun compared to the planets.

The initial significant breakthrough in this problem was due to Kolmogorov [28], with the conclusion that although a dense set of tori is indeed generally destroyed by such a perturbation, a large measure collection of the invariant tori survive, precisely those whose frequency of quasiperiodic flow (1.1.17) satisfy the Diophantine condition

(1.5.3)

for all nonzero and fixed and . The tori satisfying this Diophantine condition are said to be nonresonant.

The field of KAM theory developed from this problem as a broad class of techniques applicable to perturbation problems in classical mechanics, founded by Kolmogorov, Arnold and Moser.

More recent work by Popov [36] proved a version of the KAM theorem for Hamiltonian systems in the Gevrey regularity class, with the purpose of constructing a Birkhoff normal form. This led to a quantum Birkhoff normal form construction, and a proof of the existence of quasimodes with exponentially small error localising onto the nonresonant tori in [37].

The details of Popov’s construction are summarised in Chapter 4 and Chapter 5 for families of Hamiltonians of the form

(1.5.4)

for smooth real-valued symbols in a suitable Gevrey class in the notation of (2.2.5).

In Chapter 6, we prove the following main result. The formal statement is Theorem 6.1.3.

Theorem 1.5.4.

Suppose is a compact boundaryless Gevrey smooth Riemannian manifold, and the perturbation is such that is a positive operator and there exists a slow torus in the energy band .

Then there exists such that for almost all the quantisation of is non-quantum ergodic over the energy surface for a positive Lebesgue measure subset of energies .

A slow torus, defined formally in 6.1.1, is an invariant nonresonant Lagrangian torus in the energy surface such that the average of over is strictly smaller than the average of over . The assumption of the existence of such a torus is a mild one, and will typically be satisfied by perturbations whose symbols are nonconstant on energy surfaces.

Chapter 2 Semiclassical Analysis

In this chapter, we briefly collect some of the necessary machinery of the semiclassical pseudodifferential calculus necessary for the results in the remainder of this thesis. The Fourier transform on from classical harmonic analysis allows us to pass from the -dimensional position space to the -dimensional frequency space. Pseudodifferential operators allow us to formulate and prove statements in the full -dimensional phase space, and generalise the procedure of using a Fourier multiplier as a frequency cutoff. Some standard references for the classical pseudodifferential calculus include [18] [41] [25]. Our presentation shall be in the semiclassical formalism, for which an extensive account can be found in [50] and [12].

A key application of the pseudodifferential calculus to spectral theory is Weyl’s law, which provides an asymptotic for eigenvalues of a Schrödinger operator.

2.1 Semiclassical pseudodifferential operators

We begin by presenting the semiclassical pseudodifferential calculus on .

The semiclassical pseudodifferential calculus provides a correspondence between classical observables (smooth functions on the phase space ) and quantum observables (integral operators on position space ).

The classical observables in this correspondence are traditionally referred to as symbols, and estimates on their derivatives are required to obtain desirable mapping properties for their quantisations.

For such operators, we can define semiclassical pseudodifferential operators on .

(2.1.1)

The quantisation (2.1.1) is referred to as the standard quantisation. It is sometimes more convenient to work with a formally self-adjoint operator however. This motivates the definition of the Weyl quantisation.

(2.1.2)

which is a formally self-adjoint operator if is real.

These formally defined integral operators are clearly convergent if and are of Schwartz class, but is otherwise understood in the sense of oscillatory integrals ([50] Theorem 3.8). In this fashion, the Kohn–Nirenberg symbol class leads to a class of semiclassical pseudodifferential operators bounded on semiclassical Sobolev spaces.

The standard class of -th order Kohn–Nirenberg symbols on is given by:

(2.1.3)
Remark 2.1.1.

It is also sometimes useful to consider classes , where the right-hand side of (2.1.3) is multiplied by with each differentiation, but we shall not require these symbol classes.

This is a sufficiently broad symbol class for most applications, and includes semiclassical differential operators

(2.1.4)

as a special case by quantising polynomials in .

The index in Definition 2.1.3 corresponds to the mapping properties of the associated pseudodifferential operator.

Proposition 2.1.2.

If , then

(2.1.5)

is a bounded operator for any , where denotes the Sobolev space of order . In particular, zero-th order semiclassical pseudodifferential operators are bounded on , and negative order semiclassical pseudodifferential operators are compact on .

In practice, symbols of semiclassical pseudodifferential operators are often constructed using formal power series in . Indeed, if and for each , we introduce the notation

(2.1.6)

to mean that

(2.1.7)

for each uniformly in some interval .

The key point is that for an arbitrary formal series with symbols , we can find a -equivalent symbol .

Proposition 2.1.3.

Given an arbitrary sequence of symbols , there exists a symbol satisfying (2.1.6). We call the Borel resummation of the formal series .

A proof of Proposition 2.1.3 can be found in Theorem 4.15 of [50].

We refer the the leading term in (2.1.6) as the principal symbol of , and write . This is of course only well-defined modulo .

From repeated integration by parts in (2.1.1), we see that if the , then the function is of size for any . We denote such a size estimate by , and note that these terms can be regarded as negligible in the semiclassical limit .

At this point we introduce the notion of a semiclassical wavefront set for functions.

Definition 2.1.4.

Suppose is a collection of smooth functions on for . Then the semiclassical wavefront set is defined as follows. if there exists a with such that we have

(2.1.8)

for any .

Such a definition is possible in considerably more general classes of distributions (See Section 8.4 of [50]), but we shall not require it in this generality.

Remark 2.1.5.

In fact, it suffices to prove that for any and a single sequence .

A crucial formula in the pseudodifferential calculus is the composition formula, which assets that if and , then the composition

(2.1.9)

is a semiclassical pseudodifferential operator of order , and its symbol is given by

(2.1.10)

as is shown in Theorem 9.5 of [50].

Expanding the symbols and in (2.1.10) as semiclassical series yields the following

Proposition 2.1.6.

Given two symbols and , their composition as the Borel resummation of

(2.1.11)

where

(2.1.12)

A key feature of the pseudodifferential calculus that immediately lends itself to PDE applications is that of the invertibility of elliptic operators.

Proposition 2.1.7.

If we have a symbol with

(2.1.13)

for some , then there exists a symbol with

(2.1.14)

The proof of Proposition 2.1.7 is an application of (2.1.10) and can be found in Proposition 2.6.10 of [30].

Importantly, for an arbitrary diffeomorphism from with open, the symbols classes and invariant under the pullback of the lift of to a symplectomorphism . This invariance allows for the construction of semiclassical pseudodifferential operators on compact manifolds, as is done in ([50] Chapter 14).

Definition 2.1.8.

We write to denote the class of -th order Kohn-Nirenberg symbols on a compact manifold and we write to denote the class of Kohn-Nirenberg symbols of differential order and semiclassical order .

Definition 2.1.9.

We write to denote the class of -th order semiclassical pseudodifferential operators on in the sense of ([50] Chapter 14). We write to denote the class of semiclassical pseudodifferential operators of differential order and semiclassical order .

One significant difference between the calculus on compact manifolds and on Euclidean space however, is that the symbol of a semiclassical pseudodifferential operator is only invariantly defined modulo .

One can also define semiclassical pseudodifferential operators on the space of half-densities on a compact manifold .

Definition 2.1.10.

A half-density on an dimensional vector space is a map such that

(2.1.15)

for any is a linear transformation on . We denote the space of half-densities on by .

Definition 2.1.11.

The space of smooth half-densities on a compact Riemannian manifold is given by the collection of maps such that for each , and for any smooth vector fields .

Since half-densities are given in local coordinates on a Riemannian manifold by

(2.1.16)

where is the Riemannian volume form, we can identify half-densities with functions in this setting, however note that their pullbacks as half-densities will involve a Jacobian factor.

Thus we can locally define semiclassical pseudodifferential operators on half-densities by setting

(2.1.17)

and they can be defined globally in a similar fashion to semiclassical pseudodifferential operators acting on functions. (See Section 14.2.5 of [50]).

An advantage of working with half-densities is that principal symbols of semiclassical pseudodiffential operators on half-densities are invariantly in , and subprincipal symbols of operators are thus invariantly defined. (See Section 1.3 of [21] for a further discussion of this invariance).

In Section 5.2, we work with semiclassical Fourier integral operators, which are generalisations of semiclassical pseudodifferential operators obtained locally by replacing the phase function in the oscillatory integral expression (2.1.1) with more general phase functions .

The kernels of such operators are then special cases of Fourier integrals

(2.1.18)

with . For such a phase function, we can associate a Lagrangian submanifold of given by

(2.1.19)

Indeed, stationary phase asymptotics show that as in [24].

Defining a canonical relation to be a relation with flipped graph

(2.1.20)

a Lagrangian submanifold of , we can then define a Fourier integral operator associated to a given relation to be a finite sum of Fourier integrals associated to .

The global theory of Fourier integrals is complicated by the fact that different phase functions can parametrise the same Lagrangian manifold locally, yet for different a different symbol will be required in (2.1.18) in order to represent the same Fourier integral . In order to invariantly define the notion of a principal symbol for a Fourier integral operator, it must be defined as an object on a certain line bundle over the Lagrandigan submanifold , known as the Maslov bundle.

A thorough account of Fourier integral operators can be found in the seminal paper [24] in the classical setting, and in [21] in the semiclassical setting. We shall summarise the relevant details in our exposition of Popov’s construction of the quantum Birkhoff normal form for KAM Hamiltonians [35][37] in Section 5.2.

2.2 Gevrey class symbols

Our application of the semiclassical pseudodifferential calculus in Chapter 5 involves working with Gevrey class symbols. We outline the relevant differences from the theory in Section 2.1 here.

We suppose is a bounded domain in , and take or a bounded domain in . We fix the parameters and , and denote the triple by .

Definition 2.2.1.

A formal Gevrey symbol on is a formal sum

(2.2.1)

where the are all supported in a fixed compact set and there exists a such that

(2.2.2)
Definition 2.2.2.

A realisation of the formal symbol (2.2.1) is a function for with

(2.2.3)
Lemma 2.2.3.

Given a formal symbol (2.2.1), one choice of realisation is

(2.2.4)

where depends only on and .

Definition 2.2.4.

We define the residual class of symbols as the collection of realisations of the zero formal symbol.

Definition 2.2.5.

We write if . It then follows that any two realisations of the same formal symbol are -equivalent. We denote the set of equivalence classes by .

An important feature of the Gevrey symbol calculus is that the symbol class is closed under composition.

We can now discuss the pseudodifferential operators corresponding to these symbols.

Definition 2.2.6.

To each symbol , we associate a semiclassical pseudodifferential operator defined by

(2.2.5)

for .

The above construction is well defined modulo , as for any we have

(2.2.6)

for some constant .

Remark 2.2.7.

The exponential decay of residual symbols is a key strengthening that comes from working in a Gevrey symbol class.

The operations of symbol composition and conjugation then correspond to composing operators and taking adjoints respectively. Moreover, if , then -smooth changes of variable preserve the symbol class of . This coordinate invariance allows us to extend the Gevrey pseudodifferential calculus to compact Gevrey manifolds.

At this point we introduce the notion of a microsupport in the Gevrey sense.

Definition 2.2.8.

Suppose is a collection of smooth functions on the -manifold for . Then the microsupport is defined as follows.

if there exists a product of compact sets with inside a single coordinate chart and there exists a such that for any we have

(2.2.7)

uniformly in .

It follows from stationary phase that if a symbol is in a neighbourhood of a point , then the point lies outside the microsupport of the distribution kernel of .

2.3 Weyl law

An application of the semiclassical pseudodifferential calculus that is particularly important to us is the semiclassical Weyl law, which provides asymptotics for the counting functions of eigenvalues for suitable semiclassical pseudodifferential operators in fixed energy bands or shrinking energy bands with as .

We consider semiclassical pseudodifferential operators of the form

(2.3.1)

on a compact Riemannian manifold , where is real valued.

For each fixed , the operator is a self-adjoint operator

(2.3.2)

with compact inverse, where is the Sobolev space of order .

Basic spectral theory then tells us that the spectrum of is real and discrete, consisting of a countable orthonormal basis of eigenpairs , with as .

Weyl’s law is then the statement that

(2.3.3)

where denotes the symplectic measure on .

A standard proof relies on a trace formula for a Schwartz class functional calculus for semiclassical pseudodifferential operators. If , then we can define

(2.3.4)

The rapid decay of in fact implies that is a semiclassical pseudodifferential operator in the class

(2.3.5)

In fact it can be shown that is a trace-class operator on , with principal symbol

(2.3.6)

and trace

(2.3.7)

The equation (2.3.3) then follows from (2.3.7) and regularisation of the indicator functions. Full details can be found in Chapter 14 of [50].

Remark 2.3.1.

In the special case of the Laplace–Beltrami operator , rescaling yields the classical Weyl law which gives counting asymptotics for the Laplacian eigenvalues.

The Weyl law can also be localised in phase space by a semiclassical pseudodifferential operator. That is, for any , we have

(2.3.8)

as . The proof of this generalisation again makes use of (2.3.7), and can be found in Section 15.3 of [50].

A version of the semiclassical Weyl law was proven by Petkov and Robert [32] for boundaryless manifolds that is localised to sized energy bands. That is, for regular values of the Hamiltonian , we have

(2.3.9)
Remark 2.3.2.

This result requires the dynamical assumption that the set of trapped trajectories is of measure zero. Without this assumption, we only obtain a uniform upper bound for .

Chapter 3 Quantum Ergodicity in Mixed Systems

3.1 Introduction

In this chapter, we turn our attention to mixed billiards. We begin by recalling the relevant definitions.

If is a compact boundaryless Riemannian manifold, we define dynamical billiards on to be the Hamiltonian flow on the cotangent bundle of the manifold given by Hamilton’s equations