Pascal Prefix

Patrizia Lombardo

P Engel

Keywords:

It is often said that we can have reasons for our emotions. But can such reasons be the basis for some form of knowledge? I attempt here to give a positive answer to this question, through an examination of two negative emotions, anger, and contempt. I suggest that these emotions are apt to deliver, albeit in an indirect way, a form of moral knowledge, and examine their expression in the writings of Jonathan Swift.

Emotional Reasons and Justification

Whether or not one agrees that knowledge is justified true belief, knowledge at least involves having reasons for what one knows, whether or not we have access to these. Can emotions give us reasons and can they be reasons? The answer depends of course upon the theory of emotions that one holds. There must also be enough similarity between reasons for emotions and reasons for other familiar attitudes, such as belief and desire. I shall rely on four assumptions which have been accepted by a number of writers on emotions. The first is that emotions are intentional states or episodes, which have a specific content which is most of time propositional, but which can also be directed at an object: one is afraid that p or of someone or something, happy that p, angry that p or at something or at someone. The second is that emotions are associated with typical bodily reactions, and have a specific phenomenology or feel, although the variations can be wide (sometimes it does not 10 The Grapes of Wrath and Scorn 3 feel in a specific way to be afraid, and there are cold angers). A third assumption is that emotions involve a certain kind of appraisal, related to their valence: they are negative or positive. A fourth assumption is that emotions have a formal object, which is not the object which they have as token episodes (such as fear of this dog, anger at this person at this very moment), but the type of object at which their contents are directed. This idea is in general expressed in the following way: fear's formal object is what is fearable or frightening, anger's formal object is what is worthy of irritation. Each emotion has its own formal object. An emotion is correct when it fits its formal object. In this sense one can speak of emotional truth, when the emotion toward a certain content or object is appropriate to its formal object [START_REF] Mulligan | From Appropriate Emotions to Values[END_REF]de Sousa 2011;[START_REF] Deonna | The Emotions[END_REF]. If we think of emotions as attitudes and dispositions rather than as specific episodes, we can ascribe to them properties which they share with epistemic attitudes such as beliefs, doubts or presumptions. Thus we can also say that emotions are justified when they fit their formal object. Emotions can have reasons. We say that we have reasons for being afraid or angry. They can also be reasons for epistemic attitudes. Thus my being surprised that Mary came may be a reason for my fear that Jane would not like it. Emotions can be factive: to be disgusted at something, or to be horrified at something seem to presuppose that this something exists [START_REF] Gordon | The Structure of Emotions[END_REF].

Does the fact that emotions involve these epistemic liaisons entail that they have a genuine epistemic role, such as being justifiers for epistemic attitudes? It is one thing to say that they can have cognitive role or value, and another to say that they can be justified, possibly true, or that they can yield knowledge. If emotions can have reasons or if they can be reasons, they must have the properties usually ascribed to reasons [START_REF] Skorupski | The Domain of Reasons[END_REF]. Moreover the reasons have to be epistemic. For this there must be a relation between an emotion as reason and a certain attitude of -ing. The emotion as reason has to be a relation to a fact. If the reason is epistemic, it has to be a matter of degree, and is most of the time relative to circumstances. If it can justify an attitude, it has to be sufficient for justification. Thus my reason to be angry at the rise of the price of fuel is the fact that the taxes for fuel went up. My anger will be justified if the raise of prices of fuel is a sufficient reason for being angry. We can also say that it is correct because it is fitting to the circumstance. But what is it for an emotion to be fitting? Some angers are justified, some others are not. The price of fuel often rises up. But when does it become a reason for anger? When we are told that Achilles was angry at Agamemnon for the loss of his captive slave Briseis, we feel that the hero was angry for a bad reason, but when we are told that he was angry at Hector for the loss of his friend Patroclus, we feel that his anger was for a good reason. But where does the difference lie? Not only one can have an emotion like anger for good, bad, better or worse reasons, but it can be had for the wrong kind of reason. Suppose that a powerful demon threatens to torture me to death unless I do not become angry at someone who has been very nice and helpful to me. If I manage to obey his order, I will be angry for the wrong kind of reason. The attitude does not fit the object. In contrast being offended after having been insulted seems to be a right kind of reason for being angry. This distinction is indeed similar to the right/wrong kind of reasons distinction which has been raised for attitudes in general (see Rabinowicz and Rønnov-Rassmussen 2004;Parfit 2011: Appendix A;[START_REF] Skorupski | The Domain of Reasons[END_REF]. So if there are to be reasons for emotions, there must be a right/wrong kind of reason problem.

The reason relation for emotions actually goes into both directions: What is the basis of our reasons for emotions? Let us call this basis reasons from. How can they be reasons for other states and attitudes? Let us calls these reasons for. Both questions are epistemological. What is the kind of state which justifies an emotion? If we accept the view that emotions involve or can be grounds for, evaluative judgments, how can they justify these judgments? The four assumptions mentioned above suggest a parallel between the familiar problem of the justification of perceptual beliefs and the problem of the justification of emotions. Many writers have espoused a perceptual model of the justification of the value judgments associated to an emotion: just as our perceptual beliefs (say, my belief that this is a tree ) are based on our perceptual experiences (this looks like a tree ), our emotional experiences (say, my experiencing fear of this dog ) are the base of our evaluative judgments (this dog is dangerous ). The analogy with perception is all the more tempting that the cognitive base of the emotional experience is itself a perceptual experience

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(my seeing this big salivating dog ). So on this model emotional justification seems to involve a double link: an emotion is justified for an agent just in case (i) the agent's beliefs about the object of the emotion on which the emotion is based are (epistemically) justified, and (ii) the emotion is an appropriate or fitting response to the situation as it is experienced by the agent.

The trouble is that this model is much too simple. Is the relation between the experiential base and the emotion an evidential relation? In the case of belief, a reason is sufficient in so far as it is good enough to justify some action or belief by itself; a sufficient reason makes the belief permissible. But sufficient reasons do not make believing obligatory, nor do they entail the correctness or the truth of some belief; they thus fall short of being conclusive. So reasons can be defeasible, as the perceptual model maintains, and still be sufficient; and reasons will be sufficient to justify belief in conditions in which there are no defeaters. Many have held that the reason or justifying relation is weaker than a sufficient relation, a mere prima facie kind of justification, or an entitlement relation. This condition holds even more for emotions. My learning that the price of fuel has gone up is a defeasible reason for my being angry and for my belief that it is unjust that the price of fuel arises. And indeed it is defeasible: there are many more things worthy of being angry about, and in many cases the rise of the price of fuel is not worthy of anger.

As soon as one asks about the justification of emotions on this model in analogy with the justification of perceptual beliefs, a lot of questions arise. If emotions are supposed to be fit or correct because they are justified by the experiences on which they are based, is the justification based on evidence for the evaluative judgment which is supposed to be associated to it? The latter is clearly not inferred from the former. If, on the contrary, we take the justification to be immediate, as if it sprung directly from the emotional feel, how can it be transparent?2 In other words, can we move immediately from the content of the experience (say this is a big dog ) which produces an emotion (fear of the dog ) to the But as soon as the cognitive basis of our emotional experiences becomes belief-laden, a dilemma looms: either it is the beliefs which accompany an emotion which are justified, in which case the emotion as a feeling is not in itself justified, or it is the emotion as feeling (as phenomenal feel) which is justified, in which case the justification is based only upon an affect, hence no justification at all. For an example of the first horn, take surprise. Surprise is an emotion mediated by beliefs-antecedent belief that not p, further discovery that p, surprise that p-but if only the beliefs have a justificatory power, the emotional import of the feeling of surprise does not play any epistemological role: the evaluation is purely cognitive. On the second horn, if we say that it is only the emotional part of surprise which is justified, how can the feeling itself (the startle, the eyebrows that raise, the pupil that dilate) have any epistemological import, and if there is such an import, how can it be transparent? If it is-that is if we can move from the perception to the evaluation of surprisingness (or dangerousness in the case of fear), the justificatory link between experience and belief in the emotional case is similar to the justificatory link in the perceptual case according to the so-called "dogmatist" view of perceptual justification. So, if your emotional response to a perceived object makes it seem to you that that object or event possesses some evaluative property, then you thereby have prima facie, immediate justification for believing that that object or event possesses that evaluative property. But if this were true, then fear, guilt and anger, absent defeating evidence, would immediately justify beliefs, rationalizing avoidance, self-condemnation, and retaliation. But we rarely take our emotions at face value, or if we do we should avoid doing so: they must also be related to certain beliefs and desires, which, together with the emotion and the value judgments, constitute our reasons to act. There are further disanalogies between the perceptual and the emotional case. 4

4 See in particular Brady (2013). Now, from the fact that the perceptual model of emotional justification is hard to sustain literally at the level of the reason-from, does it follow that we should withdraw any hope of establishing a reason-for relation between emotional experience and the perception of value? No. Proponents of the perceptual model give examples such as being struck by the injustice of slavery through the emotion felt in reading Uncle Tom's Cabin or by the beauty of nature through the emotion of contemplating a sunset over the Grand Teton National Park, but the ways of the perception of values are most of the time more complex: reading Huckleberry Finn or contemplating the English countryside at daybreak on a fine summer day may give rise to more complex associations of emotional experience and value. The fact that we do not typically trust emotions as sufficient reasons to form evaluative judgments, but rather as reasons to look for non-emotional reasons to confirm our initial emotional appraisal does not show that this initial appraisal cannot justify the evaluative judgments, but just that the justificatory route is more complex. The relationship between emotional experience and evaluative beliefs need be neither direct nor foundational in the sense suggested by the simple perceptual model. It can be holistic, and such that the emotional experience and its relation to values is further confirmed by related beliefs. The fit between the perceived situation, the emotion and the issued value judgments may be more a matter of coherence than a matter of perceptual basis, and the correctness of emotions need not be based on some mysterious capacity of grasping the values within the emotional experiences. The relation between the emotion and the value need not be itself perceptual, but can be based on the idea that the correctness of the emotion is due to an appropriate response to a perceived event or action. Emotions in this sense are sources of reasons, and do not have direct epistemological impact. The latter judgment is a moral one: some injustice has been done to you. How can the emotion be correct or fit its object? A certain standard, or norm, has been violated. The question immediately arises: how can the emotion be a reason for a judgment about an objective moral subject matter? For the feeling may be transitory and the judgments can change. Anger is most often directed not at a particular action or trait, but at the whole person. In Ariosto's epic Orlando furioso, the Christian knight Roland becomes furious when he discovers Angelica's love for the Saracen Medoro. But he could have be frenzy for another reason, and his judgment that an offense has been done to him might have been based on other moral standards that those of medieval chivalry. We can have distinct views of the objectivity of the standards, and one can be a non-cognitivist or a cognitivist about the nature of values, but at least the evaluative judgment to the effect than some injustice has been committed purports to be objective. This is enough for a contemporary reader to be able to understand Roland's wrath. Contempt is, among the emotions, one of the least typical, for it seems to lack any characteristic feeling and bodily expression. It is a very cognitive emotion, which seems to be mostly constituted by a certain kind of judgment appraising the status of its object, and the curling of the lips, joined and retracted in a derisive or sardonic smile cannot be its paradigmatic expressions. The judgment is such that it is directed at the whole person which is its object (if I have scorn for the way your dress, my scorn is directed at you, not at your particular clothing) and such that the person in question is taken to be inferior. Like anger, it is a "globalist" emotion (Bell 2013, p. 37) or a "characterizing attitude" [START_REF] Darwall | Contempt as an Other-Characterizing[END_REF]). Perhaps we may call it, following Strawson (1962), a "reactive attitude", expressing a global set of emotions and feelings, which are constitutive of our status as persons and of the attitudes that we have to other persons. Strawson says that such attitudes can be either "objective"-when we distance ourselves from others and consider them as non-free, non-responsible and not liable to praise or blame-or "subjective", when we take others as free and liable to evaluative judgments. Unlike other emotions, contempt is a disposition, and rarely an episodic mental occurrence, and when it is associated to a feeling, it scope can vary, from mere amusement and derision to deep hatred. But what is it, for a contempt to be fitting as an emotion? Just as for anger, the objects of contempt can be very diverse, in the sense that the kind of behaviors which might elicit contempt and the judgment of superiority which is associated to it are themselves very diverse. If contempt is linked to the perception of a social hierarchy and direct at a social status, there will as many kinds of reasons for contempt are there are kinds of social standards within a society or from one society to another: being the object of contempt in India for being an untouchable, being the object of contempt at the court of Louis XIV in Versailles because one does not have the proper degree of nobility, or being the target of the contempt of a snob in Victorian England seem to correspond to so many situations that it is hard to say that there are precise correctness conditions for this emotion. Think for instance of the way Tocqueville describes the relationship between men and women in America as contrasted with Europe:

It has often been remarked that in Europe a certain degree of contempt lurks even in the flattery which men lavish upon women: although a European frequently affects to be the slave of woman, it may be seen that he never sincerely thinks her his equal. In the United States men seldom compliment women, but they daily show how much they esteem them. (Tocqueville 1835, Ch. 11) 7 In a democratic society, which aspires to equal status for individuals, contempt will be an emotion which is itself the object of a general negative moral judgment. [START_REF] Kant | Groundwork for the Metaphysics of Morals[END_REF] famously says that contempt violates people's basic claim of respect against others because it construes people as lacking dignity and as having no moral worth, hence as unable of improving. Kant denies that there can be any good reason for contempt, hence that the reasons for being scornful are always wrong, because they contradict a basic principle of morality. This is bad news for the perceptual theory of emotions as tracking values, because, on this view, the evaluative judgment associated to contempt would always be false, and the emotion of contempt would be always inappropriate. As Bell notes, the rejection of contempt is due to its globalist nature. In this sense, it seems to be not only to be rejected for moral reasons, but also for descriptive ones. For, as many psychologists have argued, there are no such global emotions: characters traits are inconsistent, transitory, unstable. 8 So contempt never fits its object, and is always based on a false ascription of character. The objection extends to the "fitting attitude" conception of value: there is nothing like being fit for such an emotion, because we can never specify its conditions of appropriateness. This objection presupposes that the fittingness of an emotion is a descriptive property. But this is wrong. The fitting attitude or "neo-sentimentalist" analysis of emotions, as it is sometimes called (Tappolet 2016, p. 85 sq.), 9 says that evaluative and normative concepts are essentially tied to the concepts of specific responses, although it does not say that these concepts are mere projections of our emotional responses. For instance a particular action counts as admirable if the feeling of admiration is an appropriate response to this action, of fits it. This condition of 9 Tappolet (2016, p. 85 sq.).

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7 Indeed many contemporary American feminists would balk at this. 8 See e.g. [START_REF] Doris | Lack of Character: Personality and Moral Behavior[END_REF]. This is the line taken by "situationism" about character or virtue. correctness is meant to be a conceptual analysis. But it encounters two difficulties. The first is that it seems circular: we are told that an emotion is correct when the responses are fitted to the value or norm, admiration to the admirable, anger to irascible-worthiness, contempt to the contemptible. The second is that it is formulated in normative terms, involving appropriateness and correctness: the idea is that a certain kind of feeling is required, or such that we ought to have it in order to fit the value in question. But if it is supposed to be a version of the perceptual analysis, it does not suit well with a naturalist construal of the psychological state of emotion, for it seems odd to say that we perceive normative concepts. We need not, however, accept this descriptive construal of the perceptual analysis-of if it is a consequence of it, we should reject this analysis. 10 The fittingness conditions do not refer to an actual perception of value, nor to descriptive traits of character in individuals, but to an ideal of what individuals ought to be. In this sense, the Kantian interpretation of contempt is right: it refers to a judgment about what this emotion presupposes about humanity. But does that mean, as Kant implies, that this judgment is wrong? After all, some features of humanity and some kinds of actions, might be contemptible, and it might be correct to point them out, and thus to withdraw the principled attitude of respect which Kant deemed to be the very foundation of morality.

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Contempt and Moral Knowledge: A Swiftean Story

Neither the perceptual model of emotional justification nor the fittingness account entail that our perception of value on the basis of emotions have to be direct or immediate, as if we reacted to injustice in a bout of furor, or to the vileness of a character in a hiccough of scorn. Most "moral" emotions, such as shame, pity or contempt are not episodic, but dispositional. They can become virtues or vices, in so far as virtues and vices are based on dispositions. And they involve complex judgings and appraisals. An essential feature of most emotions is the capacity to reappraise them, and to revise them in the light of further emotions and judgments. 11 When they reach a certain level of sophistication, we learn much more from literature than from neuroscience and social psychology. The thesis which I want to put forward, but cannot argue in the space of this essay, is that moral emotions like anger and contempt do not involve judgments about values but judgments about characters who instantiate, or fail to instantiate these values. In other words, they involve judgments about virtue and vice, as dispositions stemming from emotions and involving dispositions to good or bad behavior. But anger and contempt are not only complex emotions involving judgments, they are also, as dispositions and character traits, the object of our moral appraisal. This is why, in particular, contempt has a bad reputation, as it is based on the feeling of superiority toward others. Both the tradition and common sense take it as itself despicable, as based on a wrong relation to values and reasons for emotions. I want to suggest that it is not: not only there can be reasons for contempt, and there are reasons for wrath, but these are also good reasons.

The Christian tradition has taken contempt to be in its very nature opposed to the virtue of humility. Its name is superbia, a vice of superiority. But the literary tradition of comedy and of satire has promoted contempt as the proper attitude toward the vices of human nature. Jonathan Swift was an heir to both traditions: he was a devoted Christian and a stubborn satirist. He has often been described as the arch-contemptor, a master of scorn. His specific feelings have been described by Samuel Johnson (1779) as a mixture of "petulance and sarcasm", of "arrogance and raillery", of a man who has "wasted his life in discontent". [START_REF] Thackeray | Jonathan Swift[END_REF] says that if you had been his inferior "he would have bullied, scorned, and insulted you", and if you had addressed him as a Yeats calls him "intense" and "vehement". 12 Swift himself said that his Travels "are erected upon a foundation of 12 Johnson (1779) and Thackeray (1854). 11 This feature has been well analyzed in the pioneering work of [START_REF] Livet | Emotions et rationalité morale[END_REF].

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Author Pr misanthropy" and his famous epitaph at St Patrick in Dublin says that saeva indignatio cannot anymore "lacerate his breast". He has been ascribed all the traditional vices of superiority: Arrogance, Superbia, Hypocrisy, Apathy, Cruelty, Greed, Jealousy, Recklessness, Bitterness, Gluttony, Lust, Wrath. Such accusations are often addressed to the satirist. He displays contempt, scorn, haughtiness toward those that he satirizes, and his message is merely one of pride and superiority.

Those who show such superiority not only have to be blamed because they do not respect their fellow mortals, but also because they do not deserve any respect. Swift was no exception. His morality, most biographers suggest, was questionable. He was, in the words of the critic John Middleton-Murry (1954), a "hypocrite reversed", one who turns toward others the accusation of viciousness that he himself knows to deserve. Contempt and anger are the engines of Swift's satire. "In a Jest, he said us, I spend my Rage", preferring to 'encounter Vice with Mirth' (Epistle to a Lady, Poems, II, p. 218). As Claude Rawson says "The angers, of course, were all too real, but Swift was temperamentally equivocal about their display. Even when we may suppose them to have been at white heat, as in A Tale of a Tub, the brilliant aggressive vitality is designed, for all the intensity of its sting, never to lose its cool. The contemptuous energy with which he mimicked the forms of 'modern' egocentrism and the self-promoting typographical antics of what we now like to call 'print culture', is a billowing performance of indignant impersonation in which the force and incriminating accuracy of the aggression never shows loss of authorial composure" (Rawson 2014, pp. 1-2). Swift's angers are often characterized as a form of moral hatred. Swift's contempt was clearly on the subjective side of Strawson's reactive attitudes, even though he often seems to adopt the objective stance. On the one hand, the satirist's expression of contempt through irony presupposes that he distances himself from the characters that he represents, and often takes them as unfree, as mere puppets ruled by their passions. As F. R. Leavis says, Swift's ironic intensity undeniably directs itself to the defense of something that he is intensely concerned to defend, the effect is essentially negative. The positive itself appears only negatively-a kind of skeletal presence, rigid enough, but without life or body; a necessary pre-condition, as it were, of directed negation. The intensity is purely destructive. (Leavis 1952, p. 75) Leavis contrasts Swift's irony with that of Gibbons, which "insinuates a solidarity with the reader (the implied solidarity in Swift is itself ironical-a means to betrayal)" (ibid.). On the other hand, the satirist could not judge and disvalue these characters if he did not adopt the participant's attitude. His irony is directed at humans in general, although he knows that he is one of them.

When we talk of Swift's anger and of his contemptuous feelings, we are not talking about biographical or psychological facts. We are talking about what the reader can read into his prose, and the portrait that he implicitly draws of himself as a contemptuous and angry figure. Anger and contempt are not feelings or emotions which are represented within Swift prose, as features of his fictional characters. They are inferred by the reader through certain cues, the most salient being his irony.

Swift, however, is quite clear, and literally so, about his moralistic intentions:

I have been only a Man of Rhimes, and that upon Trifles, yet never any without a moral View. (Correspondence iv, p. 52) "There are two Ends that Men propose in writing Satyr," private Satisfaction and a public Spirit, prompting Men of Genius and Virtue, to mend the World as far as they are able. (The Intelligencer, vol. III, 1728, in But these edifying and moralizing intentions are also the satirist's main problem: he intends to denounce human vices, but his very denunciation is itself considered as vicious. His emotion is not appropriate. We have here an instance of the wrong kind of reasons problem for contempt: the satirist, by expressing his contempt for mankind, seems to be looking for having the attitude of contempt, and not to have what is contemptible as the proper object of his emotion. In Parfit's (2011, pp. 420-432) analysis of the wrong kind of reasons problem, the satirists reasons are state given reasons, directed at the attitude of contempt, and not object given reasons, that is reasons directed at what is worthy of contempt. Contempt based on state given reasons is a form of pretense, an emotion which is not appropriate, but phony or insincere. So the satirist's project seems to be faked, and hypocrite, as many critics of Swift have argued.

There are two dilemmas of satire, which Swift exemplifies almost paradigmatically. The first is that the satirist can deliver his moral message only by sharing with his readers emotions, such as anger and contempt, which the readers find negative and so despise. This is also why satire, and the kind of negative feelings it expresses, often self-directed. There is a second dilemma for the satirist: his main weapons are irony and fiction, but how can these modes of expression carry the weight of his reprobation? If the satirist is understood too literally, the satire is unsuccessful, and if he is too ironical and too fictional, he will not be taken seriously, and will fail to deliver his moral message. The art of fiction in general requires the appropriate tuning of the emotions. This is true of satire and of the expression of contempt. The satirist is always suspected of either pretending to be contemptuous, or if he is genuinely contemptuous, to be immoral because he does not have the proper humility and respect for humanity which are these bases of morality. If, like Swift, he is also a Christian, his case is almost desperate. The solution to this problem is to reject the common understanding of contempt, as a bad emotion, which involves an absence of respect toward its target, and a scorn for humanity. Swift is a misanthrope only as far as he directs his scorn toward people who are themselves vicious, and at the kind of situations that they create: political injustice in Ireland (The Drapier letters), famine and poverty with the children of Ireland (A Modest Proposal ), bigotry and credulousness (A Tale of a Tub ), false learning as with the Academy of Lagado (Gulliver ), and all the situations which are the object of his rage. So his contempt is for the right kind of reason: it is directed at real situations which exemplify moral wrongness, stupidity and vice. Swift's emotions of anger and contempt, as they are manifested in his satires, are aimed at the right targets. They are disrespectful, because in such situations, it correct, and fit, to "vex the rogues". He is clear on the fact that it does not amount to hate of mankind:

I tell you after all that I do not hate Mankind, it is vous autres who hate them because you would have them reasonable Animals, and are Angry for being disappointed. (Swift to Pope, 26 November 1725, Correspondence, IIII, p. 118) The satirist here relies on, and displays, a form of moral knowledge. Indeed, it is disputable what this kind of knowledge amounts to. Is it, as Swift seems to imply, a knowledge of morals truths, based on acquaintance with real values, on the basis of which the emotion of contempt produces the implicit judgments of the satirist under the guise of his "mirth", as the moral realist would be tempted to say? Or does this knowledge consist in some form of understanding which falls short of being genuine knowledge [START_REF] Brady | Emotional Insight[END_REF])? I side with the first, but indeed this claim is far from evident. A non-cognitivist about moral values and norms will claim that if there is no agreement on what the moral truths are, there cannot be moral knowledge, either in ordinary life or in its expression in satires. So the view proposed here is bound to seem question-begging. But I hope to have at least indicated how there can be room for a moral realistic view of the emotions of anger and contempt.

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  Stocker (1987) talks of "emotionally laden beliefs" which are ways the beliefs are taken. But what is justified? The beliefs, or their emotional charge?

March 2019 18:12 Page: 6/18 6 P. Engel evaluative

  judgment (this dog is dangerous )? Or is the justification mediated by beliefs? Some writers have talked of emotionally laden beliefs.3

	Chapter No.: 10	Date: 16

  5 The feeling is of pain, but the formal object, what is anger-worthy, 6 is a complex relational content directed to someone for a reason, involving a judgment, to the effect that an injustice toward oneself or one's friends. The emotion is factive, in[START_REF] Gordon | The Structure of Emotions[END_REF] sense: it is based on the knowledge that someone has done you some harm (say, insulting you), which-if something like the perceptual account is right-justifies one to judge that one has been offended by someone.

			Book ID: 461130_1_En	Book ISBN: 978-3-030-15667-1
		Chapter No.: 10	Date: 16 March 2019 18:12	Page: 8/18
	8	P. Engel
	3	The Elusiveness of Anger and Contempt
	Anger and contempt are both negative emotions which illustrate the
	features listed above almost paradigmatically. Aristotle defines anger
	as "an impulse, accompanied by pain, to a conspicuous revenge for a
	conspicuous slight directed without justification toward what con-
	cerns oneself or toward what concerns one's friends" (Rhetorics,
	1378a31-1378b9).

/18 10 The Grapes of Wrath and Scorn 15 I

  Prose Works, XII, p. 34) But if my Design be to make Mankind better, then I think it is my Duty. (Prose Works XII, p. 34) You see Pope, Gay, and I use all our Endeavours to make folks Merry and wise. (Correspondence, iv. p. 53) I have finished my Travells…they are admirable Things, and will wonderfully mend the World. (Letter to Ford, 27 August 1721, Correspondence, III, p. 87) I look upon myself, in the capacity of a clergyman, to be one appointed by Providence for defending a post assigned me, and for gaining over as many enemies as I can. (Prose Works ix, p. 262) have got Materials Towards a Treatise proving the falsity of that Definition animal rationale, and to show it should be only rationis capax. Upon this great foundation of Misanthropy (though not Timons manner) the whole building of my Travells is (Swift to Pope, 29 September 1725, Correspondence, II, p. 607)

	Author Pr	Layout: Pop_A5 Chapter No.: 10	Book ID: 461130_1_En Date: 16 March 2019 18:12	Book ISBN: 978-3-030-15667-1 Page: 15

See e.g.[START_REF] Robinson | Deeper Than Reason[END_REF] and[START_REF] Lombardo | Literature, Emotions and the Possible: Hazlitt, Stendhal[END_REF].

[START_REF] Brogaard | Against Emotional Dogmatism[END_REF] and[START_REF] Echeverri | Emotional Justification[END_REF] argue against this alleged transparency of the "dogmatist" view of emotional justification. Although they do not quote[START_REF] Pryor | The Skeptic and the Dogmatist[END_REF] (as Echeverri 2017 does), they clearly want to refer to an analogue of this view for emotions.

De Sousa says that the role of emotions is often to attract our attention: "Paying attention to certain things is a source of reasons"(1987, p. 196). So the kind of skepticism about the perceptual model expressed by[START_REF] Brady | Emotional Insight[END_REF] need not entail the falsity of the correctness account of emotions. De Sousa (2011) suggests a more coherentist model.[START_REF] Pelser | Emotion, Evaluative Perception and Epistemic Justification[END_REF] and[START_REF] Tappolet | Emotions, Values and Agency[END_REF] have proposed more sophisticated accounts of the perceptual model.

There is no lexical item in English corresponding to what is the formal object of anger, in the ways the admirable is the formal object of admiration.

I thus would disagree with[START_REF] Tappolet | Emotions, Values and Agency[END_REF], who aims to defend such a sophisticated version of the perceptual view.

Acknowledgements Without my friends in Geneva, I wouldn't have learnt anything about emotions. Four years ago, Patrizia Lombardo asked me to write on contempt, but I failed to produce anything decent. Her work on these issues, and everything that she did on literature (such as Lombardo 2014), inspired me a lot. Thanks to Santiago Echeverri for his comments. I owe a huge debt to Laura Candiotto, who not only organized a wonderful conference in 2017 in Edinburgh, but also has been so generous and patient.


M Fleury Lycée

Poitiers Camille Guérin

Calogero-Moser systems with periodic and doubly periodic interaction potentials and loop algebras

published or not. The documents may come

Calogero-Moser systems with periodic and doubly periodic interaction potentials and loop algebras

I. INTRODUCTION

Calogero-Moser systems were introduced long ago and were proven to be completely integrable by Lax pairs technics, see Ref. 1. The systems with 1 r 2 and 1 sin 2 r interaction potentials received in Ref. 2 an enlightening geometric interpretation through Marsden-Weinstein reduction of geodesic flows on Lie algebras or Lie groups. Though, the system with a Weierstrass P function interaction potential remains less understood. The two particle system has been solved in Ref. 3, and the only known integration scheme in the general case was provided by Krichever in Ref. 4, through the solitonic solutions of K-P equation.

The present work starts from the solution of the system with 1 r 2 interaction potential and proceeds by infinite replication. Calogero and Françoise had previously used a duplication scheme to produce new integrable systems and their solutions from the system on the line in Ref. 5. Here we have in mind the eulerian development of 1 sin 2 and proceed by infinite replication by translations of the system on the line. The corresponding matrices are infinite dimensional and are interpreted as distributional elements of some loop algebra. The striking fact is that these distributions are just elementary differential operators. This allows us to give a new proof of the solution of the system with 1 sin 2 r interaction potential. Then, we apply a doubly infinite replication to investigate the system with Weierstrass P function interaction potential, which leads us to a partial differential operator on the space of doubly periodic functions with values in C n . Though this operator is very simple looking and much similar to the preceding one, it is indeed much more involved to determine its spectrum and this is beyond the author's reach. We connect this problem to some finite difference equation in a space of simply periodic analytic functions.

II. INFINITE REPLICATION

It is known that the Calogero-Moser system on the real line with 1 r 2 pairwise interaction potential can be integrated in the following way [5].

Let x 1 , x 2 , . . . , x n denote the initial positions of the particles, and y 1 , . . . , y n their initial velocities.

Let

A =      ix 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 . . . 0 ix n      and B =        iy 1 1 x 2 -x 1 . . . 1 x n -x 1 1 x 1 -x 2 . . . . . . . . . . . . . . . . . . 1 x n -x n-1 1 x 1 -x n . . . 1 x n-1 -x n iy n        so that [A, B] = M =      0 -i . . . -i -i . . . . . . . . . . . . . . . . . . -i -i . . . -i 0     
The position of particles at time t are then the eigenvalues of A + tB, divided by i.

We also recall that the positions at time t of the Calogero-Moser on the circle with pairwise interaction potential 1/ sin 2 r 2 (also called Sutherland system) are the arguments of the eigenvalues of the matrix exp(A).exp (tC),

where C = (c k,l ) with c k,l = i e i(x l -x k ) -1 for k = l and c k,k = iy k ,
Our first aim is to show that the solution of the system on the circle can be deduced from a system on the line with infinitely many particles.

Let us recall the Eulerian expansion:

1 sin 2 r = k∈Z 1 (r -kπ) 2
According to this formula, one can write the hamiltonian

H = 1 2 i y 2 i + 1 2 i =j 1 4 sin 2 x j -x i 2 in the form H = 1 2 i y 2 i + 1 2 i =j k∈Z 1 (x j -x i -2kπ) 2 = 1 2 i y 2 i + 1 2 i =j lim N →+∞ 1 2N k,l∈[-N,N ] 1 [(x j + 2kπ) -(x i + 2lπ)] 2
For all i ∈ Z, let us write i = kn + r with 1 ≤ r ≤ n and set x i = x r + 2kπ and y i = y r . The hamiltonian takes thus the form

H = lim N →+∞ 1 2N 1 2 -nN ≤i≤nN y 2 i + 1 2 -nN ≤ i, j ≤ nN i ≡ j[n] 1 (x j -x i ) 2
Therefore, the n-body Sutherland system on the circle is analogous to the infinitly many body system on the line obtained by associating to each particle on the circle infinitly many particles on the line, 2π-distant of each other and animated with the same velocity.

We thus form the infinite dimensional matrices Ã, M and B indexed by Z × Z where

à =        . . . A -2iπI 0 0 0 A 0 0 0 A + 2iπI . . .        B =       
. . .

P 0 P -1 P -2 P 1 P 0 P -1 P 2 P 1 P 0 . . .        M =        . . . M M M M M M M M M . . .       
where

P k =        iy 1 δ k,0 1 x 2 -x 1 +2kπ . . . 1 x n -x 1 +2kπ 1 x 1 -x 2 +2kπ . . . . . . . . . . . . . . . . . . 1 x n -x n-1 +2kπ 1 x 1 -x n +2kπ . . . 1 x n-1 -x n +2kπ iy n δ k,0        so that [ Ã, B] = M ,

III. LOOP ALGEBRAIC INTERPRETATION

The matrices à and B will be considered as linear operators on the space of 1-periodic

C n -valued functions f : θ → k∈Z f k e 2ikπθ Writing f as f =        . . . f -1 f 0 f 1 . . .        , one gets Ãf =        . . . -2iπf -1 + Af -1 Af 0 2iπf 1 + Af 1 . . .        = f + Af
where the prime designates derivation with respect to the variable θ, and

Bf =        . . . . . . + P 0 f -1 + P -1 f 0 + P -2 f 1 + . . . . . . + P 1 f -1 + P 0 f 0 + P -1 f 1 + . . . . . . + P 2 f -1 + P 1 f 0 + P 0 f 1 + . . . . . .        = Λf
where Λ(θ) = . . . + P -1 e -2iπθ + P 0 + P 1 e 2iπθ + . . ..

We now proceed to the computation of Λ.

The equation [ Ã, B] = M can be rewritten in the following way:

∀k ∈ Z AP k -P k (A -2ikπI) = M
Multiplying each member by e 2ikπθ , we get after summation:

A k∈Z P k e 2ikπθ - k∈Z P k e 2ikπθ A + k∈Z 2ikπP k e 2ikπθ = M k∈Z e 2ikπθ
Thus [A, Λ(θ)] + Λ (θ) = M δ 0

In order to solve this equation on the circle, we first solve the associated equation on the line:

[A, Λ(θ)] + Λ (θ) = M k∈Z δ k (cf Poisson summation formula)
and we'll retain the 1-periodic solutions.

The associated homogeneous equation is

Λ (θ) = -[A, Λ(θ)]
the general solution of which is Λ(θ) = e -θA Ke θA (K constant matrix).

The solution of the complete equation is obtained by varying the constant K, which satisfies the equation e

-θA K e θA = M k∈Z δ k Therefore, K is constant on each intervale ]k, k + 1[ (k ∈ Z) and
has Heaviside singularities at integer points. All that we need is the value of K on ]0, 1[. In order to get this value, we just write

B = P 0 = 1 0 Λ(θ)dθ = 1 0 e -θA Ke θA dθ Writing K = (K k,l ) 1≤k,l≤n , we get e -θA Ke θA = e iθ(x l -x k ) K kl 1≤k,l≤n
and thus

iy k = B k,k = K k,k and for k = l 1 x l -x k = B k,l = e i(x l -x k ) -1) i(x l -x k ) K k,l which yields K k,l = i e i(x l -x k ) -1 . Thus K is the matrix C of the Kazdhan-Kostant-Sternberg the- ory [5].
and Λ(θ) = e -θA Ce θA .

Recall that the positions of the particle at time t are, in the finite case, the eigenvalues of A + tB divided by i. We conjecture that in the case at hand, they are the eigenvalues of d dθ +(A+tΛ(θ))id (divided by i), acting on the space of 1-periodic functions.

Since Λ is continuous except at integer points where it has Heaviside singularities, we look for the 1-periodic, differentiable except at integer points, but everywhere continuous solutions of this equation. This equation is satisfied by a non zero, 1-periodic and continuous function if only if e -λ is an eigenvalue of e -A e -tC Thus, the eigenvalues of Ã+t B are the (pure imaginary) complex numbers the exponentials of which are eigenvalues of e tC e A , which are the same as the eigenvalues of e A e tC . This agrees with the solution given in Ref. 2.

IV.1. The Operator

We apply the same scheme when the pairwise interaction potential is the Weierstrass P function,

P(z) = 1 z 2 + (k,l) =(0,0) 1 (z -2kπ -2ilωπ) 2 - 1 (2kπ + 2ilωπ) 2
We will use to this purpose matrices of infinite order indexed by (Z × Z) 2 instead of Z 2 , and the underlying Lie algebra will be the complexification of u(n), that is gl n (C).

Vectors of C Z×Z will be identified with C n -valued doubly periodic functions (with periods 1 and 1/ω with respect to variables θ and φ),

f (θ, φ) = k,l ξ k,l e 2ikπθ e 2ilπωφ
The matrix à has now the block decomposition

à = (A klk l ) k,l,k ,l ∈Z with A klk l = A+(2ikπ+2lωπ)I if k = k and l = l and A k,l,k ,l = 0 otherwise. It is identified with the operator ( Ãf )(θ, φ) = Af (θ, φ) + k,l ξ k,l (2ikπ -2lωπ)e 2ikπθ e 2ilπωφ = Af (θ, φ) + ∂f ∂θ - 1 i ∂f ∂φ = Af (z) + 2 ∂f ∂ z
(by writing z = θ + iφ)

In the same way, we identify B with the multiplication operator f → Λf where Λ is a doubly periodic gl n (C)-valued function.

M having the same meaning as before, Λ satisfies the partial derivative equation

[A, Λ] + 2 ∂Λ ∂ z = M δ O
where O denotes the origin of the torus.

As before, we start by solving the equation on C where K is any holomorphic matrix valued entire function.

We solve the complete equation by varying the "constant" K, which satisfies the equation 2e

-zA 2 ∂K ∂ z e zA 2 = M k,l δ k+il/ω . ∂K ∂ z = 1 2 Ω∈R δ Ω e ΩA 2 M e -ΩA 2 where R = Z + i ω Z
The solutions of this equation are the meromorphic functions whose poles are simple and are the points of the network R, the residue at Ω being 1 2π e ΩA 2 M e -ΩA 2

.

But Λ is also doubly periodic, and therefore ∀Ω ∈ R e -(z+ Ω)A/2 K(z + Ω)e (z+ Ω)A/2 = e -zA/2 K(z)e zA/2 K(z + Ω) = e ΩA/2 K(z)e -ΩA/2

Therefore:

the diagonal coefficients of K are pole free (because the diagonal coefficients of M are null) elliptic functions, and thus they are constant. In the same way as above, writing B = [0,1] 2 Λ, we get

K k,k = iy k ; tions which satisfy K k,l (z + Ω) = e Ωi(x k -x l )/2 K k,l (z)
and with simple poles Ω ∈ R and residues 1 2iπ e Ωi(x k -x l ) . As we prefer to deal with (simply) periodic functions, we set

P (z) = e -zA/2 K(z)e zA/2
One have for all Ω ∈ R, P (z + Ω) = e -(z+Ω)A/2 e ΩA/2 K(z)e -ΩA/2 e (z+Ω)A/2 = e -(Ω-Ω)A/2 P (z)e (Ω-Ω)A/2 Thus P is periodic with period 1 and

P (z + i/ω) = e -i/ωA P (z)e i/ωA
The diagonal coefficients of P are those of K. They are con-

stants: P k,k = iy k .
The off-diagonal coefficients of P are meromorphic with simple poles on the network and residue at zero: 1 2iπ They satisfy where Im(τ ) > 0 and a, b ∈ C. This defines an entire function which satisfies the fundamental relation

P k,l (z + 1) = P k,l (z) and P k,l (z + i/ω) = e x k -x l ω P k,l (z) (1
ϑ a b (z + pτ + q) = e iπ(-2pz-p 2 τ +2pb-2aq) ϑ a b (z)
∀p, q ∈ Z. and the zeros of ϑ a b are simple and located at the

(a -1 2 )τ + b -1 2 + τ b + q, p, q ∈ Z. Set τ = i/ω. The function q = ϑ a b ϑ a b satisfies q(z + i ω p + q) = e iπ(2 p ω (b-b )-2(a-a )q) q(z)
This matches with (1) if we choose a, b, a , b such that

a -a = 0 and b -b = -i x k -x l 2πω
and the location of poles matches if a = b = 1 2 . Then

P k,l
q is an entire function with at most one simple pole in the fundamental cell (since theta functions have just one simple zero in the fundamental cell), thus it is constant and equal to its value at zero.

Hence,

P k,l (z) = 1 2iπ × ϑ 1/2 1/2 (0) ϑ 1/2 1 2 -i x k -x l 2πω (0) × ϑ 1/2 1 2 -i x k -x l 2πω (z) ϑ 1/2 1/2 (z) = 1 2iπ × ϑ 1 (0) ϑ 1 (i x k -x l 2πω ) × ϑ 1 (z + i x k -x l 2πω ) ϑ 1 (z)
where we have set ϑ 1 = ϑ 1/2 1/2 (Jacobi's notation).

We have another way for expressing the off-diagonal coefficients of P .

Let δ ∈]0, 2π[.

Consider the series f (z) = j∈Z e -jδ
1-e -2jπ/ω e 2iπz . This series converges normally on all compact of C \ R and defines a meromorphic function with period 1 that verifies f (z+ i ω ) = e δ f (z). Its only poles are simple and are the points of R. Its residue at 0 is 1 -2iπ . In the same way, for δ ∈] -2π, 0[, the series

f (z) = j∈Z e -jδ
1-e 2jπ/ω e -2iπz defines a meromorphic function with period 1 that verifies f (z

+ i ω ) = e δ f (z). Its residue at 0 is 1 2iπ . write if k < l P k,l (z) = j∈Z e -j(x k -x l )/ω 1 -e 2jπ/ω e -2iπz if k > l P k,l (z) = - j∈Z e -j(x k -x l )/ω 1 -e -2jπ/ω e 2iπz

IV.2. The spectrum

We conjecture that the positions of the particles at time t are the eigenvalues of the operator

A + 2 ∂ ∂ z + tΛ(z)
acting on the space of doubly periodic functions (with periods 1 and i/ω).

Let λ be some (pure imaginary) complex number and f some doubly periodic function satisfying

Af + 2 ∂f ∂ z + tΛf = λf Set f (z) = e
λz 2 e -zA/2 g(z) The preceding p.d.e. becomes 2e λz 2 e -zA/2 ∂g ∂ z + te -zA/2 K(z)e zA/2 e λz 2 e -zA/2 g(z) = 0 which amounts to

2 ∂g ∂ z + tK(z)g(z) = 0
Recall that K is holomorphic except at the points of the network R. Set g(z) = e -tzK(z)/2 h(z).

The preceding equation amounts to ∂h ∂ z = 0, except at the points of the network.

Therefore, except at these points, f takes the form f (z) = e λz/2 e -zA/2 e -tzK(z)/2 h(z)

where h is holomorphic.

The function f is doubly periodic. This means that for all Ω in the network,

f (z + Ω) = f (z)
Since f (z + Ω) = e λ(z+ Ω)/2 e -zA/2 e -ΩA/2 e -t(z+ Ω)K(z+Ω)/2 h(z + Ω) = e λ(z+ Ω)/2 e -zA/2 e -t(z+ Ω)K(z)/2 e -ΩA/2 h(z + Ω)

we get h(z + Ω) = e -λ Ω/2 e ΩA/2 e t ΩK(z)/2 h(z)

Set then h(z) = e tzK(z)/2 k(z) and recall that K(z

+ Ω) = e ΩA/2 K(z)e -Ω/2
The preceding condition reads e ΩA/2 e t(z+Ω)K(z)/2 e -ΩA/2 k(z + Ω) = e -λ Ω/2 e ΩA/2 e t ΩL(z)/2 e tzK(z)/2 k(z)

which simplifies to k(z + Ω) = e -λ Ω/2 e ΩA/2 e t( Ω-Ω)K(z)/2 k(z)

Set then k(z) = e -λz/2 e zA/2 φ(z). The preceding condition reads thus e -λ(z+Ω)/2 e (z+Ω)A/2 φ(z + Ω) = e -λ Ω/2 e ΩA/2 e t( Ω-Ω)K(z)/2 e -λz/2 e zA/2 φ(z)

which simplifies to φ(z + Ω) = e λ(Ω-Ω)/2 e -(z+Ω-Ω)A/2 e t( Ω-Ω)K(z)/2 e zA/2 φ(z)

Therefore we look for analytic functions on C \ R satisfying φ is periodic with period 1 and φ(z + i/ω) = e iλ/ω e -(z+2i/ω)A/2 e -it/ωK(z) e zA/2 φ(z) = e iλ/ω L(z)φ(z)

where L(z) = e -(z+2i/ω)A/2 e -it/ωK(z) e zA/2 = e -i/ωA e -it/ωP (z)

We conjecture that the correct condition is that φ, which coincides with f on the real axis, should have no singularity there.

Since φ is then holomorphic on the domain |Im(z)| < 1, we can write φ(z) = j∈Z φ j e 2iπjz , this expansion being valid in the same domain.

Since L(x-i

2 ) is a smooth 1-periodic function of the real variable x, we can write L(x -i 2 ) = j∈Z L j e 2iπjx . Condition (2) is satisfied if and only if it is satisfied on the line R -i 2 (because only analytic function are involved), and this reads: ∀x ∈ R e -πj φ j = e iλ k∈Z L j-k e πk φ k that is e -iλ = e x is an eigenvalue of the infinite matrix T = (T j,k ) (block decomposition) with T j,k = e π(j+k) L j-k .

Our problem is to express the Fourier expansion of x → L(xi/2), which is beyond our reach in the general case.

Though, some remarks can be made.

ϑ 1 is odd, therefore

P k,l (-z) = -P l,k (z) if k = l
The series expansion of P yields also

P k,l (z) = -P l,k (z) For x ∈ R, we have for k = l: P k,l (x -i/2) = -P l,k (x -i/2) = -P l,k (x + i/2) = -e x l -x k P l,k (x -i/2).
and this holds also for k = l since P k,k is a pure imaginary number.

Therefore, e -iA/ω P (x -i/2) is skew-hermitian, and thus, -itP/ω is hermitian with respect to the hermitian form with matrix e -iA . Thus so is e -itP/ω and thus L = e -iA/ω e -itP/ω is hermitian (at x -i/2 for real x).

Therefore for all integer j,

L j = 1 0 e -2iπjx L(x - i 2 )dx = 1 0 e -2iπjx [ L(x - i 2 )] † dx = [ 1 0 e 2iπjx L(x - i 2 )dx] † = [ L-j ] †
where the dagger denotes transposition, and thus the infinite matrix T is hermitian.

13 Numerical simulations confirms our conjectures.

Quite surprisingly, finite submatrices of small order of the infinite matrix T give very accurate results. Using a submatrix of order 14, in the two particle case at time 1, we got a four digit agreement with the positions given by Runge-Kutta integration.

CONCLUSION

Our infinite replication scheme has succeeded in re-investigating the Calogero-Moser system with 1 sin 2 r potential, but made the system with elliptic potential more mysterious than ever by connecting it to a new finite difference equation. Many questions remain open. Does this problem have a close form solution? How is it connected with the solution given by Krichever 6 through K-P equation? And is there a geometric interpretation of the system analogous to the one given by Kazdhan, Kostant and Sternberg 5 for 1 sin 2 r potential?

1 CALOGERO (F.), Exactly solvable one-dimensional many-body problems, Lettere

  On ]0, 1[, this equation reads f + (A + e -θA Ce θA )f = λf Set f (θ) = e λθ e -θA g(θ). The preceding differential equation is equivalent to g (θ) + tCg(θ) = 0 or f (θ) = e λθ e -θA e -tθC f 0 with constant f 0 .

  τ (m-a) 2 +2(m-a)(z-b)]

  al Nuovo Cimento 13 (1975), 411-416 2 KAZDHAN (D.), KOSTANT (B.) and STERNBERG (S.), Hamiltonian Group Actions and Dynamical Systems of Calogero Type, Communications on Pure and Applied Mathematics XXXI (1978), 481-507 3 CHUDNOVSKY (D.V.), Meromorphic solutions of nonlinear partial differential equations and many-particle completely integrable systems, J. Math. Phys. 20 (1979), 2416-2422 4 KRICHEVER (I.M.), Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl. 14:4 (1980), 282-290 5 CALOGERO (F.), FRANCOISE (J.P.), Integrable dynamical systems obtained by duplication, Annales de l'Institut Henri Poincaré 57 (1992), 167-181 6 DEBARRE (O.), Tores et variétés abéliennes complexes, Société mathématique de France, EDP Sciences (1999)


Marc Fleury
email: mfleury@ccr.jussieu.fr

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO(n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices.

Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.

Introduction

Few quantum mechanics problems are exactly solvable. Except some one-body problems, the only known solvable problems are those the classical-mechanics counterparts of which are completly integrable: Toda lattices and Calogero-Moser systems [START_REF] Calogero (f) | Solution of the One-Dimensional N -body Problems with Quadratic and/or Inversely Quadratic Pair Potentials[END_REF] or similar ones, such as Sutherland's [START_REF] Sutherland | Exact results for a quantum many body problem in one dimension I[END_REF], [START_REF] Sutherland | Exact results for a quantum many body problem in one dimension II[END_REF].

The general idea of this paper comes from a description of the rational Calogero Model with an external quadratic potential as a projection of a harmonic motion in a matrix space. This idea has been used first for the explicit integration of the equations of motion in the classical case in papers by Olshaneski and Perelomov [START_REF] Olshaneski | Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature[END_REF], [START_REF] Olshaneski | Explicit solutions of some completely integrable systems[END_REF]. Kazdhan, Kostant and Sternberg's [START_REF] Kazdhan | Hamiltonian Group Actions and Dynamical Systems of Calogero Type[END_REF] proved that this projection method coincide with the Marsden-Weinstein symplectic reduction scheme (see also [START_REF] Perelomov | Integrable systems of classical mechanics and Lie algebras[END_REF] and Françoise [START_REF] Franc | Canonical Partition Functions and the Stationary Phase Formula[END_REF], who has solved a conjecture made by Gallavoti and Marchioro [START_REF] Gallavotti | On the calculation of an integral[END_REF]).

This idea is also very useful in the quantum case and sometimes gives the integral representations for wave functions and propagators (Green functions). This is emphasized in the review paper of Olshaneski and Perelomov [START_REF] Olshaneski | Quantum integrable systems related to Lie algebras[END_REF], where an analogue of the theorem 4 of this paper is given for more general cases.

Olshaneski and Perelomov [START_REF] Olshaneski | Quantum systems related to root systems and radial parts of Laplace operators[END_REF], by generalizing some results of Berezin [START_REF] Berezin | Laplace operators on semisimple Lie groups[END_REF], have noticed that the radial part of the Laplace-Beltrami operator of symmetric spaces is conjugated to the quantum Hamilton operator of the Sutherland system, which describes the motion of identical particles on a circle or a branch of a hyperbola with a pairwise interaction potential inversely proportionnal to the square of their mutual distance. They also observed that the problem on the line (Calogero's problem) was obtained as a limit when the curvature of the symmetric space goes to zero.

In this paper, we consider the problem on the line, starting not from a symmetric space, but from the space of real symmetric matrices, which can be understood as the limit of the symmetric space U (n)/SO(n) when its curvature goes to zero. Thus, taking the radial part of the Laplace-Beltrami operator for the action of SO(n), we directly obtain the problem on the line. This scheme allows us to add an external quadratic potential before reduction, that we find again in the reduced one.

The problem thus obtained is slightly different from the one Calogero has studied [START_REF] Calogero (f) | Solution of the One-Dimensional N -body Problems with Quadratic and/or Inversely Quadratic Pair Potentials[END_REF], where the quadratic potential is pairwise rather than external.

In part 1, we fix the notations in use along this paper. In part 2, we give the radial reduction leading to Calogero's problem. Part 3 is devoted to the calculation of the propagator giving the time evolution in terms of the initial wave function, by use of methods analogous to Debiard and Gaveau's ones [START_REF] Debiard | Analysis on Root Systems[END_REF]. In part 4, we give the short time approximation of the propagator by the stationary phase method. This approximation is not exact, but becomes so when the interaction potential vanishes, which allows to compute the Fourier transform of the delta function of adjoint orbits in u(n). Part 5 shows how other constants in the 1/r 2 pairwise potential may be obtained and the corresponding propagators are computed therein. The obtention of systems by reduction allows us to determine the eigenvalues of the Hamilton operator, and then to compute the trace of the propagation operator, which is done in part 6. The results we obtain deal with the propagation in a Weyl chamber. In the last part, we derive the physical propagation according to the statistics the particles obey to.

Notations

The space of n × n real symmetric matrices will be denoted by V , and will be provided with the scalar product: < X, Y >= tr(XY ). The group SO(n) operates on V by conjugacy : (g, X) → gXg -1 (g ∈ SO(n), X ∈ V ) and preserves the scalar product on V . We provide SO(n) with the bi-invariant metric induced by the invariant scalar product on the Lie-algebra so(n) : (X, Y ) → -tr(XY ).

The submanifold Λ =     h 1 h 2 . . . h n     , h 1 < h 2 < ... < h n is
We will denote by dg et dḡ the corresponding invariant measures on SO(n) and SO(n)/M .

We will also use the density fonction δ, which is the ratio between the riemannian measure induced by V on an orbit and the measure dḡ on SO(n)/M . The function δ is radial (SO(n)-invariant). Let us compute this function δ. We remark that the matrices , p < q,

A pq = 1 √ 2      . . .
(1 and -1 being the coefficients of indices (p, q) and (q, p)) form an orthonormal basis of so , p < q, pairwise orthogonal, of norms (h q -h p ), which proves that

(n) = T Id SO(n) = T o (SO(n)/M ) (o denoting the origine of SO(n)/M ). Their images tangent at H =     h 1 h 2 . . . h n     ∈ Λ to the orbit through H are [A pq , H] = h q -h p √ 2      . . . .
δ(gHg -1 ) = p<q (h q -h p ) g ∈ SO(n), H ∈ Λ.
The Laplace-Beltrami operator L V on V induces its radial part ∆L V on Λ, given by ( [START_REF] Helgason | Groups and Geometric Analysis[END_REF]):

∆L V = δ -1/2 L Λ δ 1/2 -δ -1/2 L Λ (δ 1/2 )
where

L Λ = p ∂ 2 ∂h 2 p
is the Laplace-Beltrami operator on Λ provided with the metric induced by the metric of V .

Calogero operator

It has been noticed in [START_REF] Berezin | Laplace operators on semisimple Lie groups[END_REF] that operators of Calogero-Moser-Sutherland type are obtained by conjugation of the radial part of the Laplace-Beltrami operator on a symmetric space by the square root of the density function. Let's make the computation in the case at hand:

δ 1/2 ∆L V δ -1/2 = L Λ -δ -1/2 L Λ (δ 1/2 ) = r ∂ 2 ∂h 2 r -δ -1/2 r ∂ 2 δ 1/2 ∂h 2 r But ∂δ 1/2 ∂h r = 1 2δ 1/2 ∂δ ∂h r = 1 2δ 1/2 p<r δ h r -h p - q>r δ h q -h r = 1 2 δ 1/2 p =r 1 h r -h p and ∂ 2 δ 1/2 ∂h 2 r = 1 2 ∂δ 1/2 ∂h r p =r 1 h r -h p - 1 2 δ 1/2 p =r 1 (h r -h p ) 2 = 1 2 1 2 δ 1/2 ( p =r 1 h r -h p ) 2 - 1 2 δ 1/2 p =r 1 (h r -h p ) 2 = 1 4 δ 1 2 p =r 1 (h r -h p ) 2 + p =r,q =r,p =q 1 (h r -h p )(h r -h q ) - 1 2 δ 1 2 p =r 1 (h r -h p ) 2 = 1 4 δ 1/2 - p =r 1 (h r -h p ) 2 + p =r,q =r,p =q 1 (h r -h p )(h r -h q ) Therefore : δ -1/2 r ∂ 2 δ 1/2 ∂h 2 r = 1 4 - p =r 1 (h r -h p ) 2 + p =r,q =r,p =q 1 (h r -h p )(h r -h q )
But the second sum inside the brackets is zero, since:

3 • p =r,q =r,p =q 1 (h r -h p )(h r -h q ) = p =r,q =r,p =q 1 (h r -h p )(h r -h q ) + 1 (h p -h q )(h p -h r ) + 1 (h q -h p )(h q -h r ) = p =r,q =r,p =q h p -h q -(h r -h q ) (h r -h p )(h r -h q )(h p -h q ) + 1 (h q -h p )(h q -h r ) = p =r,q =r,p =q h p -h r (h r -h p )(h r -h q )(h p -h q ) + 1 (h q -h p )(h q -h r ) = 0. Therefore δ 1/2 ∆L V δ -1/2 = r ∂ 2 ∂h 2 r + 1 4 p =r 1 (h r -h p ) 2 .
If we apply the same scheme to the SO(n)-invariant operator on V :

- 1 2 L V + λ 2 2 X 2 (X ∈ V )
which describes the quantum mechanical harmonic oscillator, we get, after conjugating its radial part, the Calogero operator on Λ with an external quadratic potential:

H = - 1 2 r ∂ 2 ∂h 2 r - 1 8 q =r 1 (h r -h q ) 2 + λ 2 2 r h 2 r (h 1 < h 2 < . . . < h n ).
3. Propagator

General case

The propagator of the N-dimensional harmonic oscillator is known. It is given by Mehler's formula (see [START_REF] Guillemin | Symplectic Techniques in Physics[END_REF]) :

K t (X, X ) = (-i) N α t e -iN π/4 | 2π sin λt λ | -N/2 e iW (X,X ) with W (X, X ) = λ 2 sin λt cos(λt) X 2 + X 2 -2 < X, X > and α t = 0 if t ∈]0, π λ [, 1 if t ∈] π λ , 2π λ [, 2 if t ∈] 2π λ , 3π λ [, etc...
Let ψ 0 be an initial wave function of the form

ψ 0 = δ -1/2 f 0 where f 0 : V → C | is a radial function. Then ψ 0 ∈ L 2 (V ) iff f 0 |Λ ∈ L 2 (Λ) since V | ψ 0 (X) | 2 dX = Λ δ(H) SO(n)/M | ψ 0 (gHg -1 ) | 2 dḡdH = SO(n)/M dḡ × Λ | f 0 (H) | 2 dH
(see [START_REF] Helgason | Groups and Geometric Analysis[END_REF] for this change of variables. g ∈ SO(n) is a representative of ḡ).

The wave function at time t is of the form ψ t = δ -1/2 f t where f t is radial, and f is the solution of Schrödinger equation i ∂f ∂t = Hf with initial wave function f 0 . But for all H ∈ Λ:

δ -1/2 (H)f t (H) = ψ t (H) = V K t (H, X )ψ 0 (X )dX = (-i) N α t e -iN π/4 | 2π sin λt λ | -N/2 × V e iW (H,X ) δ -1/2 (X )f 0 (X )dX
where

N = dim V = n(n + 1) 2 = (-i) n(n+1) 2 α t e -i n(n+1)π 8 | 2π sin λt λ | -n(n+1) 4 × Λ δ(H )δ -1/2 (H )f 0 (H ) SO(n)/M e iW (H,gH g -1 ) dḡ.dH = 1 2 n-1 (-i) n(n+1) 2 α t e -i n(n+1)π 8 | 2π sin λt λ | -n(n+1) 4 × Λ δ 1/2 (H )f 0 (H ) SO(n)
e iW (H,gH g -1 ) dg.dH

(since M has cardinal 2 n-1 )
Hence the propagation of the restriction of f to Λ :

Theorem 1: The solution of Schrödinger equation i ∂f ∂t = Hf with initial condition f |t=0 = f 0 ∈ L 2 (Λ) is given by f t (H) = Λ k t (H, H )f 0 (H )dH where k t (H, H ) = 1 2 n-1 (-i) n(n+1) 2 α t e -i n(n+1)π 8 | 2π sin λt λ | -n(n+1) 4 δ 1/2 (H)δ 1/2 (H ) × e iλ cos λt 2 sin λt ( H 2 + H 2 ) SO(n) e -iλ sin λt <H,gH g -1 > dg Remark: If λ = 0, the propagator is k t (H, H ) = 1 2 n-1 e -i n(n+1)π 8 (2πt) -n(n+1) 4 δ 1/2 (H)δ 1/2 (H ) × e i 2t ( H 2 + H 2 ) SO(n) e -i t <H,gH g -1 > dg
In order to compute < H, gH g -1 >, it is sufficient to know the diagonal entries of gH g -1 , given by : < gH g -1 e p , e p >=< H g -1 e p , g -1 e p >=< r h r (g -1 ) rp e r , r (g -1 ) rp e r > = r h r (g -1 ) rp 2 = r (g pr ) 2 h r .

Therefore < H, gH g -1 >= pq (g pq ) 2 h p h q .

Remark: The propagator is 2π λ -periodic if n is congruent to 0 or 3 modulo 4, 4π λ -periodic in the remaining cases. This property comes from the fact that the system is the reduction of a n(n+1)

2

-dimensional harmonic oscillator. The classical system is 2π λ -periodic (Zoll system), for the same reason.

The case n=2 (two-particle systems)

In the case n = 2, we show that the integral in the expression of the propagator can be expressed in terms of a Bessel function.

We parametrize SO(2) in the usual way :

g = cos θ -sin θ sin θ cos θ and we have dg = √ 2dθ since the metric chosen on SO(2) is ds 2 = -tr 0 dθ -dθ 0 2 = 2dθ 2 and < H, gH g -1 > = cos 2 θ(h 1 h 1 + h 2 h 2 ) + sin 2 θ(h 1 h 2 + h 2 h 1 ) = 1 2 (h 1 + h 2 )(h 1 + h 2 ) + 1 2 (h 2 -h 1 )(h 2 -h 1 ) cos 2θ. Therefore SO(2) e -iλ sin λt <H,gH g -1 > dg = e -iλ 2 sin λt (h 1 +h 2 )(h 1 +h 2 ) 2π 0 e -iλ 2 sin λt (h 2 -h 1 )(h 2 -h 1 ) cos 2θ √ 2dθ = √ 2e -iλ 2 sin λt (h 1 +h 2 )(h 1 +h 2 ) 2πJ 0 λ(h 2 -h 1 )(h 2 -h 1 ) 2 sin λt
(where J 0 is the zero order Bessel function).

N = dim(V ) being equal to 3, we get the following expression of the propagator

k t (h 1 , h 2 ; h 1 , h 2 ) = 1 2 √ π i α t e -i3π/4 | sin λt λ | -3/2 (h 2 -h 1 )(h 2 -h 1 ) ×e iλ 2 sin λt (h 2 1 +h 2 2 +h 1 2 +h 2 2 ) cos λt-(h 2 +h 1 )(h 2 +h 1 ) J 0 λ(h 2 -h 1 )(h 2 -h 1 )
2 sin λt .

Remark : If the system is reduced to its center of mass, it is equivalent to a (separable) system of two uncoupled particles on a line, of equal masses, independent from one another, one in the harmonic potential λ 2 2 x 2 , the other in the potential λ 2 2 x 2 -1 8x 2 . Our result for two particle sytems can thus be deduced from Khandekar and Lawande's results [START_REF] Khandekar | Exact Propagator for a Time Dependent Harmonic Oscillator with and without Perturbation[END_REF]. The reader can also refer to Schulman's book [START_REF] Schulman | Techniques and Applications of Path Integration[END_REF] which gives a list of computable propagators.

Stationnary phase approximation

In view of the expression of the propagator found in the case n = 2, which contains a Bessel function, the semi-classical approximation of the propagator is not exact for finite times. Yet, it is possible to obtain the asymptotic behavior of the propagator for short times, using the stationnary phase method.

According to this method,

(2πt) -1 2 dim (SO(n)/M ) SO(n)/M e -iλ sin λt <H,gH g -1 > dḡ = ḡ c(ḡ)e -i t <H,gH g -1 > + O(t),
the summation being made on all the critical points of

f : ḡ ∈ SO(n)/M → -< H, gH g -1 >, c(ḡ) being equal to exp i π 4 sgn Hess ḡ f | det Hess ḡ f | -1 2 .
Here Hess ḡ f denotes the Hessian matrix of f at ḡ in a basis of T ḡ (SO(n)/M ) of volume 1, and sgn its signature, i.e. the difference between the numbers of its positive and negative eigenvalues.

Let H, H ∈ Λ. The critical points of the function f are those for which H is orthogonal to the orbit through H at gH g -1 , which amounts to gH g -1 being diagonal. Therefore the critical points of ḡ are in a one to one correspondence with the matrices obtained by permutation of the diagonal entries of H . Let ḡ be a critical point, H = gH g -1 . The vectors A pq .g (p < q) form an orthonormal basis of T g SO(n) and

< H, exp XgH g -1 exp(-X) > =< H, exp XH exp(-X) > =< H, (I + X + 1 2 X 2 )H (I -X + 1 2 X 2 ) > +o( X 2 ) =< H, H + [X, H ] + 1 2 (X 2 H + H X 2 ) -XH X > +o( X 2 ) =< H, H > + 1 2 < H, X 2 H + H X 2 -2XH X > +o( X 2 )
If X = p<q s pq A pq , we get

-< H, exp XgH g -1 exp(-X) >= -< H, H > + 1 2 p<q (h q -h p )(h q -h p )s 2 pq
The hessian matrix of f at ḡ is thus diagonal. The absolute value of its determinant is δ(H)δ(H ). Its signature is n(n-1)

2

-2I(σ), where σ ∈ S n is the permutation of the set {1, . . . , n} corresponding to ḡ and I(σ) = Card {(p, q)| p < q, σ(p) > σ(q)} is the number of inversions in σ.

Thus

SO(n)/M e -iλ sin λt <H,gH g -1 > dḡ = (2πt) n(n-1) 4 δ -1 2 (H)δ -1 2 (H )e i n(n-1) 8 π σ∈S n e -i π 2 I(σ) e -i t (h 1 h σ(1) +...+h n h σ(n) ) + O(t)
Theorem 2: The short time approximation of the propagator is given by

k t (H, H ) = (2πt) -n 2 e -in π 4 σ∈S n e -i π 2 I(σ) e i 2t H-σ -1 (H ) 2 + O(t)
where I(σ) = I(σ -1 ) is the number of infinite potential wells

h i = h j across the segment [H, σ -1 (H )].
The contribution of H to the propagator for short times can be interpreted as resulting of broken straight lines from H to H, with a phase shift of -π 2 at each "reflexion" on a potential well, i.e. at each collision between two particles. Remark: By computing as in the preceding section the propagator for functions on u(n) invariants for the adjoint action of U (n), we can get the Fourier transform of the delta distribution of an adjoint orbit, a result first proved by Harish Chandra (cf [START_REF] Guillemin | Symplectic Techniques in Physics[END_REF]).

As a manifold transversal to the orbits, we chose the set Λ of diagonal purely imaginary matrices whose diagonal entries: ih 1 , . . . , ih n verify h 1 < h 2 < . . . < h n . The adjoint orbits are diffeomorphic to U (n)/M , M being the isotropy subgroup of purely imaginary diagonal matrices, i.e. the set of diagonal matrices in U (n). The radial part ∆L of the Laplace-Beltrami operator L on the euclidean space u(n) displays :

δ 1/2 ∆L • δ -1/2 = r ∂ 2 ∂h 2 r
where δ = r<l (h l -h r ) 2 is the density function, i.e. dX = δ.dh 1 . . . dh n dḡ, dḡ denoting the riemannian measure on U (n)/M . The propagator of Shrödinger equation i

∂ ∂t = -1 2 L for a free particle on u(n) is K t (X, X ) = e -iN π/4 1 (2πt) N/2 e iW (X,X ) where W (X, X ) = 1 2t X 2 + X 2 -2 < X, X > and N = dim u(n) = n 2
Let h denote the set of diagonal purely imaginary matrices. If a family of radial functions (ψ t ) t≥0 in L 2 (u(n)) verifies Schrödinger equation i ∂ ∂t = -1 2 L, the family of functions (f t ) t≥0 in L 2 (h), skew-symmetric in the variables h 1 , . . . , h n , defined by

f t (H) = δ 1/2 ψ t (H) = r<l (h l -h r )ψ t (H) verifies the equation i ∂ ∂t = -1 2 r ∂ 2 ∂h 2 r
. Hence f t is propagated by the kernel

k t (H, H ) = e -inπ/4 1 (2πt) n/2 e i 2t H 2 + H 2 -2<H,H >
and its restriction to Λ by σ∈S n (σ)k t (H, σ(H )), and also by δ 1/2 (H)δ 1/2 (H )

U (n)/M K t (H, gH g -1 )dḡ
We deduce Harish Chandra's formula for the Fourier transform of the delta distribution of an adjoint orbit

U (n)/M e -i<H,gH g -1 > dḡ = e in(n-1)π/4 (2π) n(n-1)/2 r<l (h l -h r )(h l -h r ) σ∈S n (σ)e -i<H,σ(H )>
which can be interpreted as a case of exactitude of the stationary phase approximation [START_REF] Guillemin | Quasi-classical aspects of reduction[END_REF].

Other interaction constants

Here we investigate two different schemes to obtain other interaction constants than -1 4

Eigenfunctions of orbital Laplace-Beltrami operators

Using the separation of variables, we consider fonctions which are the product of a radial function and a fixed angular function whose restriction to each orbit is an eigenfunction of the orbital Laplace-Beltrami operator.

For this purpose, we need the expression of the Laplace-Beltrami operator L V with respect to the coordinates (H, ḡ) ∈ Λ × SO(n)/M . L V splits into a transversal and an orbital part (cf [START_REF] Helgason | Groups and Geometric Analysis[END_REF]), which means that at each point X of V , for any function ψ ∈ C ∞ (V ),

(L V ψ)(X) = (L V ) T ψ (X) + (L O X ψ |O X )(X)
where (L V ) T is the transversal part of L V :

(L V ) T ψ (gHg -1 ) = (∆L V ) H ∈ Λ → ψ(gH g -1 (H),
and L O X is the Laplace-Beltrami operator on the orbit O X through X.

Let H ∈ Λ. Let us denote by A pq the infinitesimal rotations defined in part 1, as well as the vector fields that they induce on O H . The vectors B pq = A pq /(h q -h p ), p < q form at each point of the orbit an orthonormal basis of the tangent space. For every function f defined on O H , the gradient of f is given by grad(f ) = p<q (B pq f )B pq . Furthermore, the metric of the orbit and therefore its volume form being invariant under the action of SO(n), the vector fields B pq are divergence free. Hence

L O X f = div gradf = p<q [< grad(B pq f ), B pq > +(B pq f )div(B pq )] = p<q B 2 pq f = p<q 1 (h q -h p ) 2 A 2 pq f.
Let c be a function on SO(n)/M inducing on each orbit an eigenfunction of the orbital Laplace-Beltrami operator. Functions of the form ψ(gHg -1 ) = f (H)c(ḡ) conserve this form through their time evolution with respect to the harmonic oscillator problem.

The orbital Laplace-Beltrami operator is a linear combination of the squares of the infinitesimal rotations A 2 pq , p < q in the coordinates planes. Therefore, if c(ḡ) is an eigenfunction of the square of the infinitesimal rotation in each coordinates plane, then it will satisfy to the condition above .

Let χ be a representation of SO(n) on some vector space E. If u ∈ E is an eigenvector for each operator (dχA pq ) 2 , and η any linear form on E, then the representation coefficient c(g) = η(χ(g)u) is also an eigenvector of the A 2 pq 's with the same eigenvalues. Indeed:

A 2 pq c(g) = d 2 dt 2 |t=0 c(g exp(tA pq )) = d 2 dt 2 |t=0 η(χ(g exp(tA pq )u) = η χ(g)(dχA pq ) 2 u
Let us consider the irreducible representation χ of SO(n) on the space E of harmonic polynomials of n indeterminates x 1 , . . . , x n homogeneous of degree n (cf [START_REF] Vilenkin | Fonctions spéciales et théorie de la représentation des groupes[END_REF]). We have:

dχ(A pq ) = 1 √ 2 (x q ∂ ∂x p -x p ∂ ∂x q )
Note that the harmonic monomial u = x 1 x 2 . . . x n yields:

dχ(A pq )u = 1 √ 2 (x 1 . . . xp . . . x 2 q . . . x n -x 1 . . . x 2 p . . . xq . . . x n )
(the symbol ˆdesigning an omitted factor ) and: (dχ(A pq ))

2 u = -2u
Furthermore u is left invariant by the rotations of angle π in the coordinates planes (which amount to turning two of the coordinates x p and x q into their opposite), thus the coefficients c(g) = η(χ(g)u) define functions c(ḡ) on the quotient space SO(n)/M and therefore on the orbits. For instance, if η is the component on the vector x 1 x 2 . . . x n of the canonical basis of E, c(ḡ) = σ∈S n g 1σ(1) g 2σ(2) . . . g nσ(n) where g = ((g pg )) ∈ SO(n) is a representative of ḡ. These functions are eigenfunctions, and the associated eigenvalue on the orbit

O H is -2 p<q 1 (h q -h p ) 2
Remark: these functions are exceptional. The author has verified that in the case n = 3, there is no other non proportional function which is an eigenfunction for each (dχA pq ) 2 .

Therefore, for ψ :

gHg -1 → f (H)c(ḡ), L V (δ 1/2 ψ)(gHg -1 ) = δ 1/2 (H) r ∂ 2 ∂h 2 r - 3 2 p<r 1 (h r -h p ) 2 (f )(H)c(ḡ)

Hence

Theorem 3: The propagator of the Hamilton operator

- 1 2 r ∂ 2 ∂h 2 r + 3 4 p<r 1 (h r -h p ) 2 + λ 2 2 H 2 is k t (H, H ) = 1 2 n-1 (-i) n(n+1) 2 α t e -i n(n+1)π 8 | 2π sin λt λ | -n(n+1) 4 δ 1/2 (H)δ 1/2 (H ) × 1 c(1) e iλ cos λt 2 sin λt ( H 2 + H 2 ) SO(n) e -iλ sin λt <H,gH g -1 > c(g)dg
In the exceptional case n = 2, all the irreducible representations of SO(2) are unidimensionnal and each vector of the representation line is obviously an eigenvector for

A 12 = 1 √ 2 d dθ . If c(ḡ θ ) = e 2inθ
, g θ designing the rotation of angle θ and n a fixed integer (c(ḡ) is not single-valued for n half integer), A 2 pq c = -n 2 c. Hence we can obtain all the interaction constants of the form n 2 -1 4 , n ∈ I N . The integral in the propagator can be expressed using a Bessel function of order n:

k t (h 1 , h 2 ; h 1 , h 2 ) = 1 2 √ π i α t -n e -i3π/4 | sin λt λ | -3/2 (h 2 -h 1 )(h 2 -h 1 ) ×e iλ 2 sin λt (h 2 1 +h 2 2 +h 2 1 +h 2 2 ) cos λt-(h 2 +h 1 )(h 2 +h 1 ) J n λ 2 sin λt (h 2 -h 1 )(h 2 -h 1 )

Other symmetric spaces

Olshaneski and Perelomov remarked in [START_REF] Olshaneski | Quantum systems related to root systems and radial parts of Laplace operators[END_REF] that the radial parts of the Laplace Beltrami operator of symmetric spaces SU (2n)/Sp(n) and E 6 /F 4 are conjugated to Sutherland systems with interaction constants 2 et 12. The corresponding Calogero systems can be obtained by letting the curvature go to zero with homothetical transformations and by making a central extension to get rid of the constraint " h p = 0". For instance, when the interaction constant is 2, we consider the action of Sp(n) by conjugacy on the orthogonal complement of sp(n) in u(2n), which is the space of matrices of the form

Z 1 Z 2 Z2 -Z1
, Z 1 ∈ u(n), Z 2 complex skew-symmetric.

Energy levels and trace of the propagator

Let us denote by X = ((X pq )) the generic element of V and set

z pp = X pp z pq = 1 √ 2 X pq , p < q
The z pq , p ≤ q form a system of orthonormal coordinates on V . The one-dimensional harmonic oscillator with Hamilton operator -1 2

d 2 dx 2 + λ 2 2
x 2 has eigenfunctions of the form ψ(x) = cste.e -λ x 2 2 H p ( √ λx) (where H p denotes the p th Hermite polynomial) with eigenvalues (p + 1 2 )λ (see [START_REF] Landau | Physique Théorique tome 3: Mécanique quantique[END_REF] for instance). The eigenfunctions of the harmonic oscillator which hamiltonian is

- 1 2 L V + λ 2 2 X 2
are thus the functions of the form

ψ(X) = exp(-λ X 2
2 ) The eigenfunctions of the reduced operator are obtained by multiplicating by the quare root of the density function the restriction to Λ of the radial functions among the ones above.

(
Therefore each eigenfunction of the reduced system is of the form

ϕ(H) = δ(H) exp(-λ H 2 2 )P (h 1 , . . . , h n )
where P is a polynomial. Furthermore, since the invariant function ψ(gHg -1 ) = exp(-λ H 2 2 )P (h 1 , . . . , h n ) defined by ϕ is an eigenfunction of the harmonic oscillator, it is analytic and thus

ψ h σ(1) . . . h σ(n) = exp(-λ H 2 2 )P (h σ(1) , . . . , h σ(n) )
for every permutation σ ∈ S(n).

But h σ(1) . . . h σ(n) is conjugated to h 1 . . . h n
hence the left hand side of the equality above is also equal to exp(-λ H 2 2 )P (h 1 , . . . , h n ), which proves that the polynomial P is symmetric.

We will prove that the functions of the form ϕ(H) = δ a (H) exp(-λ H 2 2 )P (H), where P is a homogeneous symmetric polynomial, provide a basis of L 2 (Λ) in which the matrix of

H = -1 2 r ∂ 2 ∂h 2 r + b q =r 1 (h r -h q ) 2 + λ 2 2
r h 2 r is triangular. The exponent a > 0 will depend on the interaction constant b and be equal to 1/2 in the case of the reduction of radial functions (b = -1/8).

We have

∂ϕ ∂h r = δ a (H) exp(-λ H 2 2 ) a p =r 1 (h r -h p ) P -λh r P + ∂P ∂h r , ∂ 2 ϕ ∂h 2 r = δ a (H) exp(-λ H 2 2 ) p =r a 2 -a (h r -h p ) 2 + p =r,q =r,p =q a 2 (h r -h p )(h r -h q ) P +(λ 2 h 2 r -λ)P + ∂ 2 P ∂h 2 r + 2 a p =r 1 h r -h p -λh r ∂P ∂h r + 2a p =r 1 h r -h p (-λh r )P Since r p =r h r h r -h p = n(n-1)
2 and p =r,q =r,p =q 1 (h r -h p )(h r -h q ) = 0 (cf part 2),

Hϕ = δ a (H) exp(-λ H 2 2 ) - 1 2 r ∂ 2 P ∂h 2 r + r (λh r -a p =r 1 h r -h p ) ∂P ∂h r + a n(n -1) 2 + n 2 λP ,
where a > 0 is chosen so that (a 2 -a)/2 is equal to the interaction constant b.

If P is homogeneous of degree d, then Euler formula r h r ∂P ∂h r = d.P yields:

Hϕ = δ a (H) exp(-λ H 2 2 ) - 1 2 r ∂ 2 P ∂h 2 r -a r p =r 1 h r -h p ∂P ∂h r +(a n(n -1) 2 + n 2 +d)λP
If moreover P is symmetric, then ∂P ∂h r -∂P ∂h p is zero when h r = h p and thus is divisible by (h r -h p ).

Therefore, we obtain:

Hϕ = ( n(an + 1 -a) 2 + d)λϕ + δ a (H) exp(-λ H 2 2 )Q
where Q is a symmetric polynomial of degree at most d -2. Hence:

Theorem 4: Each eigenfunction corresponds to a symmetric polynomial, the energy levels are ( n(an+1-a) 2 + d)λ (d ∈ I N ), with multiplicity equal to the dimension of the space of homogeneous symmetric polynomials of n indeterminates of degree d. This dimension is the cardinal of the set {(a 1 , . . . , a

n ) ∈ I N n | a 1 ≤ a 2 ≤ . . . ≤ a n , a 1 + . . . + a n = d}.
We refer the reader to [START_REF] Olshaneski | Quantum integrable systems related to Lie algebras[END_REF] for an anologue of this theorem for more general cases.

Hence the trace of the propagation operator has a very simple closed form expression:

tr(e -itH ) = ) (2i) n sin(λt/2) sin(λt) sin(3λt/2) . . . sin(nλt/2) Besides, the trace can also be expressed with respect to the propagator:

tr(e -itH ) = Λ k t (H, H)dH
In the case of two-particle systems, we obtain known identities involving Bessel functions, connected with the formula (cf [START_REF] Watson | A Treatise on the Theory of Bessel Functions[END_REF])

+∞ 0 e -ax J ν (bx)dx = [ √ a 2 + b 2 -a] ν b ν √ a 2 + b 2
More complex identities may be obtained by considering systems with three particles or more.

Bosons and Fermions

The propagators computed so far allow to describe the motion of n particles whichever statistic (bosonic or fermionic) the particles obey.

The wave function f t at time t of such a system is not defined on Λ, but on the spce of diagonal matrices with distinct entries :

Λ = σ∈S n σ(Λ)
where a permutation σ ∈ S n de {1, 2, . . . , n} acts on the space of diagonal matrices by :

σ     h 1 h 2 . . . h n     =      h σ -1 (

  transversal to the action of SO(n) and orthogonal to the orbits since the tangent space to the orbit through H ∈ Λ is {[Z, H] = ZH -HZ|Z skewsymmetric}, the tangent space to Λ is the space of diagonal matrices H and < H , [Z, H] >= tr(H ZH -H HZ) = 0 because HH = H H.We denote by M = an even number of -1 the isotropy subgroup in SO(n) of matrices in Λ.

Theorem 5 :

 5 ia 2 λt) . . . ∞ a n =a n-1 exp(-ia n λt) But ∞ a n =a n-1 exp(-ia n λt) = exp(-ia n-1 λt) ∞ a n =0 exp(-ia n λt) = exp(-ia n-1 λt) 1 1 -exp(-iλt) = exp(-ia n-1 λt) exp(iλt/2) 2i sin(λt/2)Thus the trace of the propagation operator tr(e -itH ) is equal to exp(-i n(an + 1 -The trace of the propagation operator tr(e -itH ) is equal to exp(i (1-2a)n(n-1)λt 4

  a pq ) p≤q ∈I N where the c a 11 a 12 ...a nn are arbitrary constants. The corresponding energy level is (r + N/2)λ = [r + n(n + 1)/4]λ.

				√
			c a 11 a 12 ...a nn	H a pq (	λz pq )
		n(n+1) 2	,	p≤q
	p≤q	a pq =r

  In both cases, the propagation of ft is given by the one of f t : the solution of the Schrödinger equation -h p ) 2 + b

	i	∂ ft ∂t	= -	1 2	∆ ft + a	p =q	1 (h q p	h 2 p ft ,
	with initial wave function f0 , is given by :
			ft σ(H) =	k t (H, H )f 0 (H )dH
						Λ
						(for bosons)
					ou	(σ)	k t (H, H )f 0 (H )dH
							Λ
						(for fermions)
	Bibliography :
								1)	
									h σ -1 (2)	. . .	   	.
									h σ -1 (n)
	But ft is determined by its restriction to f t à Λ :
			ft σ(H) = f t (H)	for bosons
			ft σ(H) = (σ)f t (H)	for fermions
							H ∈ Λ, σ ∈ S n
						(σ) denotes the signature of σ.