Amaury Mouchet Soy

Esos Otros

Jorge Luis

Borges Al

Identity and difference: how topology helps to understand quantum indiscernability

 comfort that topology (of the braids) provides a fertile soil for our understanding of quantum particles.

1 Old puzzles Among the most common primary concepts that are jeopardised by quantum physics, the related notions of identity, individuality or discernability are not the least 2 . The question of identifying a material object, even if it is immediately accessible to the common sense, has always raised many philosophical issues once we consider it as the set of its replaceable constituents. Heraclite's thoughts on the dynamical changes and the paradox of talking about the "same river" while "waters flows" (Graham, 2008, chap. 5, § 2) has never ceased to nurture Western philosophy (and Borges in particular). Another variation on these questions goes back to an even more remote era when some Greek founding myths were forged: is it justified to talk about the ship of Theseus after some or even all of her original parts have been replaced? However one can nevertheless suspect that all these issues can be reduced to a matter of semantics (what is meant by "same"), keeping in mind the danger of the inevitable tautology that plagues ontology while trying to explain what "existence" signifies.

Amaury Mouchet

Institut Denis Poisson de Mathématiques et de Physique Théorique, Université François Rabelais de Tours 37200 Tours, France 1 I am those others too. The eternity is in the things of time, which are precipitate forms (trad. am).

2 See for instance the contributions in [START_REF] Castellani | Interpreting bodies. Classical and quantum objects in modern physics[END_REF].

An attempt to give some empirical flesh to the question of indiscernability can be found in the writtings of Leibniz who reports an observational test of the principle of the identity of the indiscernibles which is now attached to his name3 [START_REF] Pesic | Leibniz and the leaves: Beyond identity[END_REF]:

Philalethes. A relative idea of the greatest importance is that of identity or of diversity. We never find, nor can we conceive it 'possible, that two things of the same kind should exist in the same place at the same time[. That is why, when] we demand, whether any thing be the same or no, it refers always to something that existed such a time in such a place; from whence it follows, that one thing cannot have two beginnings of existence, nor two things one beginning...in time and place'.

Theophilus. In addition to the difference of time or of place there must always be an internal principle of distinction: although there can be many things of the same kind, it is still the case that none of them are ever exactly alike. Thus, although time and place (i.e. the relations to what lies outside) do distinguish for us things which we could not easily tell apart by reference to themselves alone, things are nevertheless distinguishable in themselves. [. . . ] If two individuals were perfectly similar and equal and, in short, indistinguishable in themselves, there would be no principle of individuation. I would even venture to say that in such a case there would be no individual distinctness, no separate individuals. That is why the notion of atoms is chimerical and arises only from men's incomplete conceptions. For if there were atoms, i.e. perfectly hard and perfectly unalterable bodies which were incapable of internal change and could differ from one another only in size and in shape, it is obvious that since they could have the same size and shape they would then be indistinguishable in themselves and discernible only by means of external denominations with no internal foundation; which is contrary to the greatest principles of reason. In fact, however, every body is changeable and indeed is actually changing all the time, so that it differs in itself from every other. I remember a great princess, of lofty intelligence, saying one day while walking in her garden that she did not believe there were two leaves perfectly alike. A clever gentleman who was walking with her believed that it would be easy to find some, but search as he might he became convinced by his own eyes that a difference could always be found. One can see from these considerations, which have until now been overlooked, how far people have strayed in philosophy from the most natural notions, and at what a distance from the great principles of true metaphysics they have come to be. [START_REF] Leibniz | New essays on human understanding[END_REF]Leibniz ( /1996) ) We learn from a letter written by Leibniz to Sophie, Electress of Hanover (dated october, 31rd, 1705), that the challenge took place in the Herrenhäusen gardens of Hanover between princess Sophie and M . Carl August von Alvensleben, the "clever gentleman". Sure the conclusion of the "experiment" would have been less straightforward if instead of leaves, the bet had concerned bees or ants (since clones are ubiquitous in one hive or in one anthill). However, in the same letter, maybe remembering the geometric patterns of the garden itself (Fig. 1

), Leibniz writes

There are actual varieties everywhere and never a perfect uniformity in anything, nor two pieces of matter completely similar to each other, in the great as in the small. [. . . ] Therefore there is always actual division and variation in the masses of existing bodies, however small we go. Perfect uniformity and continuity exists only in ideal or abstract things, as are time, space, and lines, and other mathematical beings in which the divisions are not conceived as all done, but as indeterminate and still feasible in an infinity of ways. Leibniz (2011, § 68. pp. 327-328, notes 669, 670) Retrospectively, and somehow ironically, the above argument where perspires Leibniz' aversion against atomism was founded. The existence of atoms not only do undermine the principle the identity of the indiscernibles but the refutation is much stronger than Leibniz could have thought: even the difference in position between atoms-a classical notion, on which, as Leibniz explains it, one may always rely to make a distinction between similar material objects-becomes irrelevant at the quantum level as long as it is not measured.

Quantum abandon of individuality

After a long maturation, mainly done in the first quarter of xxth century [START_REF] Dover | The bose-einstein statistics: Remarks on debye, natanson, and ehrenfest contributions and the emergence of indistinguishability principle for quantum particles[END_REF][START_REF] Saunders | The concept 'indistinguishable[END_REF], starting with Planck's 1900 work on the blackbody radiation that can be seen as the foundation stone of quantum physics, our concept of indistinguishability of quantum particles proceeds from the quantum theory of fields. Quantum particles, whether considered as elementary or composite, appears as elementary excitations with respect to a ground reference state (the vacuum of the considered particles) that are characterised by a handful of well-determined values which are the only observable quantities that can be attributed simultaneously to each of them: the mass, the electric charge, the spin and few other "flavours". For instance, an electron is the particle whose mass is 9.109 • • • × 10 -31 kg, whose charge is e = -1.602 • • • × 10 19 C, whose spin is 1/2, etc. Other individual observable quantities, eventhough they remain reasonably stable because of some conservation laws, for instance the linear momentum, may be affected by an individual measurement of a non compatible quantity (the two observables do not commute in some precise algebraic sense) including, notably, the position of the particle. In fact, the orthodox interpretation leads to quantum properties that cannot be attributed simultaneously without raising contradictions with observations. According to the famous Heisenberg's inequalities, after an arbitrarily precise measurement of its momentum, it is not that we do not know the position of the particle, it is just that it does not have a precise position at all. In other words, a measurement (say, the component J z of the angular momentum along one direction z) does not affect the value of the previously measured non compatible quantity (say the component J x of the angular momentum along an orthogonal direction x), it completely erases its existence: once J z is measured, one cannot attribute an even unknown value of J x to the particle anymore. Therefore any quantum particle cannot have any history nor accidental properties nor contingent secondary qualities that would still allow to individualise it, including its position4 .

These completely counter-intuitive properties reflect all the more the stangeness of the quantum world that the number of particles is itself a quantity that maybe incompatible with other observables5 . There are quantum pure states (that is on which we have the maximal possible amount of information we can conceivably get) where the number of its constituents cannot be attributed.

Even in the cases where the number N of the particles can be attributed and maintained constant because the available energy is not sufficient to create or destroy some of them, the particles of the same species (electrons, neutral alcaline atoms, etc.) still cannot be numbered, even in principle, for such a numbering is nothing but the attribution of a discriminative quantity (essentially based on a position in a given configuration). The consequences are considerable if one wants to study the statistical properties of a set of a such identical particles. We all know that the odds (and the gains) to win at a trifecta horse race are significantly different if we decide to take into account or not the finishing order of the top three horses. In statistical physics, it is energy (not money) that is distributed according to the odds of the configurations and the distinguishability of the particles has observable consequences even at the macroscopical level. The organisation of nucleons in the nuclei, of the electrons on the atoms which explain the chemical Mendeleiev classification as well as the stability of matter, could not be explained if the particles were distinguishable. The conducting properties of materials, notably the superconducting ones, their thermal response, the superfluidity, the behaviour of photons in a laser beam, or the existence of states of matter like a Bose-Einstein condensate are emergent properties coming out from purely collective effect of some set of particles where only the number of its constituant has a physical meaning.

In fact, all quantum particles we know up to now fall into two families according to their collective behaviour. The fermions, whose spin is half an integer, design particles that cannot share two identical states (the Pauli exclusion principle) whereas the bosons, of integer spin, can condensate into the same individual state. All the particles that constitute ordinary matter that we consider to be fundamental are fermions (mainly quarks, electrons). An assemblage of an odd number of fermions remains a fermion whereas any particle made of an even number of fermions (a Cooper pair of electrons, an Helium 4 atom for instance) follows a bosonic statistics.

One remarkable thing is that this dichotomy can be understood with some topological arguments and the following tries to give some hints about how this works. As a bonus, I will try to explain that for models in condensed matter where we can consider that the dynamics lies in a layer whose effective dimension is D = 2, the same topological arguments offers more possibilities. In two dimensions, one may consider some identical particles whose statistical behaviour is characterised by a continuous parameter θ that allows to interpolate continuously between bosons (say for θ = 0) to fermions (say for θ = π). These existence of these anyons (any-ons)-the word was coined by Frank [START_REF] Wilczek | Quantum mechanics of fractional-spin particles[END_REF]-has been proposed theoretically by Jon Magne Leinaas and Jan Myrheim (1977) forty-five years ago but it is only last year that they received a first experimental confirmation [START_REF] Bartolomei | Fractional statistics in anyon collisions[END_REF].

Superpositions, interferences and phases

To understand better, the reason why we must give up the systematic attribution of some properties to quantum objects is that their pure states are described in terms of a (linear) superposition of states having definite properties. There is an experimentally accessible manifestation of these superpositions: the interferences they can produce. Think of the most important Young experiment on light diffracted by two holes on an opaque screen that produces an interference pattern. Keeping a pure wave interpretation, the interference pattern is due to the superposition of two waves, each one being diffracted by one hole (the other being closed). But once the two holes are opened it is meaningless to say that the resulting wave has passed though one hole rather than the other. Rather than getting a fuzzy or an unknown path, we actually completely loose the possibility of attribution of a path. These Young-like configuration, as well as others interference experiments, have been set up for individual quantum particles (photons but also electrons, atoms and even organic molecule made of hundred of atoms). One crucial quantity that is measured in all these interference experiments is the relative phase ϕ between the states being superposed, that is essentially an angle given by the time delay between two periodic oscillations expressed in unit of their common period (like an angle defined modulo one turn, only a delay modulo a period unit can be measured, see fig 2). These relative phases have been identified by Chen Ning Yang as one of the three melodies of theoretical physics in the xxth century [START_REF] Yang | The thematic melodies of twentieth century theoretical physics[END_REF] and Dirac, looking back on the development of quantum physics that he contributed to shape, writes

The question arises whether the noncommutation is really the main new idea of quantum mechanics. Previously I always thought it was but recently I have begun to doubt it and to think that maybe from the physical point of view, the noncommutation is not the only important idea and there is perhaps some deeper idea, some deeper change in our ordinary concepts which is brought by quantum mechanics. [. . . ] [Following Heisenberg and Schrödinger], the probabilities which we

ϕ=180°ϕ ϕ=0°= = = + + + Figure 2:
The resulting superposition of two waves with the same amplitude and frequency 1/T is governed by their relative phase ϕ. When there is no dephasing (upper case with ϕ = 0), two maxima (or minima) of the superposed waves coincide and they add constructively into a wave of maximal amplitude. Conversely, when the two waves have opposed phase (lower case with ϕ is half a turn), each "bump" is compensated by a "hollow"; the two cancel one with the other and the resulting amplitude is almost zero. The relative phase is very much like an angle defined modulo one turn (or 360 • ), if the horizontal axis stands for the time, and ∆t the time delay between two maxima, then ϕ = (∆t/T ) 360 • and one cannot distinguish between two delays differeing by an integer multiple of T . To follow Dirac's argument in mathematical terms, the square modulus of the sum of two complex numbers

z 1 = |z 1 |e iϕ 1 and z 2 = |z 2 |e iϕ 2 differs from |z 1 | 2 + |z 2 | 2 because of the relative phase ϕ = ϕ 1 -ϕ 2 : |z 1 + z 2 | 2 = |z 1 | 2 + |z 2 | 2 + 2|z 1 ||z 2 | cos ϕ.
The last term in the right-hand side is precisely the interference term.

have in atomic theory appear as the square of the modulus of some [complex] number which is a fundamental quantity. [. . . ] I believe that this concept of probability amplitude is perhaps the most fundamental concept of quantum theory.

[. . . ] The immediate effect of the existence of these probability amplitudes is to give rise to interference phenomena. If some process can take place in various ways, by various channels, as people say, what we must do is to calculate the probability amplitude for each of these channels. Then add all the probability amplitudes, and only after we have done this addition do we form the square of the modulus and get the total result for the probability of this process taking place. You see that the result is quite different from what we should have if we had taken the square of the modulus of the individual terms referring to various channels. It is this difference which gives rise to the phenomenon of interference, which is all pervading in the atomic world [. . . ]. So if one asks what is the main feature of quantum mechanics, I feel inclined now to say that it is not noncommutative algebra. It is the existence of probability amplitudes which underlie all atomic processes. Now a probability amplitude is related to experiment but only partially. The square of its modulus is something that we can observe. That is the probability which the experimental people get. But besides that there is a phase, a number of modulus unity which can modify without affecting the square of the modulus. And this phase is all important because it is the source of all interference phenomena but its physical significance is obscure. So the real genius of Heisenberg and Schrödinger, you might say, was to discover the existence of probability amplitudes containing this phase quantity which is very well hidden in nature and it is because it was so well hidden that people hadn't thought of quantum mechanics much earlier. (Dirac, 1972, pp. 154-158)

Feynman's paths

Half a century after Dirac showed the equivalence of Schrodinger's "wave mechanics" and the Heisenberg's "matrix mechanics" in a unified formalism, Feynman proposed in his thesis of 1942 [START_REF] Feynman | The principle of least action in quantum mechanics[END_REF] a third way of computing quantum predictions. Also equivalent to the first ones, Feynman's formalism, at the price of introducing a subtly new mathematical concept of functional integration, gives to the quantum interferences the first role 6 : the probability amplitude Z i→f for a system to evolves from an initial state i to a final state f are explicitely written as the result of the interference between all the possible histories the system may follow between the two states: up to a normalisation factor we can write 7

Z i→f = all possible histories h connecting i to f

|z h | e iϕ h (1)
Each virtual history h is weightened by a complex number z h whose phase is ϕ h . Only for a subset of histories that interfer constructively, we can think of a quasi-classical evolution-but still fuzzy at Planck's constant scales. Most of virtual histories, even when they have the same amplitude |z h |, have a phase that differ too much from the average of the classical bunch and their contribution is mostly destroyed by their neighbours (Fig. 3). One possible history of one particle is just given by one continuous path made of all possible positions in ordinary space (its trajectory) and the constructive interference occurs precisely when it satisfies Newton's (or Euler-Lagrange's, or Hamilton's) classical equations. However as soon as two or more particles evolve, even if non interacting, the appropriate place to describe histories is not the ordinary space anymore but a more abstract space, the configuration space (the same space on which the Schrödinger wavefunctions are defined). To simplify the discussion, we will assume that the initial state has been 6 The Dirac's quotation given at the end of section 3 is the transcription of a conference he gave in April 1970 for a general audience. Though he talks about the recent development in quantum electrodynamics, and though he defends the idea that this concept of probability amplitude is perhaps the most fundamental concept of quantum theory, surprisingly enough, Dirac does not mention Feynman at all in his text. The only very vague allusion I could find lies, perhaps, in the sentence immediately following the quotation above: If you go over the present day theory to see what people are doing you find that they are retaining this idea of probability amplitude (Dirac, 1972, p. 158).

7 The writing is simplified (it hides that the sum covers an infinite functional continuum) but captures the spirit of the famous Feynman path-integrals.

i f

Figure 3: For one particle starting at initial position i, its probability amplitude to find it in the final position f is given by the sum (1) over all histories i.e. all the continuous (non necessarily differentiable) possible trajectories connecting i to f . Only a bunch of trajectories in a neighbourhood-whose size is governed by Planck's constant-of the classical Newtonian trajectory (here the black parabola for a particle in a uniform constant force field) will contribute with a constructive interference (the phases are proportional to the classical action which is stationary). Any other bunch of trajectories far from the classical one (say around the blue lower trajectory) will bring a negligible contribution because their phases ϕ h vary extremely rapidly and provoque a destructive interference.

prepared with a determined number N of particles of the same species and this number will be maintained all along the evolution. In that case 8 , the configuration space has N D dimensions where D is the dimension of ordinary space (most of the time, obviously, it is 3 but in some condensed matter models, we shall see that D can be lowered to 2 or even 1). To try to visualise the evolution of such a system, one may come back to the ordinary D-dimensional space where inevitably each particle can be individualised by a numbering that is continuously followed as they evolve from a given initial state to a given final state. But as we explained in the previous section, such a numbering has no physical basis at the quantum level and any continuous permutation among them not only can but must be considered among the Feynman's histories. It is also crucial to keep in mind in the following that the inevitable interactions between particles (the photon being aside, see the last footnote) exclude the histories where two of them overlap (for the corresponding energy of such a configuration diverges which make the phase oscillating infinitely quickly which destroy the superposition).

8 When particles can be created or annihilated, one cannot avoid working with fields whose configuration space is infinitely dimensional. This is almost unavoidable when dealing with quantum electrodynamics since photons are massless particles and thus can have an arbitrary low individual energy; therefore are cheap to create and easy to absorb. It requires a tremendous experimental skill to preserve a fixed number of photons for a while (in a superconducting cavity for instance).

Where the topological properties come from

It is to [START_REF] Dirac | Quantised singularities in the electromagnetic field[END_REF] that we owe the first identification of a quantum property of topological origin, namely a universal constraint on the electric charges if there would exist magnetic monopole 9 . By Feynman's own admission, [START_REF] Dirac | The lagrangian in quantum mechanics[END_REF] was also an inspiring source for the implementation of the ideas of the sum over histories briefly sketched in the previous section. Indeed, the sum over histories in (1) probes directly the topology of the space where the histories take place, namely the configuration space. In particusince the histories are necessarily continuous they can be classified according to the so-called first homotopy group of the configuration space, each homotopy class being made of all the histories that can be continuously deformed one into another.

For one particle in ordinary space, most often, the homotopy properties are trivial in the sense that only one class generally exists. Like in Fig. 3, all the paths can be continously deformed one into another 10 . But as soon as at least two identical particles are involved more than one homotopy class should be considered.

Whenever several homotopy classes are present, some new possibilities are offered in the Feynman's approach [START_REF] Mouchet | Path integrals in a multiply-connected configuration space (50 years after)[END_REF], and its references for a rigourous justification). Because of the well-behaved composition law of histories together with some conservation of probabilities, one can attribute to each class c a phase χ c that depends only on c and not on any specific choice of one of its member. This is why they are called topological phases and then (1) can be extended to (2)

Z i→f =
Of course when only one class is present or if we take χ c = 0 for all classes one recovers (1) but it happens that other choices are actually realised.

Permutations and braids

Consider for instance N = 3 identical particles. Figure 4 provides two examples of histories represented in ordinary space with the time being given by the vertical axis.

As explained above, because the particles cannot overlap, we can follow each individual trajectory by a thread that allows to keep their individualisation (by a numbering or, more visually, a color). In the two examples drawn, the two histories differ by a permutation of particles: in the final states the end position of the "green" thread has been exchanged with the end position of the "red" one. And precisely because of this permutation, one cannot deform one history into another without cutting two threads before gluing the pieces appropriately which would break the continuity. The representation chosen here can involve two spatial dimensions only (the horizontal plane). When D = 3 one can imagine that the motion in the third spatial dimension is represented by a change of darkness in the color of the thread. With this image in mind, one can understand that when D = 3 one can have the thread crosses one with an 9 Monopoles would come with a violation of the Maxwell equation divB = 0. 10 One can manufacture on purpose some holes in space by creating zones that are forbidden to the paths by a magnetic field. This is the celebrated geometry proposed by Ehrenberg-Siday-Aharonov-Bohm [START_REF] Ehrenberg | The refractive index in electron optics and the principles of dynamics[END_REF]Aharonov andBohm, 1959, 1961) where quantum topological phases are involved even for just one particle. other even though two particles still cannot be at the same place at the same moment (Fig. 5). This latter constraint forbids two threads to cross at some point where their darkness is the same (all the three coordinates would coincide) but one can always bypass this restriction by changing the the darkness of one thread at a point of crossing (that is moving it in the third spatial dimension) then cross the threads at this point and restore the original darkness after this operation. In summary, for D = 3, when using the graphical representation of an history given in figure 4 or 6, the thread can be crossed but not in D = 2. This latter statement makes all the topological difference between D = 2 and D = 3. In the latter case, on can see that each homotopy class is in fact an ordinary permutation, whereas the structure of the classes in D = 2 is much richer. By concatenating the threads in order to represent the composition of two evolutions, we introdcuce naturally an algebraic internal law that makes the set of classes a group (the so-called fondamental homotopy group of the configuration space). Then, by a straightforward generalisation to N identical particles, in D = 3 the topological group is simply the permutation group of N elements whereas in D = 2, the group is called the braid group of N strands. Moreover, it can be shown that the topological phases must naturally compose accordingly: if c • c ′ stands for the class obtained by concatenating the N threads in c with the N threads in c ′ , then, is can be shown that we must have e iχ c•c ′ = e iχc e iχ c ′ .

(

) 3
For D = 3, the latter relation leaves us with a simple alternative since the (2p + 1) th iteration of a transposition of two threads, p being an integer, leaves us with the trans-

for all the classes c associated with a transposition of two identical particles. The first choice corresponds to bosons and the second to the fermions. For D = 2, if c is a bread with two threads associated with a transposition one can iterate the concatenation of the same bread made of two threads an arbitrary number of times and the result is always a different braid because of the interdiction of crossing the threads (Fig. 6). The algebraic constraints on the bread group are much less restrictive that for the permutation group and in particular we can take consistently e iχc = e iθ [START_REF] Leibniz | New essays on human understanding[END_REF] where θ is an angle, that characterises the species of the identical particles under consideration: it continuously interpolates between the bosons (θ = 0) and the fermions (θ = π). These is precisely this quantity that characterises an anyon. This phase beeing relative to some terms in a quantum amplitude of probability, they have observational consequences through interferences. This is the way we can connect the statistical physical properties of N identical quantum particles (the bosonic, fermionic or even anyonic character under permutations) and some topological properties (the first homotopy group of their configuration space).

Figure 1 :

 1 Figure 1: On the left: The Herrenhäusen Gardens around 1708 (wikipedia from Gottfried-Wilhelm-Leibniz-Bibliothek, Kartensammlung C Mappe 18 Nr. 178 b). On the right: Leibniz with Duchess Sophie, Carl August von Alvensleben and two ladies-inwaiting in the Herrenhäusen Gardens (wikipedia from a biography of Leibniz by Johann August Eberhard published in 1795).

  all homotopy classes c of histories connecting i to f e iχc all histories h in c |z h | e iϕ h .

Figure 4 :

 4 Figure 4: Representation of two Feynman histories (or paths in the configuration space) for N = 3 particles. Time axis is chosen to be vertical while ordinary D-dimensional space is perpendicular to it. Since on these examples, the red and green threads do not connect the smae initial and final points, these histories cannot be continously deformed one into the other and therefore paths a) and b) belongs to two different homotopy classes.

Figure 5 :

 5 Figure5: In these graphical representations of a path for N = 2 particles, the time axis is plotted vertically while the ordinary space is perpendicular. When D = 3, moving in the third spatial dimension can be thought as changing the darkness of the threads. Because no particle can be at the same place at the same time, in D = 2 the threads cannot cross. In D = 3, one can always continuously separate them in the third dimension without any cut: in step 1 a darkening of a portion of one thread (here the green one) and a lightening of a portion of the other (the red one). Then, a crossing of these two portions in the two other dimensions is possible and, eventually, in step 3 one can restore the initial value of the third spatial coordinate.

  time

Figure 6 :

 6 Figure 6: In D = 3, the crossing of two threads being possible, one can always entangle a succession of an arbitrary number of exchanges of two particles, leaving us with two homotopy classes only (the identity and the transposition). In D = 2, the braid such obtained is always different from the previous ones and the homotopy classes can be labeled by a unbounded integer.

plato.stanford.edu/entries/identity-indiscernible

Some experiments have succeeded the technical challenge of keeping for several weeks one particle sufficiently localised away from the others. However, on the other hand, the correlations of entangled pairs over long distances show that the individuality cannot be based on spatial separation. Connected to the subject of the present text, the quantum contribution to the Western philosophical analyticreductionnist/holistic-emergentist dialectics concerning the relationships between the part and the whole is fascinating(Maudlin, 1998, for instance).

[START_REF] Leibniz | New essays on human understanding[END_REF] Some other interpretations try to fix what happens to be, as a last resort, a question of interpretation of quantum probabilities: alike what occurs in the classical world, do the latter reflect a lack of information or not? But as far as I know, these de Broglie-Bohmian points of view[START_REF] Bricmont | Making Sense of Quantum Mechanics[END_REF], for a particularly interesting plea) concern a fixed number of particles only (generally one) and do not venture in the quantum field arena where even elementary particles can be created or annihilated.


Keywords: Duct acoustics, guided waves, exceptional point, Puiseux series, Green's function, Non-Hermitian physics

This study focuses on sound attenuation in a two-dimensional waveguide with arbitrary admittance boundary conditions on both sides of the guide. The emphasis is on understanding the formation and potential applications of the exceptional points (EPs) which arise when two (EP2) or three (EP3) modes degenerate into a single mode. A perturbation approach is used to obtain asymptotic expressions for the trajectories of the axial wavenumbers in the complex plane as they coalesce to form an EP. The numerical results presented herein suggest that the first triple root (EP3) assures maximum modal attenuation along the waveguide. Further, it is demonstrated that the classical Green's function is degenerate at an EP. Modified Green's functions which are valid at EP2 and EP3 are presented.

Mode coalescence and the Green's function in a two-dimensional waveguide with arbitrary admittance boundary conditions E. Perrey-Debain * , B. Nennig + , J.B. Lawrie ++

Introduction

In his seminal article [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF], Tester considered sound attenuation in a duct with one rigid wall and one lined. He built upon the work of Cremer [START_REF] Cremer | Theory regarding the attenuation of sound transmitted by air in a rectangular duct with an absorbing wall, and the maximum attenuation constant produced during this process[END_REF] who showed that the optimum attenuation within such a duct occurs when a mode pair degenerates into a single mode (an exceptional point). Tester derived the Green's function for this situation and showed that computed results for the attenuation compared favourably with experimental data. In the same decade, Zorumski [START_REF] Zorumski | Multiple eigenvalues of sound-absorbing circular and annular ducts[END_REF] extended Tester's analysis to the cases of circular ducts with locally reacting liners and Koch [START_REF] Koch | Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow[END_REF] used a generalised Wiener-Hopf approach to study attenuation in a duct with different sections of lining (on both walls). Surprisingly, it took many years before the concept of 'optimal impedance' became a subject of intensive research. Recently, however, interest in the topic has burgeoned and a number of papers have appeared in the literature. Most of these research works deal with circular and annular ducts with flow and have potential application to noise suppression within ducts in aero-engines, gas turbines, blowers and various mufflers [START_REF] Kabral | Optimum sound attenuation in flow ducts based on the "exact" Cremer impedance[END_REF][START_REF] Qiu | The Cremer concept for annular ducts for optimum sound attenuation[END_REF][START_REF] Zhang | The Cremer impedance: An investigation of the low frequency behavior[END_REF][START_REF] Xiong | Sound attenuation optimization using metaporous materials tuned on exceptional points[END_REF][START_REF] Spillere | Optimum acoustic impedance in circular ducts with inviscid sheared flow: Application to turbofan engine intake[END_REF][START_REF] Sánchez | Theoretical and numerical investigation of optimal impedance in lined ducts with flow[END_REF]. Additionally, [START_REF] Bi | New insights into mode behaviours in waveguides with impedance boundary conditions[END_REF][START_REF] Kelsten | Modeling of acoustic waves in pipes with impedance walls and double roots[END_REF] consider mode-matching for finite lined region, [START_REF] Guo | Spatial transient behavior in waveguides with lossy impedance boundary conditions[END_REF] examines the relationship between the the nature of the source and the transmitted power whilst [START_REF] Shenderov | Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls[END_REF] discusses the form of the additional wavefunctions required at an exceptional point. More recently, the present authors have proposed a numerical algorithm which enables them to explore the trajectories of the eigenvalues in the vicinity of an exceptional point in a systematic way [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF]. Despite the recent progress a rigorous description of the mode coalescence, the associated wavefunctions and the modified the Green's functions is still missing from the literature. With the view both to extending the work of Tester and filling this gap, the present study focuses on sound attenuation in a two-dimensional waveguide with arbitrary admittance boundary conditions on both sides of the guide.

In section 2 the boundary value problem is stated and the wave functions associated with mode coalescence are introduced. The analysis starts with the classical solution method which involves expanding the solution in terms of normal modes of separable form: Y (s, µ, y)e isx-iωt where s is the axial wavenumber and ω = ck in which c is the fluid sound speed and k the wavenumber. For convenience and to aid future manipulations, the lower wall admittance µ is included as a variable in the transverse wavefunction. As usual the precise form of Y (s, µ, y) is obtained by solving a non-Hermitian eigenvalue problem [START_REF] Morse | Theoretical acoustics[END_REF] in which the dispersion relation is even in the transverse wavenumber α = (k 2 -s 2 ) 1/2 . It is well known, however, that there are values of the duct wall admittance(s) for which this approach breaks down due to the existence of non-separable solutions associated with multiple-roots of the dispersion equation [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF][START_REF] Zorumski | Multiple eigenvalues of sound-absorbing circular and annular ducts[END_REF][START_REF] Shenderov | Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls[END_REF][START_REF] Kelsten | Modeling of acoustic waves in pipes with impedance walls and double roots[END_REF]. These roots correspond to the coalescence of two (EP2) or three (EP3) acoustic modes and special care must be taken to derive the corresponding waveform. It is shown that the location of EPs in the complex-plane can be described as a function of the product of the two admittances, and the notation ᾱ is used to denote the transverse wavenumber in this situation. The asymptotic forms (first presented by Tester [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF] and Koch [START_REF] Koch | Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow[END_REF]) for the value of ᾱ in the case of one lined and one rigid wall are stated and compared with precise numerical values. A mapping of the complex ᾱ plane is presented for the fully lined situation.

Section three presents an analysis of the behaviour of the roots coaelescing in the vicinity of an optimal point (ie. when wave attenuation is maximized). The lower wall admittance (μ) is kept fixed whilst the wavenumber s and the upper wall admittance are perturbed from their EP values. A double Taylor series is used to expand the dispersion relation and this is then inverted using a Puiseux series. This approach enables the trajectories of the coalescing roots to be plotted in the complex plane for both EP2 and EP3. The link with Cremer's optimum impedance [START_REF] Cremer | Theory regarding the attenuation of sound transmitted by air in a rectangular duct with an absorbing wall, and the maximum attenuation constant produced during this process[END_REF] based on the axial attenuation rate is established. It is also shown that there are situations where the acoustic field can exhibit a linear amplification along the waveguide axis with no dissipation at all, that is with purely real axial wavenumber. Further, the numerical results presented herein suggest that the first EP3 (resulting from the coalescence of the first three modes) provides maximum attenuation along the waveguide.

Although interest in exceptional points (EPs) has recently arisen in a variety of physical situations [START_REF] Midya | Modes and exceptional points in waveguides with impedance boundary conditions[END_REF][START_REF] Doppler | Dynamically encircling exceptional points in a waveguide: asymmetric mode switching from the breakdown of adiabaticity[END_REF][START_REF] Ding | Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization[END_REF][START_REF] Heiss | Greens functions at exceptional points[END_REF], the Green's function for a lined duct at an EP has not, as far as the authors are aware, appeared in the literature except for the case of one hard wall investigated by Tester. This deficiency is addressed in section four where an analysis of the Green's function is presented for EP2 and EP3 using the same methods as in section three. It is demonstrated that the results of Tester are retrieved for the hard wall case. The main results of this article are summarised in section five, together with a brief discussion of their implications for noise control and some suggestions for further work.

Analysis of the dispersion equation

Using the usual two-dimensional Cartesian frame of reference, we consider a two-dimensional duct of unit width and local boundary conditions:

∂ y ψ = -µψ, at y = 0 and ∂ y ψ = νψ, at y = 1, (1)
where µ, ν are the wall admittances. Note that harmonic time dependence e -iωt is assumed and thus ψ(x, y) is the reduced fluid velocity potential. Solutions of the Helmholtz equation

(∂ 2 xx + ∂ 2 yy + k 2 )ψ = 0, (2)
have the general separable form ψ = Y (s, µ, y) e isx where Y (s, µ, y) = cos(αy) -µ α sin(αy) with α = k 2 -s 2 .

(3)

On using the second condition of Eq. ( 1), it is clear that

(νψ -∂ y ψ) y=1 = K e isx , ( 4
)
where function K is interpreted as a function of s, µ and ν and is defined by

K(s, µ, ν) = (ν + µ) cos α + α - νµ α sin α. (5)
Thus, ψ is a solution provided α is a root of the dispersion equation

K(s, µ, ν) = 0. ( 6
)
Note that, whilst α depends on both µ and ν, it is not directly dependent on frequency. Also, it should be observed that functions K and Y are both holomorphic with respect to all their arguments. Because these are even functions of α, they are also holomorphic in s and, when necessary, we can apply the chain rule ∂ s ≡ -s/α ∂ α . Now, by differentiation, we can define a new wave function

ψ = ∂ s ψ = (Y + ixY )e isx (7)
with

Y = ∂ s Y = sy sin(αy) α + sµy cos(αy) α 2 - sµ sin(αy) α 3 ( 8
)
and it is easy to see that

∂ s (νψ -∂ y ψ) y=1 = (νψ -∂ y ψ ) y=1 = (K + ixK)e isx . ( 9
)
Thus ψ (which is not of separable form) is also solution if the double root problem K = 0 and K = ∂ s K = 0 is satisfied. In this case, after some manipulation, we arrive at the set of equations

p = μν = ᾱ2 2ᾱ + sin(2 ᾱ) 2ᾱ -sin(2 ᾱ) and q = μ + ν = -tan ᾱ ᾱ - p ᾱ . ( 10
)
Wall parameters which are associated with an exceptional point of order 2 (EP2) are recovered via

μ = q -q2 -4p 2 and ν = q + q2 -4p 2 . ( 11
)
Note that (μ, ν) as defined by Eqs (11) form a continuous set since ᾱ can be regarded as an arbitrary complex-valued parameter. From Eq. (10), we recognize the two limit cases corresponding to one hard wall condition p = 0 (where the reader is reminded that p is not the pressure) and one pressure release condition |p| → ∞, given by the relations

2ᾱ ± sin(2 ᾱ) = 0. ( 12
)
Approximate solutions have been given independently by Tester and Koch [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF][START_REF] Koch | Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow[END_REF] and these are presented here for completeness:

2ᾱ m ≈ 2m + 3 2 π -i ln((2m + 3)π) m = 0, 1, 2, . . . , (13)
where m even corresponds to the hard wall condition μ = 0 giving ν = -ᾱ tan ᾱ and m odd to the pressure release condition giving ν = ᾱ cot ᾱ. Approximate and exact (by this we mean that results were computed with sufficient accuracy) values of the roots ᾱm , m = 0, 1, 2, . . . are presented in Tables 1 and2.

Only roots associated with absorbing boundary conditions (i.e. the imaginary part of ν or equivalently the real part of the wall impedance must be positive) are shown. Other solutions associated with 'active' boundary conditions exist and are obtained from ᾱ by taking the complex conjugate. To illustrate this in a more systematic way, it is convenient to study the values of the wall admittances on both sides in the complex-plane as a function of ᾱ = (Re ᾱ, Im ᾱ) as shown in Figure 1. This can be m Approx.

Exact ν = -ᾱ tan ᾱ 0 2.3561 -1.1216i 2.1061 -1.1253i 1.6506 + 2.0599i 2 5.4977 -1.5453i 5.3562 -1.5515i 2.0578 + 5.3347i 4 8.6393 -1.7713i 8.5366 -1.7755i 2.2784 + 8.5226i 6 11.780 -1.9263i 11.699 -1.9294i 2.4311 + 11.688i regarded as a representation of the EP2 complex ᾱ-plane since associated values of the wall parameters satisfy the double root condition and to each value of ᾱ corresponds an unique pair of complex-valued wall-parameters (ν, μ) through Eqs [START_REF] Shenderov | Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls[END_REF]. Three regions can be clearly identified: (i) a lower region with absorbing walls on both sides, (ii) an upper region with active walls on both sides and (iii) an intermediate region with both active and absorbing walls. Particular roots corresponding to hard wall and pressure release conditions belong to a continuous curve that delimitates these regions. Dissipative scenarios of region (i) are usually discussed in the scientific literature as they can lead to practical realisations. Note that a similar description can be found in Kelsten's MSc Thesis (see Fig. 3

.3 p. 18 in [14]).

There is a further set of interesting solutions satisfying the triple root condition:

K = K = K = 0. ( 14
)
Following the previous derivation given by Eqs ( 7)-( 9), it is easy to see that this condition leads to the existence of another wavefunction:

ψ = ∂ 2 s ψ = (Y + 2ixY -x 2 Y )e isx . ( 15
)
The additional constraint leads to the following equation for ᾱ where the double overbar is used to indicate that this is an EP3:

4 cos ᾱ - sin ᾱ ᾱ2 (2 ᾱ + sin(2 ᾱ)) = 0. ( 16
)
It should be noted that whilst ᾱ is a continuum, ᾱ comprises a discrete set of values. Koch gives an approximation for these roots [START_REF] Koch | Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow[END_REF]:

2 ᾱm ≈ (2m + 3)π -i[ln 2 + 2 ln((2m + 3)π)] m = 0, 1, 2, . . . ( 17
)
There exit also purely real solutions which are not mentioned in Koch's paper, probably because they do not correspond to absorbing walls. These roots admit the following approximation:

2 ᾱm ≈ (2m + 3)π - 2 (2m + 3)π , m = 0, 1, 2, . . . ( 18
)
Approximate and exact values of the triple roots are presented in Tables 3 and4 and also in Fig. real eigenvalues for non-Hermitian operators and this is typical of PT -symmetric systems which have attracted intensive research interest in recent years (although the discussion is outside the scope of the present paper, we can cite for instance [START_REF] Schnabel | PT-symmetric wave guide system with evidence of a third-order exceptional point[END_REF]). Finally, we note in passing that the case p = q = 3, gives rise to a fourth order root at α = 0, this case is not considered in this paper. Before we end this section we should note that numerical solutions of the dispersion equation, Eq. ( 6), as well as double and triple roots solutions of Eqs ( 12) and ( 16) are computed with the method proposed in [START_REF] Delves | A numerical method for locating the zeros of an analytic function[END_REF] and implemented for a dissipative silencer by the present authors in [START_REF] Nennig | A mode matching method for modelling dissipative silencers lined with porous elastic materials and containing mean flow[END_REF]. The approach which is based on Cauchy's theorem allows the zeros of analytic functions to be located within a closed path in the complex plane. The quality of the solution can be checked by substituting the roots into the dispersion equation and a refinement strategy is also possible around each root in order to increase the accuracy.

m

Perturbation analysis and optimality

In the dissipative scenario where both walls are absorbing, it is advocated in many papers, that double root solutions are optimal in the sense that coalescence of the first two modes coalesce yields optimal treatment in the sense that this should provide the best attenuation, i.e. the associated axial wave number has the strongest imaginary part. The reason for this can be found by looking at the behaviour of the solution in the vicinity of the optimal point. To do this, we introduce parameters δ = ν -ν and = s -s and try to identify the relationship between these two quantities. The starting point for the analysis is to consider a Taylor expansion of K around (s, μ) while the admittance of the lower wall μ is kept fixed. Thus,

K(s, μ, ν) = ∞ i=0 ∞ j=0 i δ j i!j! ∂ i s ∂ j ν K(s, μ, ν). (19)
Note that, this series simplifies significantly since

∂ i s ∂ j ν K(s, μ, ν) = 0 for j ≥ 2 and Y (s, µ, 1) = ∂ ν K(s, µ, ν). ( 20
)
Further, we can make the analysis tractable by adopting the following notations:

K i = ∂ i s K(s, μ, ν) i! and Y i = ∂ i s ∂ ν K(s, μ, ν) i! = ∂ i s Y (s, μ, 1) i! ( 21
)
and this gives

K(s, μ, ν) = K 0 + K 1 + 2 K 2 + 3 K 3 + 4 K 4 + 5 K 5 + . . . + δ(Y 0 + Y 1 + 2 Y 2 + 3 Y 3 + 4 Y 4 + . . .) (22)
By construction K(s, μ, ν) is zero everywhere. Thus, the double root condition, K 0 = K 1 = 0, leads to the equation

2 K 2 + 3 K 3 + 4 K 4 + 5 K 5 + δ(Y 0 + Y 1 + 2 Y 2 + 3 Y 3 + 4 Y 4 ) + . . . = 0. ( 23
)
This Taylor series can be inverted, at least locally by assuming small perturbations, using a Puiseux series expansion. To do this, we shall assume for the moment that K 2 = 0 so that we can formally write = a

√ δ + bδ + . . . ( 24
)
which, after substitution, gives the explicit form for the expansion coefficients

a 2 = - Y 0 K 2 and b = Ba 2 with B = 1 2 Y 1 Y 0 - K 3 K 2 . ( 25
)
Eqs ( 23) - [START_REF] Lawrie | On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries[END_REF] indicate that ν is a branch point in the complex ν-plane so the variation in s is infinitely sensitive as (νν) tends to zero. That this should provide an optimal value, at least locally, can be understood from a graphical point of view as the two eigenvalues, call them s 1 and s 2 , coalesce in opposite directions and in fact

n = s n -s = (-1) n a √ δ + Ba 2 δ + . . . , n = 1, 2. ( 26
)
The result is a generalization of Tester's formula corresponding to the specific case of one rigid wall, i.e. μ = 0. When the triple root condition is satisfied, K 2 = 0 and the Puiseux series Eq. ( 24) breaks down. To remedy this, one needs to seek an alternative expansion of the form

= aδ 1/3 + bδ 2/3 + cδ + . . . ( 27
)
which, after substitution, gives the explicit form for the expansion coefficients

a 3 = - Y 0 K 3 , b = Ba 2 and c = Ca 3 (28)
where

B = 1 3 Y 1 Y 0 - K 4 K 3 (29) C = 1 3
K 4 K 3 2 + Y 2 Y 0 - Y 1 Y 0 K 4 K 3 - K 5 K 3 (30)
In the close vicinity of the triple root, the three eigenvalues coalesce as follows

n = s n -s = z n a δ 1/3 + z 2n Ba 2 δ 2/3 + Ca 3 δ + . . . , n = 1, 2, 3 (31)
where z 3 = 1 is the cube root of unity. In order to illustrate this, it is convenient to consider the variation of the transverse wavenumber α. It is found that, to leading order

α n = ᾱ + z n δ 1/3 -3! ∂ ν K( s, μ, ν) ∂ 3 α K( s, μ, ν) 1/3 + O(δ 2/3 ), n = 1, 2, 3. ( 32
)
Take the first triple root ᾱ0 = 4.1969 -2.6086i given in Table 3 for instance, we find that

α n ≈ ᾱ0 + z n (ν -ν0 ) 1/3 (0.6886 -0.7678i), (33)
with ν0 = 3.1781 + 4.6751i. In Fig. 2, computed roots of the dispersion equation Eq. ( 6) showing the coalescence of 3 classical modes in the vicinity of the triple root ᾱ0 are shown. The real part of α, which gives the attenuation of waves at low frequency, reaches a maximum value at the EP3 which means that the latter corresponds to an optimal treatment. Another way to observe the crossing of modes is shown in Fig. 3 where trajectories of the roots in the complex plane are illustrated (left) by solving Eq. ( 6) (recall that ν = δ + ν0 ) by gradually varying δ on the real axis from -0.1 (indicated by the "+") to 0.1 (indicated by the "o"). The right hand figure shows the trajectories using the leading order expression Eq. (33) which gives a reasonable approximation. Values, represented as black squares, are calculated using the same incremental step for both figures and this shows how sensitive modal solutions are near EP3.

The question now arises as to whether the exceptional points EP3 give best attenuation in all cases. To investigate this, we consider the following linear trajectory for the tranverse wavenumber in the EP2 ᾱ complex plane as defined earlier: where parameter η is a positive real number and ᾱ0 corresponds to the first solution of the hard wall case (ν = 0) (see Table 1). Thus, when η = 0 we are at the red star in Figure 5 a). As η is incrementally increased the corresponding values of the wall admittances μ and ν, which must be regarded as functions of ᾱ, are calculated via Eq. ( 11); their trajectories are represented in Figure 4 and show a cusp at the triple root (EP3). For each pair (μ, ν), the first four roots of the dispersion relation

ᾱ = ᾱ0 (1 -η) + ᾱ0 η, η ≥ 0, (34)
(ν + μ) cos α + α - ν μ α sin α = 0, (35)
(where it is understood that ν = ν(ᾱ) and μ = µ( ᾱ)) are calculated and their trajectories are plotted in Figure 5 (a). One recognizes the linear path given by Eq. ( 34) and other trajectories correspond to classical modes (i.e. single roots). The coalescence of one of these modes with the double root occurs at ᾱ0 as expected. Note the coalescence is somehow atypical as is does not correspond to the merging of three classical modes and the description made earlier in Eq. (32) does not hold in this case. Corresponding axial wavenumbers are shown in Figures 5 (b)(c) and (d) for three frequencies k = π/10, π and 3π. At low frequency, s ≈ iᾱ which corresponds to a rotation of π/2 as shown in Figure 5 (b) and the gain, in terms of attenuation, is quite substantial when compared to the first Tester's optimal value ᾱ0 for the hard wall case, i.e. the imaginary part of the axial wavenumber is nearly twice as large at EP3. At higher frequency, the attenuation decreases but in every case considered, EP3 leads to best attenuation. In order to see this in more systematic way, let us consider again the EP2 ᾱ complex plane (we shall limit the analysis to the dissipative region (i) here). For each value of ᾱ, discrete solutions of Eq. ( 35) are computed and the imaginary part of the axial wavenumber s corresponding to the least attenuated mode is reported in Fig. 6. Results clearly suggest that the EP3 for the first triple root ᾱ0 provide the optimal pair of admittances ( ν0 , μ0 ) given in Table 3. As already discussed earlier (and also in Tester's original paper [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF] for the hard wall case µ = 0), this optimum condition is offset by an amplification rate which varies linearly and/or quadratically with distance and in fact |ψ | ∼ |xψ| and |ψ | ∼ |x 2 ψ| where |ψ| = |Y | exp -x Im (k 2 -ᾱ2 0 ) 1/2 . To the authors' knowledge this result has never been presented in the scientific literature. The attenuation rate of the mean square modal amplitude (the amplification being now ignored) is proportional to the imaginary part of the axial wavenumber, and in the usual units the attenuation is 8.69 Im s dB per duct width. For the sake of illustration, numerical values are reported in Fig. 7 for three cases of special interest (note k = π corresponds to the first cut-off frequency in a duct with rigid walls).

Green's function

Classical case

It is well known that the Green's function, which satisfies

(∂ 2 xx + ∂ 2 yy + k 2 )G(x, x 0 ) = δ(x -x 0 )δ(y -y 0 ) (36)
where x = (x, y) and x 0 = (x 0 , y 0 ) are the evaluation and the source point, can be constructed elegantly using a Fourier-transform and subsequent application of the residue theorem in the complex plane [START_REF] Rienstra | An analytic Green's function for a lined circular duct containing uniform mean flow[END_REF]. Simple poles are associated with classical duct modes and the occurrence of a higher order pole signifies that these modes are of a different nature. The procedure used in the present work, however, follows Tester's derivation [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF]. The starting point is to express the Green's function as an infinite series of the form

G(x, x 0 ) = ∞ n=1 Ψ(s n , x, x 0 ) 2iP n s n (37) where Ψ(s, x, x 0 ) = Y (s, µ, y)Y (s, µ, y 0 )e is|x-x0| (38)
with the requirement that Im s ≥ 0 in the dissipative case. In Eq. ( 37), the α n 's are the discrete roots of the dispersion equation ( 6) and the ordering in the summation is by increasing imaginary part of axial wavenumber, s n . Thus, s 1 is the least attenuated mode. The associated eigenfunctions, i.e. the duct acoustic mode are given explicitly by Eq. ( 3). The quantity P n is calculated from the integral

P n = 1 0 Y 2 (s n , µ, y) dy. ( 39
)
Note that following Lawrie's work [START_REF] Lawrie | On eigenfunction expansions associated with wave propagation along ducts with wave-bearing boundaries[END_REF][START_REF] Lawrie | On acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings[END_REF], it can be shown1 that where the reader is reminded that the prime indications differentiation with respect to s. This result represents one of the main ingredients of the procedure as it establishes the deep connection between P n and the dispersion equation of the waveguide eigenvalue problem. Thus, the series, which can take the alternate form

P n = - Y (s n , µ, 1) 2s n K (s n , µ, ν), (40)
G(x, x 0 ) = ∞ n=1 iΨ(s n , x, x 0 ) Y (s n , µ, 1) K (s n , µ, ν) , ( 41
)
is only valid if K = 0 for all modes involved in the series. The fact that P n can be zero whenever we are facing the existence of an exceptional point, stems from the fact that Eq. (39) does not define a norm and in this case EP modes are said to be self-orthogonal.

Case of an EP2

In this scenario, the wall parameters (μ, ν) correspond to the existence of an EP2 which means that one of the integrals in Eq. ( 39) is zero. To make progress, we must identify the pair of classical modes which lie close to the exceptional point. Without loss of generality, we label these two modes with indices n = 1 and n = 2 (although the reader is reminded that coalescence could occur with higher order modes). Consider a slight deviation of the optimal admittance ν = ν + δ while μ is fixed. In the spirit of the previous section, we can use the regularity of function K and expand it as a Taylor series around s and ν (recall that = s -s):

K (s, μ, ν) = 2 K 2 + 3 2 K 3 + 4 3 K 4 + 5 4 K 5 + . . . + δ(Y 1 + 2 Y 2 + 3 2 Y 3 + 4 3 Y 4 + . . .), (42)
and For the moment we may assume that K 2 = 0 so we can use the Puiseux series expansion Eq. ( 26). Application of Taylor's expansion with respect to (μ, ν), yields the asymptotic form

similarly with µ = μ Y (s, μ, 1) = Y 0 + Y 1 + 2 Y 2 + 3 Y 3 + 4 Y 4 + . . . ( 43
) 0 5 10 15 -6 -5 -4 -3 -2 -
Y (s n , μ, 1) K (s n , μ, ν) = (-1) n Λ 1 √ δ + Λ 2 δ + . . . ( 44
)
where

Λ 1 = 2aY 0 K 2 and Λ 2 Λ 1 = a Y 1 Y 0 + K 3 K 2 . ( 45
)
Function Ψ can also be regarded as regular function with respect to s and it can be expanded around s as

Ψ(s n , x, x 0 ) = Ψ(s, x, x 0 ) + n Ψ (s, x, x 0 ) + . . . = Ψ(s, x, x 0 ) 1 + (-1) n Φ 1 a √ δ + . . . ( 46
)
where, to ease the notation we put Ψ (s, x, x 0 ) = Φ 1 Ψ(s, x, x 0 ) with

Φ 1 = Y (s, μ, y) Y (s, μ, y) + Y (s, μ, y 0 ) Y (s, μ, y 0 ) + i|x -x 0 |. ( 47
)
Figure 7: Comparison of attenuation rate (in dB per duct width) for three scenarios: (i) optimal condition (with ᾱ0 , see Table 3) (solid line); (ii) optimal condition with one rigid wall (with ᾱ0 , see Table 1) (dash-dot); (iii) optimal condition with pressure release condition (dashed) at one wall (with ᾱ1 , see Table 2).

We can now calculate the limit for the first two terms of the series as follows

G # EP2 (x, x 0 ) = lim δ→0 n=1,2 iΨ(s n , x, x 0 ) Y (s n , μ, 1) ∂ s K(s n , μ, ν) = iΨ(s, x, x 0 ) lim δ→0 n=1,2 1 + (-1) n Φ 1 a √ δ + . . . (-1) n Λ 1 √ δ + Λ 2 δ + . . . ( 48
)
After some manipulation, this can be given in the more compact and tractable form:

G # EP2 (x, x 0 ) = iΨ(s, x, x 0 ) Y 0 K 2 Φ 1 - Y 1 Y 0 - K 3 K 2 . ( 49
)
Thus, the Green's function for this case can be expressed as

G EP2 (x, x 0 ) = ∞ n=3 Ψ(s n , x, x 0 ) 2iP n s n + G # EP2 (x, x 0 ). ( 50
)
As far as the authors' are aware, formula Eq. ( 50), which is one of the main results of this paper, does not appear in the scientific literature except for the hard wall scenario (μ = 0), investigated by Tester. Tester's result can be retrieved from Eqs (49)-(50), on noting that that Y 0 = cos ᾱ and Y 1 = -γ sin ᾱ (here we defined γ = -s/ᾱ). Further, it can be shown that (see details in Appendix):

K 2 = γ2 cos ᾱ and K 3 K 2 = 1 + γ2 s - γ 3 tan ᾱ. ( 51
)
So, from the characteristic equation associated with the hard wall condition, see Eq. ( 12), we obtain the following equalities: ᾱ2 (cos ᾱ) 2 = -ᾱ t and t2 + t ᾱ + 1 = 0 with t = tan ᾱ.

(52) Thus, we can calculate

i Y 0 K 2 - Y 1 Y 0 - K 3 K 2 = i ᾱ s2 t 1 + γ2 s - 4γ 3 t = i s3 s2 3 t2 + ᾱt -s2 . ( 53
)
Now, from Eq. ( 8) (with µ = 0), we have Y = cos(ᾱy) and Y = -γy sin( ᾱy). Thus,

Φ 1 = -γy tan( ᾱy) -γy 0 tan( ᾱy 0 ) + i|x -x 0 |, (54)
and we find

G # EP2 (x, x 0 ) = cos(ᾱy) cos(ᾱy 0 )e is|x-x0| (55)
× t is2 (sy 0 tan( ᾱy 0 ) + sy tan( ᾱy) + i ᾱ|x -

x 0 |) + i s3 s2 3 t2 + ᾱt -s2 ,
which is the exact form as reported (see Eq. ( 28) in [START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF]) 2 .

Case of an EP3

In this scenario, K 2 = 0 and expression Eq. ( 49) is no longer valid. We need to use the fact that three eigenvalues, labelled n = 1, 2, 3, coalesce at EP3 associated with the pair of wall parameters ( ν, μ) and their trajectories are described by the Puiseux series Eq. ( 31) with δ = µ -μ. Following the previous derivation, we find

Y (s n , μ, 1) K (s n , μ, ν) = z 2n Λ 1 δ 2/3 + Λ 2 δ + z 4n Λ 3 δ 4/3 + . . . , (56)
where

Λ 1 = 3a 2 Y 0 K 3 , Λ 2 Λ 1 = - 2a 3 2 Y 1 Y 0 + K 4 K 3 , ( 57
) and Λ 3 Λ 1 = a 2 9 7 Y 1 Y 0 2 + 7 Y 1 Y 0 K 4 K 3 -5 K 4 K 3 2 + 3 Y 2 Y 0 + K 5 K 3 . ( 58
)
Again, we shall use the fact that function Ψ is a regular function of s and we can expand it around s as (we need to expand the function up to second order here):

Ψ(s n , x, x 0 ) = Ψ( s, x, x 0 ) + n Ψ ( s, x, x 0 ) + 1 2 2 n Ψ ( s, x, x 0 ) + . . . . (59)
To ease the notation, we introduce function Φ 2 defined such that Ψ = Φ 2 Ψ and

Φ 2 = Y ( s, μ, y) Y ( s, μ, y) + Y ( s, μ, y 0 ) Y ( s, μ, y 0 ) -(x -x 0 ) 2 + 2 Y ( s, μ, y) Y ( s, μ, y) Y ( s, μ, y 0 ) Y ( s, μ, y 0 ) + 2i|x -x 0 | Y ( s, μ, y) Y ( s, μ, y) + Y ( s, μ, y 0 ) Y ( s, μ, y 0 ) . ( 60
)
We finally find the following expansion of Puiseux type

Ψ(s n , x, x 0 ) = Ψ( s, x, x 0 ) 1 + z n Φ 1 aδ 1/3 + z 2n BΦ 1 + Φ 2 2 a 2 δ 2/3 + . . . , (61)
where coefficient B is given explicitly in Eq. ( 29) and function Φ 1 is defined in Eq. (47) (with s, μ replaced by s, μ). We can now calculate the limit

G # EP3 (x, x 0 ) = iΨ( s, x, x 0 ) lim δ→0 3 n=1 1 + z n Φ 1 aδ 1/3 + z 2n BΦ 1 + Φ2 2 a 2 δ 2/3 + . . . z 2n Λ 1 δ 2/3 + Λ 2 δ + z 4n Λ 3 δ 4/3 + . . . = iΨ( s, x, x 0 ) Λ 1 lim δ→0 δ -2/3 3 n=1 z -2n + δ -1/3 3 n=1 Φ 1 az -n - Λ 2 Λ 1 z -4n + 3 n=1 (BΦ 1 + Φ 2 2 )a 2 - Λ 3 Λ 1 + Λ 2 2 Λ 2 1 z -6n -Φ 1 a Λ 2 Λ 1 z -3n . ( 62
)
After using the cubic root identity (j and p are arbitrary integers):

3 n=1 z jn = 3δ j,3p , (63)
(where z 3 = 1 and δ j,m is the usual Kronecker delta function), the singular terms disappear and this finally leads to

G # EP3 (x, x 0 ) = 3iΨ( s, x, x 0 ) Λ 1 Λ 2 2 Λ 2 1 - Λ 3 Λ 1 -Φ 1 a Λ 2 Λ 1 + BΦ 1 + Φ 2 2 a 2 . ( 64
)
This can be recast in the final form

G # EP3 (x, x 0 ) = iΨ( s, x, x 0 ) Y 0 K 3 Λ -Φ 1 Y 1 Y 0 + K 4 K 3 + Φ 2 2 , ( 65
)
where

Λ = Y 1 Y 0 + K 4 K 3 2 - Y 1 Y 0 K 4 K 3 + Y 2 Y 0 + K 5 K 3 (66)
which has a similar structure to Eq. (49). Closed forms expressions for all variables involved Y i and K i can be found in the Appendix. Finally, the Green's function for the EP3 scenario can be expressed as

G EP3 (x, x 0 ) = ∞ n=4 Ψ(s n , x, x 0 ) 2iP n s n + G # EP3 (x, x 0 ). (67)

Conclusions

A comprehensive analysis of the first two exceptional points (EP2 and EP3) has been presented for a two-dimensional waveguide with arbitrary admittance boundary conditions on both sides of the guide. It has been confirmed that ᾱ (the double root corresponding to EP2) is a continuum, that is to each value of ᾱ there corresponds a unique pair of complex-valued wall parameters (ν, μ), whilst ᾱ admits only discrete values. An EP is formed by the coalescence of two (or three) nearby eigenvalues (axial wavenumbers) and this process has been studied using a perturbation method. The dispersion relation was expressed as a double Taylor series which was then inverted using a Puiseux expansion to obtain approximate expressions for the coalescing eigenvalues in terms of s and s for fixed μ as ν → ν (see Eqs [START_REF] Rienstra | An analytic Green's function for a lined circular duct containing uniform mean flow[END_REF] and (31)). The trajectories predicted by Eq. (31) have been compared with those obtained by numerical solution of the dispersion relation and good agreement has be shown (Figure 3). Further, Figures 5 and6 suggest that optimum attenuation is achieved at EP3, although there is scope for further analytic/numerical work to verify this. The Green's function for the eigensystem has been considered. It has been demonstrated that the classical expression for the Green's function is degenerate at an EP. An analogous process to that used to study the coalescence of the eigenvalues, has enabled the authors to present modified Green's functions which are valid at EP2 and EP3. These contain additional terms which ensure the completeness of the eigenfunctions in these cases. It is worth commenting that standard modematching methods do not satisfactorily address the cases of EP2 and EP3 (such methods are, however, valid close to EP and have been used to investigate problems in this context). It is anticipated that the additional terms seen in the Greens functions must be incorporated into, and thus modify, the mode-matching procedure at EP. This is a topic of current investigation by the authors.

To conclude, the methods and results presented in this paper pave the way for complementary studies, for example, of the eigensystem underpinning the boundary value problem studied herein and/or of exceptional points in waveguides with wavebearing boundaries. It is anticipated that the results will be of use in various applications where noise control is desirable.

with h(α) = g (α)/α, here symbol means differentiation with respect to α and this notation is only adopted in this Appendix. Finally, straightforward calculations yield where it is reminded that the overbar symbol relates to EP2.

In the scenario of EP3 (identified with double overbar symbols), the calculations are somewhat tedious but it can be shown that:

K 3 = γ3 3
[h ( ᾱ) psin ᾱ], (A.9) which is required for the EP3 Green's function.

K 4 K 3 = γ 4

1 .Figure 1 :

 11 Figure 1: Representation of the EP2 complex ᾱ-plane. Isovalue of magnitude of the product of the two wall admittances in logarithmic scale (log 10 |p|) is shown. Zeros and poles of p correspond to hard wall (see Tab. 1) and pressure release conditions (see Tab. 2) at one wall. The dash-dot lines correspond to Im μ = 0 or Im ν = 0. Star symbols correspond to EP3 points associated with the existence of wavefunctions satisfying the triple root condition (14) (see Tab.3 and 4).

Figure 2 :

 2 Figure 2: Computed roots of the dispersion equation Eq. (6) showing the coalescence of 3 classical modes in the vicinity of an EP3. The real part of α is given in (a) and the imaginary part in (b).

Figure 3 :

 3 Figure3: Trajectories of the roots on the complex plane in the vicinity of an EP3. Values are either computed from the dispersion equation (left) or given by the leading order approximation Eq. (33) (right).

Figure 4 :

 4 Figure 4: Trajectories of the wall admittances associated with Eq. (34). The red stars indicate the η = 0 points and the black stars indicate the EP3 as given in first line of Tab. 3.

Figure 5 :Figure 6 :

 56 Figure 5: Trajectories of the first transverse (a) and first axial wavenumbers for three frequencies (b)-(d).

∂ 3 α

 3 K = -∂ α K -2 sin α + 2h (α)p, (A.4) ∂ 4 α K = -∂ 2 α K -2 cos α + 2h (α)p, (A.5) ∂ 5 α K = -∂ 3 α K + 2 sin α + 2h (α)p. (A.6)Function K can also be regarded as a regular function of the axial wavenumber s. By applying the chain rule: ∂ s ≡ γ ∂ α where γ = -s/α repeatedly, we can calculateK 2 = γ2 [h(ᾱ) p + cos ᾱ]

Y 0

 0 = cos ᾱ -g(ᾱ)μ, (A.12)Y 1 = s (g(ᾱ) + h( ᾱ)μ) ᾱ) + h( ᾱ) μᾱγ 2 (g ( ᾱ) + h ( ᾱ) μ) , (A.14)

Table 1 :

 1 First 4 double roots (EP2) for the hard wall condition (lower wall).

	m	Approx.	Exact	ν = ᾱ cot	ᾱ
	1 3.9269 -1.3770i 3.7488 -1.3843i 1.8952 + 3.7194i
	3 7.0685 -1.6709i 6.9499 -1.6761i 2.1802 + 6.9329i
	5 10.210 -1.8548i 10.119 -1.8583i 2.3605 + 10.107i
	7 13.351 -1.9889i 13.277 -1.9915i 2.4929 + 13.268i

Table 2 :

 2 First 4 double roots (EP2) for the pressure release condition (lower wall).

Table 3 :

 3 First 5 triple roots (EP3) associated with absorbing conditions.

Table 4 :

 4 First 5 triple roots (EP3) associated with active and absorbing conditions on the real axis. Large order values can be estimated simply via νm ≈ 1 + i ᾱm.

The specific case corresponding to the pressure release condition Y (sn, µ, 1) = 0 is not included in the present analysis.

In Tester's paper, the width of the waveguide is h and Eq. (28) is recovered by simply replacing transverse and axial wavenumbers ᾱ and s by kyn h and kxn h. Note that the two formulae differ by a factor 2 which is due to the form of the Green's function series given in Eq. (20) in[START_REF] Tester | The optimisation of modal sound attenuation in ducts, in the absence of mean flow[END_REF].

Appendix A. Derivatives of the dispersion equation

We recall that (p = νµ and q = ν + µ)

where g = sin(α)/α stands for the cardinal sine function. Following standard algebra, we obtain

and ∂ 2 α K = -K + 2 cos α + 2h(α)p, (A.3)