Construction of an approximate solution of the Wigner equation by uniformization of WKB functions

Construction of an approximate solution
of the Wigner equation
by uniformization of WKB functions

Konstantina-Stavroula Giannopoulou

November 2015


Mathematics & Applied Mathematics


Prof. G.N. Makrakis


A. Athanassoulis, University of Leicester
S. Filippas, University of Crete
G. Karali, University of Crete
T. Katsaounis, King Abdullah University of Science and Technology (KAUST)
G. Kossioris, University of Crete
P. Rozakis, University of Crete








Επιβλέπων Καθηγητής: Γεώγιος Ν. Μακράκης






Abstract The Wigner equation is a non-local, evolution equation in phase-space. It describes the evolution of the Weyl symbol of the density operator which, in general, is governed by the Liouville-von Neumann equation of quantum mechanics. For pure quantum states, the Wigner equation is an equivalent reformulation of the standard quantum-mechanical Schrödinger equation, and it could be also derived in an operational way by considering the Wigner transform of the quantum wave function, without using the Weyl calculus.

In this thesis, we construct an approximate solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. Although the WKB approximations of Schrödinger eigenfunctions have non-physical singularities at the turning points of the classical Hamiltonian, the phase space eigenfunctions provide bounded, and correctly scaled, wave amplitudes when they are projected back onto the configuration space (uniformization).

Therefore, the approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy.

The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well. In principle, this construction could be extended to higher dimensions using canonical forms of the Hamiltonian functions and employing the symplectic covariance inherited by the Weyl representation into the Wigner equation.


Schrödinger equation, Wigner equation, semiclassical limit, geometric optics, caustics, Weyl quantization, Weyl operators, Wigner transform, uniform stationary phase method

AMS (MOS) subject classification: 78A05, 81Q20, 53D55, 81S30, 34E05, 58K55

This thesis has been partially supported by the “Maria Michail Manasaki” Bequest Fellowships.


Περίληψη Η εξίσωση Wigner (Wigner equation) είναι μια μη-τοπική (non-local) εξίσωση εξέλιξης στον χώρο των φάσεων (phase space). Περιγράφει την εξέλιξη του Weyl συμβόλου του τελεστή πυκνότητας (density operator) ο οποίος, εν γένει, διέπεται από την εξίσωση Liouville-von Neumann της κβαντομηχανικής. Για καθαρές κβαντικές καταστάσεις (pure states), η εξίσωση Wigner είναι μια ισοδύναμη αναδιατύπωση της βασικής εξίσωσης της κβαντικής μηχανικής, της εξίσωσης Schrödinger και θα μπορούσε επίσης να παραχθεί με έναν τελεστικό τρόπο, θεωρώντας τον μετασχηματισμό Wigner (Wigner transform) της κυματοσυνάρτησης, χωρίς τη χρήση του λογισμού Weyl (Weyl calculus).

Σε αυτήν τη διατριβή, κατασκευάζουμε μια προσεγγιστική λύση της εξίσωσης Wigner εκφρασμένη σε όρους συναρτήσεων Airy (Airy function), οι οποίες συγκεντρώνονται ημικλασικά πάνω σε κάποιες Λαγκραντζιανές καμπύλες (Lagrangian curves) στον διδιάστατο χώρο των φάσεων. Οι καμπύλες αυτές ορίζονται από τις ιδιοτιμές και την Χαμιλτωνιανή συνάρτηση (Hamiltonian function) του συσχετιζόμενου μονοδιάστατου τελεστή Schrödinger, και παίζουν κρίσιμο ρόλο στον μηχανισμό της κβαντικής αλληλεπίδρασης (quantum interference mechanism) στον χώρο των φάσεων. Δεχόμαστε ότι το δυναμικό του τελεστή Schrödinger είναι ένα μονό πηγάδι δυναμικού (single-well potential) τέτοιο ώστε το φάσμα (spectrum) να είναι διακριτό.

Η κατασκευή ξεκινάει από ένα ανάπτυγμα ιδιοσυναρτήσεων (eigenfunction series expansion) της λύσης, το οποίο παράγεται εδώ με έναν συστηματικό τρόπο για πρώτη φορά, συνδυάζοντας την στοιχειώδη τεχνική του χωρισμού μεταβλητών με φασματικά αποτελέσματα για τον εκθετικό τελεστή Moyal (Moyal star exponential operator). Οι ιδιοσυναρτήσεις της εξίσωσης Wigner είναι οι μετασχηματισμοί Wigner των ιδιοσυναρτήσεων του τελεστή Schrödinger και προσεγγίζονται με όρους της συνάρτησης Airy, από μια προσέγγιση ομοιόμορφης στάσιμης φάσης των μετασχηματισμών Wigner των αναπτυγμάτων WKB των ιδιοσυναρτήσεων του τελεστή Schrödinger. Μολονότι οι προσεγγίσεις WKB των ιδιοσυναρτήσεων Schrödinger έχουν μη-φυσικές ιδιομορφίες (non-physical singularities) στα σημεία καμπής (turning points) της κλασικής Χαμιλτωνιανής (classical Hamiltonian), οι ιδιοσυναρτήσεις στον χώρο των φάσεων δίνουν φραγμένα και σε σωστή κλίμακα κυματικά πλάτη (wave amplitudes) όταν αυτά προβάλλονται πίσω στον εποπτικό χώρο (configuration space) (ομοιομορφοποίηση (uniformization)).

Επομένως, η προσέγγιση της σειράς ιδιοσυναρτήσεων είναι μια προσεγγιστική λύση της εξίσωσης Wigner, η οποία μέσω της προβολής στον εποπτικό χώρο δίνει ένα προοσεγγιστικό κυματικό πλάτος χωρίς ιδιομορφίες. Εν γένει, αναμένεται ότι, το παραγόμενο κυματικό πλάτος είναι φραγμένο και σε σωστή κλίμακα ακόμα και επάνω στις καυστικές, αφού, μόνο πεπερασμένοι όροι των προσεγγίσεων είναι σημαντικοί για αρχικές κυματοσυναρτήσεις WKB (WKB initial wave functions) με πεπερασμένη ενέργεια.

Οι λεπτομέριες των υπολογισμών παρουσιάζονται για το απλό δυναμικό του αρμονικού ταλαντωτή , ώστε να είναι δυνατόν να ελεγχούν οι προσεγγίσεις μας αναλυτικά. Όμως, η ίδια κατασκευή μπορεί να εφαρμοστεί για οποιοδήποτε πηγάδι δυναμικού το οποίο συμπεριφέρεται όπως ο αρμονικός ταλαντωτής κοντά στον πάτο του πηγαδιού. Σε γενικές γραμμές, η κατασκευή αυτή θα μπορούσε να επεκταθεί σε υψηλότερες διαστάσεις χρησιμοποιώντας κανονικές μορφές (canonical forms) των Χαμιλτωνιανών συναρτήσεων και τη συμπλεκτική συνδιακύμανση (symplectic covariance) που προκύπτει από την αναπαράσταση Wey στην εξίσωση Wigner.

Λέξεις κλειδιά

Εξίσωση Schrödinger, εξίσωση Wigner, ημικλασικό όριο, γεωμετρική οπτική, καυστικές, κβάντωση Weyl, τελεστές Weyl, μετασχηματισμός Wigner, μέθοδος ομοιόμορφης στάσιμης φάσης 81Σ30, 34Ε05, 58Κ55

Η διατριβή αυτή έχει χρηματοδοτηθεί κατά ένα μέρος από τις υποτροφίες του κληροδοτήματος ‛‛Μαρίας Μιχαήλ Μανασάκη’’.



the Moyal product,
the solution of Schrödinger’s I.V.P.
the Schrödinger operator,
the eigenvalue of the harmonic oscillator, ,
the eigenfunction of the harmonic oscillator, ,
the semiclassical Wigner function of ,
the Wigner transform of and ,
WKB-approximation of eigenfunction ,
the Wigner transform of and ,
limit Wigner distribution,
Table 1: Notation

Chapter 1 Introduction

1.1 Schrödinger equation and high-frequency waves

High-frequency wave propagation is a fundamental problem which arises in quantum mechanics, and in classical wave theories such as acoustics, seismology, optics and electromagnetism. Wave equations modelling energy propagation including diffraction and scattering effects, in many interesting cases, as, for example, the propagation in infinite domains with non-compact boundaries, or in media with complex inhomogeneous structure, are difficult to be treated either analytically or numerically. For this reason in several important applications, as in underwater acoustics, or in propagation of laser beams in the atmosphere and the propagation of radio waves near Earth’s surface , it has been proposed, and it has been successfully implemented for practical purposes, the parabolic approximation method [Flat, Tap1, Foc1, Tap2]. The main idea is that, under certain assumptions, instead of solving the wave equations, someone approximates the slow varying component of the wave field by a wave function which solves an initial value problem for the quantum mechanical Schrödinger equation

being an appropriately modeled initial wavefunction. Note that in classical wave propagation, the variable is also spatial variable, usually representing the direction of long-distant one-way propagation, and not the physical time as in quantum mechanics.

The parameter is connected with Planck’s constant in quantum mechanics. In classical waves it is connected with the frequency of the waves, and it is small in many interesting case where the wavelength is small compared with the scale of spatial variations of the properties of, and the size of the domain occupied by, the propagating medium.

The potential encodes the properties of the propagating medium in classical waves, while it describes the forces acting on moving particles in quantum mechanics. There is, however, a certain relation between the classical waves and the quantum particles, which is described by the particle-wave duality and the correspondence principle. This duality becomes mathematically apparent in the classical limit (or, high-frequency limit) where classical mechanics and geometrical optics emerge from quantum mechanics and wave theory, respectively, in the form of the Hamilton-Jacobi and transport equations [Rob, MF, NSS].

Geometrical optics, caustics and phase-space methods

The most interesting and difficult problems concern the propagation of highly oscillatory initial data of the form

Mathematically these problems have traditionally been treated using the WKB method, which is also known as the Geometrical Optics technique, [BLP, BB]. According to this method one seeks for an approximating solution for the problem of the form

where the amplitude and the phase satisfy the transport and Hamilton-Jacobi equations, respectively.

However, the method fails on caustics and focal points where the solution of the Hamilton-Jacobi equation becomes multi-valued and the solution of the transport diverges and it predicts physically meaningless infinite wave amplitudes. Also, the method fails in shadow regions (i.e. regions devoid of rays), where it yields erroneous zero fields. Formation of caustics and shadows is a typical situation in optics, underwater acoustics and seismology, as a result of multipath propagation from localized sources, [TC, CMP]. In quantum mechanics the formation of such singularities are connected with the classically forbidden regions, where classical particles cannot penetrate, and the tunneling effects [Raz].

Uniform asymptotic expansions near simple caustics have been constructed , assuming that the multivalued function is known away from caustics and using boundary layer techniques and matched asymptotic expansions [BB, BaKi]. However, these analytical techniques are very complicated to apply in specific problems, since the matching procedure depends on the form of the particular caustic and it requires delicate geometrical constructions.

A different category of methods for the construction of uniform wave fields near caustics, is based on representations of the solutions in terms of phase-space integrals. The main and most known methods in this category are Maslov’s canonical operator [MF, MSS], and the Lagrangian integrals (Kravtsov-Ludwig method) [Kra, Lu, Dui1, Dui2, KO]. Both representations are special cases of Fourier integral operators [Dui3, Tr].

All the above described techniques assume an ansatz for the wave field, which for the final determination requires the knowledge of the multivalued phase functions, or, geometrically, of the Lagrangian manifold generated by the bicharacteristics of the underlying Hamiltonian system in phase space.

1.2 Wigner transform and Wigner equation

An alternative approach is to reformulate the evolution equations in configuration space to kinetic-type equations in phase space by using phase space transforms of the wave functions (see, e.g., [MCD] for a review of this idea in several wave problems of classical physics).

The most popular phase space transform appears to be the Wigner transform. This is a function defined on phase space as the Fourier transform of the two-point correlation of the wave function,

This object was introduced by E.Wigner [Wig] for modeling purposes in quantum thermodynamics, and recently, it has been successfully used in semiclassical analysis for the reformulation of wave equations as non local equations in phase space and the study of homogenization problems in high-frequency waves [GMMP].

In the case of the Schrödinger equation, the corresponding Wigner function satisfies the semiclassical Wigner equation,

where is the classical Hamiltonian, and the Moyal product is defined by

This is a linear evolution equation in phase space, and non locality stems from the Moyal star product coupling the Hamiltonian with the Wigner function. This non-commutative product, which is the Weyl image of operator composition, encodes all the important features of quantum interferences, and it is interpreted in a physically plausible and mathematically elegant way in the framework of deformation quantization (Bopp shift) of the classical Hamiltonian mechanics [Gro, BaFFLS1].

The basic and most interesting derivation of the Wigner equation in quantum mechanics comes from the Weyl representation of the Liouville-von Neumann equation for the evolution of quantum density operator, and, for pure quantum states, it is equivalent to the Schrödinger equation [Gro, CFZ2]. There are a few basic theoretical for the Wigner equation, mainly the works of P.Markowich, et al. [MA, SMM] for the equivalence of Wigner and Schrödinger equations, and the related asymptotic analysis, and also some results on scattering theory by H.Rogeon & P.Emamirad [EmRo1, EmRo2]. However, there are not efficient constructive techniques for solving the Wigner equation, probably because of the complicated and rather unusual characters of this equation, that are described in the sequel.

For smooth potentials, the non local operations in the equation can be reformulated as an infinite order singular perturbation (with dispersion terms with respect to the momentum of the phase space) of the Liouville equation of classical mechanics. In particular, for the Schrödinger equation, which, as it will be explained later, is the quantization of standard particle Hamiltonian , the Wigner equation can be written as

It becomes now apparent that the Wigner equation has non-constant coefficients, and, at least formally, it combines two different features, namely those of transport and of dispersive equations. The first one arises the Hamiltonian system of the Liouville equation in the left hand side it is related the underlying classical mechanics of the problem. The second one arises from odd derivatives with respect to momentum in the right hand side. It is related to the quantum energy dispersion inside a boundary layer around the Lagrangian manifold of the Hamiltonian system and the modification of local scales according to the geometry of the evolving manifold.

At the classical limit formally the series in the right hand side of the equations disappears and the equation becomes the classical Liouville equation of classical mechanics. In a fundamental work, P.L.Lions & T.Paul [LP] have shown that in fact the solution of the Wigner equation has a weak solution, to the so called Wigner measure, and this measure solves the Liouville equation. For relatively smooth initial phase functions , the weak limit is equivalent to the single phase geometrical optics, in the sense that it produces the same results with the WKB method for the energy density.

In the case of multi-phase optics and caustic formation, the Wigner measure is not the appropriate tool for the study of the semiclassical limit of waveamplitudes, that we are interested in. It has been shown by S.Filippas & G.N.Makrakis [FM1, FM2], by solving analytically certain simple porblems for the case of time-dependent Schrödinger equation that the Wigner measure () cannot be expressed as a distribution with respect to the momentum for a fixed space-time point, and thus it cannot produce the amplitude of the wavefunction, and () is unable to “recognize” the correct frequency scales of the wavefield near caustics. It was however explained that the solutions of the integro-differential Wigner equation are able to capture the correct frequency scales, and therefore it became promising to look for asymptotic approximations of the solution of the Wigner equation.

Therefore, the deeper study of asymptotic solutions of the Wigner equation for small seems to be promising both for understanding the structure of the solutions of the equation, and for computing energy densities and probability densities, in multiphase geometrical optics, shadow zones and classically forbidden regions.

Before proceeding to the review and discussion of asymptotic solutions we would like to mention some interesting and useful numerical approaches for the Wigner equation. Such methods have been mainly proposed in quantum optics and quantum chemistry. First of all, it has been proposed to construct solutions of classical Liouville equation, as an alternative to apply the WKB method, by attempting to capture numerically, in a kinetic way, the multivalued solutions far from the caustic (see, e.g., [JL, Ru1, Ru2]). In order to apply this technique, it is necessary to introduce an priori closure assumption for a system of equations for the moments of the Wigner measure, which essentially fixes a finite number of rays passing through a particular point. Such a closure condition leads to systems which have many similarities with incompressible hydrodynamics and then several popular methods of computational fluid mechanics can be used. Among many numerical solutions of the Wigner equation which have been developed for specific applications in physics, chemistry and quantum chemistry, we would like to mentions some of them which we believe that give also some mathematical insight to the problem. First, the spectral method of M.Hug, C.Menke & W.P.Schleich [HMS1, HMS2], which is based on the approximation of Wigner functions by Chebyshev polynomials in phase space. Second, the particle technique which was proposed by A.Arnold & F.Nier [AN]. This technique was applied for the numerical investigation of simple problems with the presence of caustics by E.Kalligiannaki [Kal1], where the results for the wave amplitude are compared with those obtained by a finite element code for the Schrödinger equation and E.Fergadakis [Ferg] who applied a particle in cell method for the same problem in the case of the propagation of plane waves in a linear layered medium (where caustics coincide with the turning points of rays). Third, and most important, is the numerical solution of the smoothed Wigner equation proposed by A.Athanassoulis [Ath1, Ath2] for the accurate and efficient numerical treatment of wave propagation problems. The key point of his approach is that the smoothed Wigner transform can be used to compute the solution at a chosen spatio-spectral resolution. Thus leading to some degree of averaging. The novel idea, which is not common in more traditional techniques, is that the separation of the averaging operator from the error, leads to an approximate coarse-scale solution.

1.3 Asymptotic solutions of the Wigner equation

Asymptotic solutions of the Wigner equation have a relatively short history, and most of them are based on perturbation methods, guided partly from the mathematics of the equations and partly from the physics of the wave problem. The techniques that have been proposed so far, could be classified as follows 111This classification reflects somehow our point of view for the problem, and it is guided by the idea of deformation quantization mentioned above..

(a) Distributional expansions: H.Steinrück [Ste] & M.Pulvirenti [P] have constructed perturbation distributional expansions near the solution of the classical Liouville equation, by expanding the initial data in a distributional series with respect to the small parameter. The first work seems to be the first formal attempt in this direction, while the second one initiated the rigorous investigation of such expansions.

(b) Semiclassical expansions using modified characteristics: E.J.Heller [He], guided by his deep insight from quantum chemistry, noted that the distributional expansions are not physically appropriate for studying the evolution of singular (with respect to the semiclassical parameter ) initial conditions. Instead, he proposed a different expansion where the first order term is the solution of a classical Liouville equation but with an effective potential. The use of modified characteristics resulting from the effective potential, aims to include indirectly some quantum phenomena. This is a general philosophy for the treatment of the quantum Liouville equation by physicists (method of Wigner trajectories, see H.W.Lee [L]), and it has been motivated by quantum hydrodynamics (Bohm equations) and the technique of Gaussian beams. For similar reasons, F.Narcowich [N] has proposed a different expansion near the classical Liouville equation, somehow smoothing the problem by imposing where -dependent initial data, instead of distributional ones, to the Liouville equation, thus avoiding the distributional expansions of H.Steinrück and M.Pulvirenti.

(c) Airy-type semiclassical expansions: More recently, S.Filippas & G.N.Makrakis
[FM1, FM2], used asymptotic expansions of Airy type for multivalued WKB functions as asymptotic solutions of the Wigner equation and they showed that such solutions are the correct approximations at least near simple caustics. The basic tool for the construction of the asymptotic solution is Berry’s semiclassical Wigner function [Ber]. This function is an Airy type approximation of the Wigner transform of a single-phase WKB function, which has been derived by the Chester-Friedman-Ursell technique of uniform stationary phase for one-dimensional Fourier integrals [CFU]. The key observation for the construction of this asymptotic solution is that when the semiclassical Wigner function is transported by the Hamiltonian flow, it remains a good approximation of the Wigner equation at least until critical time when the Lagrangian manifold develops singularities and the first caustics appears. However, it turned out a posteriori, that the asymptotic formula remains meaningful even after the critical time. In the simplest case of fold and cusp caustics the asymptotic formula reproduces the already known asymptotic approximations of the wave field and it captures the correct dependence of the wave function on the frequency, near and on the caustics.

(d) Semiclassical expansions near harmonic oscillator: In a different direction,
E.K.Kalligiannaki [Kal] (see also [KalMak]), has proposed in her doctoral dissertation a new strategy for the construction of asymptotic approximations of the Wigner equation, departing from the eigenfunction expansion of Schrödinger equation’s solution, in the case of single potential well. The interrelation between the solutions of the Wigner equation in phase space and those of the Schrödinger equation in the configuration space has bee very important for understanding the structure of the phase space solutions. This idea goes back to the pioneering work of J.E.Moyal [Mo] for statistical interpretation of quantum mechanics, and it has been rigorously exploited for first time by P.Markowich [MA] using the functional analysis of Hilbert-Schmidt operators. The interrelation of the spectra of the Wigner and Schrödinger equations has been considered first by H.Spohn [SP], and it has been further clarified by I.Antoniou et al. [ASS]. At least in the case when the Schrödinger equation has purely discrete spectrum it is clear that the Wigner equation has discrete spectrum too, and we can construct the phase space eigenfunctions of the Wigner equation as the Wigner transforms of the eigenfunctions of the Schrödinger equation in the configuration space. Kalligiannaki’s construction relies on the combination of the Wigner transform with the perturbations expansion of the eigenfunctions of the Schrödinger equation about the harmonic oscillator derived by B.Simon [Si]. It turned out that the expansions of the Wigner eigenfunctions satisfy approximately the pair of equations which govern the phase space eigenfunctions. On the basis of these expansions she proposed an asymptotic ansatz of the solution of Wigner equation. The time-dependent coefficients of the expansion were computed from a hierarchy of equations arising through a regular perturbation scheme. The asymptotic nearness of the expansion to the true solution is studied for small times by using the technique developed by A.Bouzouina & D.Robert [BR] for the uniform approximation of quantum observables. Although she did not proved estimates for large time, she explained that the application of the expansion to caustic problems for the harmonic and quartic quantum oscillator leads to reasonable numerical approximations of the wave function near and on some caustics. An approximation of this type naturally produces expansions which are compatible and come from the “exact” solution of Schrödinger equation, which compared with the WKB solution does not reveal caustic problems (which anyway appear exactly when we use Geometrical Optics technique). Under certain conditions on the potential function the produced expansion (“harmonic approximation”–“harmonic expansion”) is in accordance with the expansion near classical Liouville equation’s solution (“classical approximation”–“classical expansion”), even at caustics in the appropriate semiclassical regime for the parameters of the problem.

1.4 Approximation by uniformization of WKB functions

Inspired by the general idea of deformation quantization 222This idea, although not mentioned explicitly, naturally underlies the construction of semiclassical Wigner function [Ber], and the approximation of the Wigner equation in [FM1]., we have explained in the works [G, GM], how the Wigner transform can be used to uniformize two-phase WKB functions near turning points (caustics). A two-phase WKB function is the sum of two WKB functions, corresponding to the two geometric phases near the turning point. The Wigner transform of this function consists of four Fourier integrals, which are approximated by Airy functions either by the uniform formula of Chester-Friedman-Ursell technique [CFU], or by the standard stationary phase formula, depending on the phase involved in each integral and the position in phase space. The approximations were combined and matched appropriately (asymptotic surgery) in order to derive a uniform Airy expression of the Wigner function the over the whole phase space 333We occasionally use the term Wigner function instead of solution of the Wigner equation, when this is clear from the context.. Then, by projecting the Wigner function onto the configuration we derive wave amplitudes that are bounded at the turning points and they have the correct scale with respect to the semiclassical parameter .

The details of the calculations were presented for the particular example of two-phase WKB approximation of the Green’s function for the Airy equation with an appropriate radiation condition at infinity. This is a simple scattering problem with continuous spectrum, which models the propagation of acoustic waves emitted from a point source in a linearly stratified medium. For this model the WKB solution has a simple fold caustic. The applicability of the uniformization technique for a general fold caustic was asserted by exploiting the geometrical similarity of a folded Lagrangian manifold with the simple parabolic manifold of our model problem.

It was also shown that the derived Wigner function is an approximate solution of the semiclassical Wigner equation corresponding to the Airy equation. Note that Airy equation is the simplest stationary Schrödinger equation, with linear potential and continuous spectrum.

Therefore, we have naturally raised the following question: Can we construct approximate solutions of the semiclassical Wigner equation by uniformization of WKB functions, when the Schrödinger operator has discrete spectrum as, for example, happens in the case of potential wells? This problem is much more complicated than the above described model one, and we expect to face a very complicated structure of the Wigner function. The reason is that instead of a single Lagrangian manifold, now we have a fan of closed Lagrangian manifolds corresponding to the energy levels (bound states) of the anharmonic oscillator. Accordingly, it is expected that the Wigner function must contain new terms due to the interaction between the manifolds, in addition to the interactions between the branches of each individual manifold.

1.5 The scope and the contents of the thesis

The present thesis aims to the construction of an approximate solution of the Wigner equation by uniformization of WKB approximations of the Schrödinger eigenfunctions in the case of the quantum harmonic oscillator. We have chosen to work with this simple potential, in order to minimise the geometrical details, and also in order to be able to check some of our approximation results analytically. The same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well, by employing perturbations of the Schrödinger eigenfunctions as was done in [Kal].

The construction of the approximation of the Wigner function departs from an eigenfunction expansion of the solution of the Wigner equation and it relies on Airy type approximations of the eigenfunctions of the equation (from now on referred as the Wigner eigenfunction). The approximate Wigner eigenfunctions are constructed by uniformization of the two-phase WKB eigenfunctions of the corresponding Schrödinger operator. It turns out that approximation series of the Wigner function is an approximation of the solution of the Wigner equation, and it is well behaved everywhere in the phase space. The projection of the approximation series of the Wigner function onto the configuration space, provides an approximate wave amplitude. This amplitude is uniformized in the sense that it is bounded on the turning point of the WKB eigenfunctions of the corresponding Schrödinger operator.

The contents of the thesis are as follows. In Chapters 2-4 and in Chapter 6,, we collect some background or known advance material, which, however, has been considerably elaborated for use in the developments of Chapters 5, 7 & 8.

In Chapter 2 we give a short description of classical and quantum mechanics. We introduce the Schrödinger equation as the basic dynamic law governing the evolution of quantum states (wavefunctions), and the more general formulation based on the Liouville-von Neumann equation governing the density operator [NSS]. We explain how E.Schrödinger [Schr] invented the Schrödinger equation by quantizing the classical Hamiltonian function. This fundamental idea had far-reaching consequences in quantum physics but also in functional analysis where it has offered crucial motivation for the development of pseudodifferential operators [BaBMo]. Also we explain how the semiclassical parameter follows from physical Planck’s constant when we use dimensionless time and coordinates, which gives meaning to the notion of semiclassical regime.

In Chapter 3 we collect some basic results about the Cauchy problem for the Schrödinger equation. First, we describe the construction of the asymptotic solution when is small by the WKB method, we introduce the ideas of geometrical optics and we explain how the method fails when caustics, shadow zones or other singularities of the ray congruence develop. Second, we describe the eigenfunction series expansion of the solution, which in principle is valid for any value of and free of caustics. This solution is derived by standard method of separation of variables. Assuming that the potential is a single well, growing without bound at infinity, so that the spectrum of the Schrödinger operator to be purely discrete. For we present asymptotic approximations of the higher eigenvalues as derived from the Bohr-Sommerfeld rule, and the WKB expansions of eigenfunctions [Fed]. Also, for later use, we present explicit formulas for the eigenvalues and WKB eigenfunctions of the quantum harmonic oscillator.

In Chapter 4 we present some ideas from the theory of pseudodifferential operators, mainly the Weyl method of quantization of symbols (classical observables). We present in details the construction of the Weyl operator following the excellent exposition of F.A.Berezin & M.A.Shubin [BS], and we introduce the notion of star product utilizing the Weyl symbol of the composition of operators. Then, we introduce the semiclassical Wigner function as the Weyl symbol of the pure state density operator, and we derive the Wigner equation, by applying the Weyl correspondence rule on the Liouville-von Neumann equation governing the evolution of the density operator. In the case of pure quantum states, it turns out that the Wigner equation can be derived also by applying the Wigner transform in a operational way directly onto the Schrödinger equation.

In Chapter 5, by exploiting the linearity of the Wigner equation we construct directly in phase space an eigenfunction series expansion of the solution of the Cauchy problem for the Wigner equation. The separation procedure leads to a pair of phase-space eigenvalue equations, one corresponding to the Moyal (sine) bracket and the other to the Baker (cosine) bracket. The emergence of these phase space eigenvalue equations is natural when we derive the Wigner equation from the Heisenberg equation by Weyl calculus, in contrary to the most familiar derivation, when one starts from configuration space and transforms the Cauchy problem for the Schrödinger equation by introducing the Wigner transform of the wavefunction in an operational way.

In the study of the eigenvalues equations we used known fundamental results for the spectra of quantum Liouvillian. More precisely, for the eigenvalues of the first (Moyal (sine) bracket) equation we used the results of H.Spohn [SP] and I.Antoniou et al. [ASS], while for the eigenvalues of second (Baker (cosine) bracket) equation we used the results derived in the thesis by E.Kalligiannaki [Kal] (see also [KalMak]), for the special case when the Hamiltonian operator has discrete spectrum. Then, the representation of the Wigner eigenfunctions as Wigner transforms of the Schrödinger eigenfunctions was proved by using the recent spectral results by Gosson & F.Luef [GL] for the -genvalue equation.

Our construction of the eigenfunction series expansion of the solution of the Wigner equation by the elementary method of separation of variables, directly in phase space, seems to be a new one. It is a folk statement of physicists that phase-space formulation of quantum mechanics is an independent approach. However, they always work with Wigner equation by using formal star calculus [CFZ1], and transferring by Wigner transform spectral results for the Schrödinger equation from configuration space into phase space, whenever they need to complete the phase picture.

In Chapter 6 we present some results for the Wigner transform of WKB functions, and we explain why it is necessary to work with the semiclassical Wigner transform in order to be consistent with geometrical optics. It is very interesting to note that the semiclassical Wigner transform has already bounced independently from Weyl calculus. The main purpose of this chapter is to present the details of Berry’s construction of the semiclassical Wigner function of a single-phase WKB function [Ber]. Also we present the Wigner transform of multi-phase WKB functions and its limit Wigner distribution.

In Chapter 7 we construct Airy-type asymptotic approximations of the Wigner eigenfunctions from the uniform asymptotic approximation of the Wigner transform of the two-phase WKB eigenfunctions of the harmonic oscillator. This is the main achievement of the thesis, since these approximations are the basic ingredients for the construction of the the approximate solution of the Wigner equation.

The technique of the construction is somehow an extension of the uniformization procedure developed in [G, GM] for the semiclassical Airy equation. However, there are certain fundamental differences between the two problems, which cause many new technical difficulties. These differences basically arise from the fact that the spectrum of the semiclassical Airy function is continuous, while the spectrum of the harmonic oscillator is discrete. They crucially affect the geometry of the Lagrangian manifolds (actually Lagrangian curves, since the phase space is two dimensional). The Lagrangian curve of the semiclassical Airy equation is an open curve extending to infinity (parabola). The Lagrangian curves of the harmonic oscillator form an infinite fan of closed curves (circles), corresponding to the eigenvalues of the oscillator. Although the local behaviour at the turning points is qualitatively the same in both cases (namely the behaviour of a fold singularity), in the case of the harmonic oscillator the presence of a second turning point creates new interactions between the upper and lower branches of the Lagrangian curves. Moreover, there are additional new interactions between the Lagrangian curves of the fan.

Thus, we have two groups of Wigner eigenfunctions. The diagonal Wigner eigenfunctions, associated with a particular Lagrangian curve, are semiclassically concentrated as Airy functions on this curve. The off-diagonal Wigner eigenfunctions, associated with a pair of Lagrangian curves are again semiclassically concentrated as Airy functions, but on an effective curve between the Lagrangian curves of the pair, and they are modulated with an oscillatory factor depending on the angular direction of phase space. This factor is responsible for the fact that the net contribution to the wave energy of the off-diagonal Wigner eigenfunctions is zero, and therefore the interactions between different Lagrangian curves of the oscillator, are responsible only for the energy exchange between different modes in the solution of the Cauchy problem for the Wigner equation.

The Airy approximations of the Wigner eigenfunctions have been compared in certain regimes of the parameters with the exact Wigner eigenfunctions of the harmonic oscillator, which are expressed through Laguerre polynomials. This comparison partially confirms the validity of our approximation technique. Moreover, the classical limits of the Airy approximations of the Wigner eigenfunctions are solutions of the classical limits of the Moyal and Baker eigenvalue equations, and this observation provides a further confirmation of the validity of our approximation technique.

In Chapter 8 we combine the eigenfunction series expansion of the solution of the Wigner equation, the approximate Wigner eigenfunctions and Berry’s semiclassical function for the WKB initial datum, to derive an approximate solution of the Cauchy problem for the Wigner equation. This is the second main achievement of the thesis. The expansion is the sum of a coherent (time-independent) part spanned by the diagonal Wigner eigenfunctions, and an incoherent (time-dependent) part, spanned by the off-diagonal Wigner eigenfunctions. The coefficients of the expansion of the coherent part (and in special case also of the incoherent part) of the solution are evaluated either analytically for quadratic initial phases, or approximately by combining a novel asymptotic decomposition of the Airy function with analytic integration, for more general phases. The calculations show that the dependence of the coefficients on the semiclassical parameter , is crucially dependent on the initial phase. The derived approximate Wigner function is well-behaved everywhere in the phase space. Moreover, its integration with respect to the momentum results in a wave amplitude which is meaningful even on the turning points of the WKB approximations of the Schrödinger eigenfunctions. Therefore, the approximate Wigner function derived by the uniformization procedure is a promising tool to smooth out the caustic singularities in time-dependent problems.

Finally, in Chapter 9 we give a short discussion of the main results and the achievements of the thesis, and we discuss about a new question raised by our investigation. This concerns the asymptotic nearness between two approximate solutions of the Wigner equation, that is the time-dependent semiclassical Wigner function constructed in [FM1] and the series approximation constructed in Chapter 8 of this thesis.

Chapter 2 Classical and Quantum mechanics

A classical or quantum mechanical system is described by defining certain basic mathematical objects which necessarily include: states, observables, dynamical law and transformations. At each instant of time the system resides in a particular state which is an element of an appropriate linear space (state space). The nature of the states depends essentially on the adopted physical modelling of the system, and they can be elects of finite-dimensional vector spaces, function space or even more general linear spaces. The experimental observations of the system provide measurements whose interpretation leads to the notion of observable (observable quantity). The evolution of the system is described by the dynamical law which defines the state of the system at any particular time. Finally, both for physical and mathematical reasons, we need to define certain transformations acting on states, which provide equivalent descriptions of the system.

2.1 Classical mechanics

The simplest system in classical Newtonian mechanics has degrees of freedom which form the dimensional configuration space . For example, if the system consists of particles moving in three-dimensional space under the action of certain forces, and without any kinematical constraints, we have . The state of such a system is determined uniquely by specifying the (generalised) position and momentum vectors of each particle. Therefore, a state of is a point , where is the vector of coordinates and is the vector of momentums. The even-dimensional space is called the phase space of the mechanical system . For a more complicated mechanical systems the phase space is a symplectic manifold, namely a smooth manifold with a non-degenerate closed two-form defined on it. In this case the phase space is the tangent space [ARN1].

An observable of the system in a state , is a function , and the measurement of gives invariably the same value . The simplest observables are the position and the momentum, i.e. and , respectively. The most important observable is the total energy of the system. In the state the energy has the value , where is the Hamiltonian of the system [ARN1, GOLD].

In general, scalar functions on the phase space can be multiplied pointwise, i.e. where the multiplication fulfills the following properties: commutativity, linearity and associativity. So that the observables equipped with the addition and the pointwise multiplication form a commutative algebra of observables.

The dynamics of the system is uniquely determined by the Hamiltonian function. More precisely, it is postulated that the evolution of the state (phase point) is governed by the Hamiltonian system of ordinary differential equations

for .

The Hamiltonian system , under certain smoothness assumptions a map , , of the phase space into itself, The map , which is known as the Hamiltonian flow, preserves the symplectic 2-form , i.e. , and the Poisson bracket, and therefore it is an admissible canonical transformation of the phase space [GOLD].

By using the Hamiltonian system, we can compute the time derivative of a time-independent observable along the Hamiltonian flow, as follows

Therefore, the evolution of is governed by the Liouville equation


where is the Poisson bracket of the Hamiltonian and the observable .

The Poisson bracket of the functions , is defined by


Sometimes it is written symbolically in the form


where the vector arrows indicate in which direction the differentiation acts. For example, and .

The Liouville equation is a fundamental equation for the description of general mechanical systems, and it lies in the heart of more general theories, as for example the classical statistical mechanics [B]. Most important, it has an analogue in the phase space description of quantum mechanics, the Wigner equation (see, Chapter 4, below), and, in this connection, it lies also in the heart of quantization and deformation theories [BaFFLS1, BaFFLS2].

2.2 Quantum Mechanics

2.2.1 Quantum states and observables

States. A quantum mechanical system is described by postulating that the states are elements of an infinite-dimensional, separable, complex Hilbert space with inner product . We assume that the states are normalized .

In Dirac’s notation [D], which is very popular in quantum mechanics [B] 111although someone can find many books on quantum mechanics which avoid this notation. See, e.g., [S, GR] and [Tak], we denote the vectors by the symbols which are called kets. The linear functionals in the dual space , are called bras and they are denoted by . The numerical value of the functional is denoted as This notation is justified by the
Riesz theorem. There is a one-to-one correspondence between linear functionals , in the dual space , and vectors in , such that

Indeed, this theorem implies that there is a one-to-one correspondence between bras and kets 222In his original presentation, Dirac had assumed this correspondence between bras and kets, but it is not at all clear if this was a mathematical or physical assumption. But by Riesz theorem, there is no need, and indeed no room, for such an assumption., and therefore we can write the equation

which says that we can think of Dirac’s ”bracket” simply as the inner product . Moreover, by Riesz theorem, the correspondence between bras and kets is antilinear, therfore

for any complex number , being its complex conjugate. Finally, by the normalization condition .

Observables. The most striking feature of quantum mechanics, as opposed to classical mechanics where the observables are functions of the phase point, is that the quantum observables are represented by linear self-adjoint operators

which are, in general, unbounded.

This choice of the observables is intimately related with the fundamental fact that the measurement of an observable, while the quantum system resides in a particular state, is not uniquely determined by the state, but it provides various values, that are modelled as a random variable obeying a certain distribution law [NEUM, BS].

More precisely it is postulated that any measurement of the observable yields a number belonging to the spectrum of . For simplicity, we assume that the spectrum is discrete with eigenvalues and let be the corresponding orthonormal basis (ONB) of eigenfunctions of . In other words and satisfy the eigenvalue problem


The expansion of the state in the ONB of the eigenfunctions is . Then the probability of obtaining the value in the measurement of the observable of when the system resides in the state , is given by


Since , it follows that these probabilities sum to the unit.

The expectation value (mean value) of the random variable provided by the measurement of the observable when the system is in the state , is given by


since . By this result, it is also referred as the mean value of the observable .

By standard results from functional analysis, the expectation satisfies the following conditions

Here denotes the identity operator and is the adjoint operator of .

We must note that when the spectrum is continuous, then admits generalised eigenfunctions indexed by the point in the continuous spectrum. Then, the sums in the above formulas are substituted by integrals and the probabilities are substituted by probability measures . Also, when the spectrum is mixed, someone must further modify the formulas by combining sums and spectral integrals.

We must also note that because we are interested for the mean values of the observables, and not for the values of the particular realisation of the random variable provided by the measurement process, and the expectation is the same for any two states and differing by a unimodular factor, it becomes clear why it is enough to work with the projective space .

2.2.2 Evolution of quantum states and observables

The evolution of a quantum system, just in classical mechanics, is determined by the distinguished observable of energy which is represented by the energy operator . It will be explained later that by the process of quantization, this operator corresponds to the Hamiltonian (total energy) of the corresponding classical system, and for this reason we will call it the Hamiltonian operator. Due to the quantization process the operator depends on Planck’s constant . This is a physical constant 333Actually is the re-scaled physical Planck’s constant which has dimensions of action and its numerical value is . which has its roots in the fundamental physics of quantum theory [Foc]. There are several different, in some sense, equivalent pictures of quantum mechanics. The most known is the Schrödinger picture which looks for the evolution of the wavefunction, and the Heisenberg picture, which looks for the evolution of the observables.

In the Schrödinger picture the state vector (a wavefunction) changes in time according to the Schrödinger equation


where is considered as a vector of which varies with time.

If the Hamiltonian function is not explicitly dependent on time, the solution of this equation with initial value is given by


where is the one-parameter group of unitary operators, which generated by the self-adjoint operator .

However, we can think in terms of the initial data and a time-dependent version of the observable, when we compute mean values of observables. The mean value of at time is given by

Thus, we are led to define the observable

such that . Then, the mean value is expressed by

as the expectation of expectation of , when the system is in state . This approach to the description of quantum systems is referred as the Heisenberg picture.

By direct computation, using , we find that satisfies the operator equation


where , denotes the commutator of and .

2.2.3 Density operator

So far we have considered that the quantum system is described by a single state. More general systems have been considered by J.von Neumann [NEUM] in his statistical theory of measurements on quantum systems. The basic idea is the following. Let an ensemble of quantum states (wavefunctions) and let be the probability of finding the system in the state . Then, we consider that the system resides in a mixed state (somehow in a mixture of states) which is described by the self-adjoint density operator


It turns out that are eigenfunctions and eigenvalues of .

  • Mixed states arise in the description of quantum systems when they are considered as subsystems of larger systems with more degrees of freedom as it is necessary to do in scattering problems and open systems. In all these cases the mixing of the sates comes from the averaging with respect to the additional degrees of freedom of the larger system with respect to the smaller one [NSS], Ch.1, and [BP].

Contrary to mixed states, when a state of a system is accurately known, we say that it resides in pure state. This means that the probability for some , and all others are zero. For this case the density operator takes the simple form


The expression shows that the density operator for a pure state is a projection operator. We refer to it as the density matrix corresponding to the state . In this sense, we say that a mixed state is a collection of different pure states.

  • Let be an operator on , and an ONB of . Then, its trace is defined by the formula

    assuming that the series converge. If is finite, we say that the operator is of trace class.

Then, by the definition of pure state density operator, we have , and therefore


Thus, by , we can write the expectation in the form


The formula provides the reason why we want to introduce the density operator in the description of quantum systems. Density operator is the tool for computing the expectation values, which are the only source of information about the system.

The density operator as introduced above has the following properties 444On the basis of these properties, we can generally define as density operator , any selfadjoint, positive semi-definite operator with .:

  1. The density operator is self-adjoint ,

  2. The trace of the density operator is equal to 1,

  3. The eigenvalues of a density operator satisfy

  4. For a pure state , and

  5. For a mixed state

The simplest (mathematical) example of mixed state is constructed as follows.

  • Let a sequence of states in and a sequence of nonnegative numbers suche that . The operator is the density matrix of a mixed state. This simple mixed state is the mixture of pure states and, roughly speaking, it has the physical interpretation that the system resides in state with probability . Then, it is consistent with the interpretation of as the expectation of in the state represented by .

We close our brief discussion on density operator, by clarifying its connection with coherence properties in quantum systems.

  • Let us consider a pure state expanded in the form


    with respect to an ONB for all of the Hilbert space . By we write the pure state density operator as follows as follows


    The first term in relates to the probability of the system being in the state , since . For the second term, we write the complex number in polar form, thus . The phase difference in the exponential expresses the coherence of terms in the state to interfere with each other.

2.2.4 The time-evolution of density operator

Let the density operator


Since each state satisfies the Schrödinger equation,

and also

By differentiating , we have

Therefore, we conclude that satisfies the Liouville-von Neumann equation


2.2.5 Schrödinger equation in coordinate representation

Schrödinger’s derivation

In the simplest case of a quantum mechanical particle (e.g. atom), we deal with the quantum analogue of a classical particle. By classical particle we mean a point mass moving in Euclidean configuration (position) space according to Newton’s second law of classical mechanics under the action of a potential . This point moves according to the Hamiltonian system , with Hamiltonian function


which is the total energy , i.e. the sum of kinetic and potential energy, of the particle.

In dealing with this simple quantum system, it is convenient to work with the standard coordinate representation, in which the state space is just the space . This is the space of functions which are continuous, infinitely differentiable, and square integrable, i.e. , with inner product . In this representation, a ket is a function . The correspondence between a ket and its associated wavefunction is , where is the bra corresponding to .

Then, it turns out that the fundamental observables are the position and momentum operators and , respectively. The operators act as multiplication by position, while the operators act by differentiation, being the Planck’s constant. These observables satisfy the relations


Then, the energy operator


is obtained from the classical particle Hamiltonian by the formal substitution


The substitution , which associates to a scalar function defined on classical phase space , a partial differential operator acting on the Hilbert space , is referred in the literature as the Schrödinger quantization. It has been introduced in 1926 by E.Schrödinger [Schr], who also assumed that the classical particle energy must be associated with the time derivative , in order to derive form the energy conservation the Schrödinger equation governs the evolution of the quantum-particle wavefunction (see [Z], Ch. 5.11.4 , for a concise description of the derivation). This equation reads as


where is the Laplacian operator. Note that is the equation written in the coordinate representation.

Dimensionless form of the Schrödinger equation

We now transform the Schrödinger equation into a dimensionless equation. In order to do this we define dimensionless coordinates , where is an arbitrary length and with is a unit of time, that we will choose to make simpler. This change of variables gives , and then, the Schrödinger equation for the dimensionless potential takes the form


The dimensionless parameter into can be written as

where is the speed of a classical particle of mass and expresses its momentum. In quantum mechanics we assume that , this means that the Broglie wavelength of the particle is small compared to the length scale . Thus we can say that the parameter in controls the quantum effects. This means that as becomes small, the quantum effects become negligible, and classical mechanics dominates the particle motion.

The regime of physical quantities where the dimensionless parameter is nonzero, but it can be arbitrarily small: , is referred as the semiclassical regime and is referred as the semiclassical parameter. The limit is designated as the classical limit and it is associated with the transition from quantum mechanics to classical mechanics.

In the sequel we use the Schrödinger equation in the dimensionless form


where .

Introducing the Schrödinger operator


we sometimes write in the symbolic form

  • Apart from quantum mechanics, the Schrödinger equation arises in many contexts in classical wave propagation problems, as the paraxial approximation of forward propagating waves [Flat, LF]. Thus, it is of practical importance for computing wave intensities in many applied fields such as radioengineering [Foc1, Foc2], laser optics [Tap1], underwater acoustics [Tap2], the investigation of light and sound propagation in turbulent atmosphere [Tat1], and seismic wave propagation in the earth’s crust [SF], to mention but a few. In these cases, the potential is explicitly related to the refraction index of the propagating medium.

Chapter 3 QM in configuration space: Schrödinger equation

As we have seen in Section 2.2 the evolution of the wave function in quantum mechanics is governed by the Schrödinger equation . From now on we will consider only the one-dimensional case .

A problem of primary interest, both for quantum mechanics and for the classical wave problems that we mentioned at the end of previous chapter, is the evolution of highly oscillatory wavefunctions. Therefore, we consider the Cauchy problem with WKB initial data


We assume the potential is real valued, and is some positive constant. Moreover, we assume that and .

We are interested for solutions in , for any fixed . This choice is motivated by the conservation of quantum energy (probability), that is, the conservation of -norm of the wavefunction . Indeed, by direct calculation, using the Schrödinger equation, it folows that , which implies that .

In the sequel we present two different constructions of the wavefunction. The first one is the WKB method (geometrical optics) which deals with the construction of asymptotic solution when is small. The second one leads to an eigenfunction series expansion of the solution, and, in principle, is valid for any value of . However, such series are, in general, very slowly convergent when is small, and therefore are not efficient for solving the Schrödinger equation in the semiclassical regime. We will use it as an intermediate tool for the investigation of the Wigner equation in phase space (see Section 5.2 ).

3.1 The time-dependent WKB method

When the semiclassical parameter is small, highly oscillatory solutions of the problem - have been traditionally studied by WKB method (Wentzel-Kramers-Brillouin), known also as geometrical optics (see, e.g., [BB, BLP, KO]).

The WKB method seeks for an approximate solution of the form ,


where , are real-valued functions 111extensions of the method for complex-valued phases have been also developed, but we do not consider them in this work.

For the moment we assume that the amplitude is sufficiently smooth and we expand it as a power series

for small .

3.1.1 Hamilton-Jacobi and transport equation

Substituting into , and retaining the terms of order and , we obtain that the phase function satisfies the Hamilton-Jacobi equation (also called the eikonal equation)


and that the (zeroth order) amplitude satisfies the transport equation


3.1.2 Bicharacteristics, rays and caustics

The Hamilton-Jacobi equation is a first-order nonlinear partial differential equation, and a standard way for solving it is based on the method of bicharacteristics (see, e.g., [HO], Vol. I, Chap. VIII, and [Jo], Chap. 2).

Let be the Hamiltonian function 222Note that this is the Hamiltonian of a classical particle of unit mass, exhibiting one-dimensional motion under the action of the potential . . Let the trajectories

be obtained by solving the Hamiltonian system


with initial conditions


The trajectories which solve the initial value problem -, , in the phase space are called bicharacteristics, and their projection onto are called rays.

The phase is obtained by integrating the eikonal equation along the rays, that is to integrate the ordinary differential equation


with initial condition . Note that along the bicharacteristics .

On the other hand, the solution of the transport equation for the amplitude along the rays, is obtained by applying divergence theorem in a ray tube,




is the Jacobian of the ray transformation (see, e.g., [BB, Zau]).

Since is a non-linear equation, it has, in general, a smooth solution only up to some finite time. The points