Charles T Wolfe
email: charles.wolfe@ugent.be

MATERIALISM: A HISTORICO-PHILOSOPHICAL INTRODUCTION

Chapter 1 (

materialism in the nineteenth century, physicalism in the twentieth, and so on. But the challenges posed by materialism as a historico-philosophical object are not just an effect of shifting meanings, historical and/or scientific contexts, or even its selfunderstanding (consider that anti-clerical materialism will have, at least for the most part, different goals and criteria of validity than neurophilosophy). They include the significant fact that unlike, say, 'idealism', it is at first a polemical term, primarily defined by its opponents, including authors who subdivide materialism into many more precise genres, the better to refute them; indeed, apologeticists often prove to be excellent guides to the internal structures of heterodox thought (leaving aside the question of whether these apologetic texts actually invent these argument structures, as is claimed in Kors 1990).

So materialism first appears on the scene as an articulated philosophical position defined by anti-materialists, one which was indeed primarily or even exclusively used to disqualify the opponent. Gradually, some nuances appear, as when the Cambridge Platonist Henry More allows, in his 1668 Divine Dialogues, for a distinction between good and bad kinds of materialists, where the former defend a form of mechanism, without holding that everything reduces to matter, as the latter do. 5 It seems that it was (appropriately) in the context of the Radical Enlightenment that the term 'materialist ' was first used by a thinker, La Mettrie, to describe himself, rather than strictly as a term of opprobrium (Bloch 1995). Thereafter, in the second half of the eighteenth century, it starts to be used positively but still with a polemical charge, so that authors have to defend themselves, e.g., against the charge of libertinage: thus Diderot, responding to the Dutch natural philosopher Hemsterhuis' request for commentary on his manuscript, observes that Hemsterhuis reasons "as if libertinage was a necessary consequence of materialism, which seems to me to match neither reason nor experience."6 That Diderot went to prison for his Letter on the Blind of 1749, and La Mettrie had to flee, not just to the Low Countries but ultimately to the exile of Frederick the Great's court in Potsdam, only adds some bitterness to such sentiments.

In the nineteenth century, materialism takes on a meaning familiar to us today, as the science-friendly doctrine, the ideological combatant for science but also its 'valet', leading to a rather pronounced split in possible meanings, between the negative usage (often with ethical overtones, as discussed in Chapter 5) and a positive usage that overemphasizes this connection to science, neglecting some differences between philosophical concepts and empirical claims (as discussed with regard to brain-mind identity theories in Chapter 7). In some cases materialism is also located somewhere in between these extremes, in a sort of transitional understanding, as in when the young Charles Darwin, toying with implications of 'animal minds' and seeking to go beyond Locke, suddenly comments to himself in (mock?) horror, "Oh you materialist!" 7 How do we handle such a shifting word? How do we analyse a doctrine which -aside from the rather banal fact that it had, and has partisans and detractors, doubtless like many other doctrines although in an especially charged mannerseems to be an alluring vision, "the most seductive philosophy"8 in Diderot's words, the most liberating, in an Epicurean sense, but also, to others, the most sterile and inhuman philosophy? (Or, which does not match either of those very normatively invested visions, a kind of reasonable, naturalistically oriented vision of the world?) In the Marquis de Sade's novels, the most explicit scenes are often precipitated by a libertine character delivering a short, emphatic petitio principii of materialism, as a kind of particularly refined stimulus for erotic activity (Warman 2002). In contrast, Raymond Ruyer (a philosopher of the 1940s-1960s whose influence on Deleuze means that he is being rediscovered today), suggests a thought-experiment in an article entitled 'What is Living and What is Dead in Materialism', which has gone rather unnoticed (it appeared in 1933…). Ruyer suggests that we imagine a law court as seen through the eyes of a materialist: "The halo of meanings, essences and values," in other words, everything relevant about the scene, vanishes, and what is left is the "functioning of a sort of complicated mechanics" whereby brains produce articulations, which in turn generate vibrations in the air, and thereby modify other nervous systems (Ruyer 1933, 28). Everything takes place in the present, which is made up of strictly quantifiable events; psychological or social reality is an emanation which can always be reduced to physical processes. Basically, materialism in this argument ad absurdum is a strange kind of reductionism which denies the reality of social institutions, values, and of course minds.

In the first case, materialism is a process of elimination of superstition and the forces which constrain the pleasure of life, as La Mettrie would write in his scandalous Anti-Seneca (also published as Discourse on Happiness). In the second, it is a theory of reality which seeks to apply the 'rigor' or 'quantification' of physics to all aspects of reality -but in a kind of illegitimate transposition or category mistake, which we will encounter in another form with regards to the identification of mental life with cerebral processes, in Chapters 6-8. Its most classic form was represented by the German 'vulgar materialist' Carl Vogt's slogan stating that all the properties we refer to as the activity of the soul are just functions of cerebral substance, and to put this more crudely, thought is (more or less) to the brain what bile is to the liver and urine to the kidneys. It is absurd to allow for an independent soul using the brain as an instrument…9

That the biochemical reductionism of the Vulgärmaterialisten was meant to be part of a socialist program of equality and elimination of class differences, does not really enter into this story, although it fits with the often expressed fear that materialism meant a kind of 'downwards' reduction, not just at the metaphysical level, but also in terms of basic human (and social) values: in 1873, one Doctor Desgrange asserted, in an address to the Société de médecine of Lyon, that "the most fearsome enemy of society today is the materialist School, whose doctrines begin at the highest levels of science, and then descend towards the lower classes, warping their ideas with breathtaking speed." 10 What is more relevant in Ruyer's rather feverish denunciation of materialism as a reduction of the world of symbolic value to a set of vibrations in the air caused by solid objects -a vision in which, to quote Father Dominique Dubarle, "The material world is what remains of reality once one forbids oneself from including in it anything vital or mental" 11 -is that it captures two recurrent definitional and polemical problems of materialism, visible already in Aristotle's critique of the atomists (Chapter 2), and in different forms in debates over early modern materialism (Chapters 4 and 5) but also the Identity Theory in the twentieth century (Chapter 7): the problem of reductionism, and by extension, the question, reduction to what?, which opens onto the issue that will be termed physicalism. Is materialism a reduction to physics? I do not pretend to answer this question in this book, especially not in an Introduction, but the reader may make her judgment based on some of these chapters (and differently put, physicalism is something of a negative Leitfaden in my story). In the name of completeness, it is worth considering an answer to Ruyer's objection, not to him in particular, but to this type of antimaterialist argument. The answer is Quine's:

Send a man into another room and have him come back and report on its contents. He comes back and agitates the air for a while, and in consequence of this agitation we learn about objects in the room which are very unlike any agitation of the air. Selected traits of objects in that room are coded in traits of this agitation of the air. The manner of the coding, called language, is complicated and far-fetched, but it works; and clearly it is purely structural, at least in the privative sense of depending on no qualitative resemblances between the objects and the agitation. Also the man's internal state, neural or whatever, in which his knowledge of the objects in that room consists, presumably bears none but structural relation to those objects; structural in the privative sense of there being no qualitative resemblances between the objects and the man's internal state, but only some sort of coding, and, of course, causation. ... I do think there is a substantial resemblance between our internal state ... and the man's internal state ... This I find plausible on broadly naturalistic grounds. 12 The force in Quine's account lies, at least for my purposes, in how open it is: the mere 'physics' of vibration or here, agitation, seems to open onto the vast vistas of naturalism.

But now we run the risk of succumbing to a classic temptation, by entering onto the terrain first staked out by Friedrich Lange in his History of Materialism back in the 1860s: Lange sought to produce an exhaustive presentation of materialism in all its historical forms, in order to refute it (the book is subtitled, after all, 'critique of its present-day significance'). To enter on such terrain means producing militant defences of the 'truth' of materialism, in a mirror image of Lange's refutation. 13 It is important, on methodological but perhaps also on fundamental philosophical grounds, to see that an attempt to understand what might be in common in the diverse forms of materialism, does not necessarily mean to assert its 'truth' in some meta-historical sense (including as a purportedly 'scientifically founded' truth), nor to propose one of the various post-May 68 'war machines' in the history of philosophy, intended to roll back forms of power, domination and repression. 14 To put it differently, the following chapters are not presented as a watertight, militant counter-history of philosophy in which Lucretius, La Mettrie, Diderot, Vygotsky, Quine and Dennett form a counter-narrative against a history in which Aristotle, Descartes, Kant and Hegel are the heroes. They do seek to do justice to such figures over and against the impoverished and sometimes downright false presentations given both in histories of philosophy and in canonical works of philosophy (with an exception being Aristotle's critique of materialism, discussed in Chapter 2, which is not a mere attempt at disqualification but a serious engagement with competing explanations of natural processes). In order to do justice to this historical complexity, I wish to return to the fact I mentioned at the outset, that materialism is born as a 'labelled' philosophical movement in an atmosphere of opprobrium. For if we leave this out, we then retreat behind a catalogue of historical definitions. Yet I should also like to set out some typological elements concerning 'forms of materialism', before turning to my particular cases in the following chapters.

Dead matter and the opprobrium of materialism

Materialism has long had a bad reputation, on two distinct yet related grounds: that it reduces everything to 'dead' matter, and that it eliminates the 'higher', intellectual or spiritual parts of life, and thereby cannot but be immoral. This set of accusations came to a head in the period we now know as the Radical Enlightenment,15 when, building on Paduan Averroist Aristotelianism (e.g. Pomponazzi), neo-Epicureanism and other partly clandestine elements, thinkers first assert themselves as materialists, boldly and confidently. One may ask (as I do in Chapters 4 and 5) whether these materialists, preachers of the pleasures of the flesh and otherwise deniers of an immortal or any other transcendent source of normativity (and thus basis for reward or punishment) were as coldly mechanistic and immoral as we are often told.

It has been said that the history of philosophy is the history of idealism. This is of interest, less as a truth claim (surely dependent on all sorts of presuppositions about the nature of philosophy, among others), and more because of it what it reveals. The import of this revelation is twofold: philosophy frequently and canonically has understood itself as idealism, both because of its opprobrium against materialism, and because of the reflexive belief -inseparably systematic and historical -that from Plato and Aristotle to Descartes, Kant and Hegel (and beyond), a philosophy is at its core a system of interlocking principles with a rational foundation. On this view, it cannot be an appeal to merely empirical, contingent properties, and still less a 'reductionist' explanation of the higher-level (consciousness, intentionality, action overall) in terms of the neuronal or biochemical properties of nematodes, sea slugs, macaques or orang-outangs. All true philosophies are then forms of idealism, while materialism is Unphilosophie, nonphilosophy (Colletti 1969, 10, 35-36) -a position that has a Hegelian ring to it (after all, for Hegel, "Every philosophy is an idealism" 16 ), but that extends beyond: Schopenhauer had declared that the "true philosophy" was in any case idealism, while materialism is the philosophy of "the subject who forgets to account" for herself. 17 The opposition between idealism and materialism certainly runs deep. The eminent scholar of French materialism, Olivier Bloch, has recalled that Plato, in the Sophist (246b-c), features a 'battle of giants' (gigantomachia) between the Lovers of Forms and those he calls the Sons of the Earth, his early version of the figures we might call the 'crude materialists'. The latter come in different guises, for Plato: there are those who explain everything about our bodies and life in terms of the Earth, and thereby confuse human life with the existence of trees and stones; there are those who obsessively take apart reality into tiny atomic components and view the universe as perpetually changing. This contrasts with Aristotle's extensive (and, in my view, more sophisticated) presentation of and 'debate' with materialism, discussed in Chapter 2. The stupidity of such thinkers is mirrored (or matched) by the purported stupidity of matter itself, a motif much belabored in early modernity: e.g., the Cambridge Platonist Ralph Cudworth spoke of "stupid and senseless Matter" in his posthumous Treatise concerning Eternal and Immutable Morality (Cudworth 1897, I, chapter II, § 8, 839).

This theme of the 'stupidity' of matter reaches something of a fever point in the seventeenth and early eighteenth centuries: for Cudworth, "Mind and Intellect are a higher, more real and substantial Thing than senseless Body and Matter," and he of course denied that "all Being and Perfection that is found in the World" could "spring up and arise out of the dark Womb of unthinking Matter" (slightly mixing metaphors, one might say: op. cit., § 13, 846). The great Jansenist Pierre Nicole, who significantly influenced Locke, also wrote around the same time that one cannot conceive of "this dead and unfeeling mass we call matter" as being "an eternal being"; it is clear, Nicole continues, that "matter lacks any internal cause of its existence ... it is ridiculous to attribute to the most vile and despicable of all beings, the greatest of perfections, which is to exist by oneself [d'être par soi-même]" (Nicole 1671, in Nicole 1714, 27). The Jesuit Dictionnaire de Trévoux (first edition, 1704) does not insult matter or materialism for what it does to "perfections" such as intelligence or autonomous existence, but opts for an equally successful strategy of discrediting it -here in dictionary entry form, in the entry 'Matériel':

Material also means massive, gross. ... These walls, these foundations are too material. This watch is not subtle, it is too material. One also says of a witless man, or one who is too fond of the pleasures of the senses, that he is quite material, he has a thick and material physiognomy (Dictionnaire de Trévoux 1704, II, n.p.).

The Enlightenment anti-materialist writer Denesle (no first name known) wrote that "matter was the most vile of all beings" 18 ; as late as 1873, Doctor Desgrange, as I mentioned earlier, called materialism "the most fearsome enemy of society today." Sometimes this pathos of hatred for matter can, surprisingly, alternate within the same author with a passion for its vitality: Alexander Pope, for instance, exclaimedquite conventionally -that "There's nought in simple Matter to delight / 'Tis the fair Workmanship that takes the Sight," so that "Where Mind is not, there Horror needs must be /For Matter formless, is Deformity," but also, closer to the 'vital' materialism discussed in Chapter 5, insists on matter as inherently alive: "All matter quick, and bursting into birth." 19 There would be more to say about this sometimes accidental, sometimes deliberate slippage between the hatred for matter and the hatred for the thinkers who 'defend' it, but this is not the place. Indeed, in a remarkable display of continuity despite highly diverse intellectual, theological and political contexts, this contempt for 'crude materialism' runs at least as far as the twentieth century, via Hegel, Engels and Sartre. Moreover, it crosses between a 'metaphysical' form of contempt (as in Cudworth, Nicole or the Dictionnaire de Trévoux cited above), a more historicized form, which becomes canonical for a certain brand of Marxism, as presented notably by Engels in the late nineteenth century, and which I shall have more to say about in Chapters 4 and 5, and even a more scholarly form, which we shall encounter again, as part of my aim will to be rebut it: this critique insists that Enlightenment materialism was necessarily a "mechanistic materialism." Here is Engels' canonical statement, which is well-known and cited in papers including mine, but it deserves a place in any attempt to give an introduction to problems in the history of materialism.

The materialism of the past century was predominantly mechanistic, because at that time ... only the science of mechanics ... had reached any sort of completion. ... For the materialists of the eighteenth century, man was a machine. This exclusive application of the standards of mechanics to processes of a chemical and organic nature -in which the laws of mechanics are also valid, but are pushed into the background by other, higher lawsconstitutes the specific (and at that time, inevitable) limitation of classical French materialism 20 18 Denesle (1754), I, 33n. 19 Respectively, Pope (1735), 345; Pope (1958), Epistle I, section VIII, 44. 20 Engels (1888), in Marx & Engels (1982), 278 (translation mine); in English in Marx & Engels (1959), 211. What I called the 'scholarly' form of this rather overdetermined presentation can be found, e.g., in a description from a study of physical concepts in the Encyclopédie, from the 1950s: the strongest, most pronounced characteristic of the metaphysics we find in the materialism of the 'encyclopédistes', is the reduction of all forms of the motion of matter to mechanical motion, and of all changes in the universe to the merely 'local changes' of a permanently self-identical and unchangeable matter. It is a mechanistic materialism. 21 It is a flagrant mistake to describe eighteenth-century French materialism as 'mechanistic', for many reasons that I shall not discuss here (for some elements see Chapter 4), ranging from the specifically organic or organismic focus of works like L'Homme-Machine to the quite real obsession with the chemistry of matter in such authors. But the aspect I wish to highlight here does not concern matter theory. Rather, it is the way such analyses frequently appeal to a somewhat unargued-for concept of interiority or selfhood which they oppose to this world of mechanistic matter, as when Suzanne Necker asserts in vulgarized Cartesian language that "half of a self is a contradictory absurdity, just like a parcel of matter that cannot be divided is also a contradiction…" (Necker 1798, III, 88).

I observe that this contempt (which here presents itself as merely a constat of historical limitations) crosses between metaphysics, history of philosophy and scholarship, also because the more strongly normative language we encountered in the seventeenth-century texts cited above somehow returns also in Marxist humanist form, e.g. in Sartre's well-known 1946 essay "Materialism and Revolution." Sartre describes materialism here as "the subjectivity of those who are ashamed of their subjectivity" (Sartre 1990, 99); materialism claims to be all about reason, but within the materialist perspective, reason is "captive, governed from outside, manipulated by blind causal chains" (86). Nature here is "pure externality" (94), purely mechanical (89-90), in sharp contrast with the world of values and action: "a causal chain can lead me to a movement, a behavior but not … to my grasping of my situation as a totality. It cannot … account for revolutionary class-consciousness" (120). In sum, materialism is Taylorism: "materialism, by decomposing man into rigorously defined behaviors like in Taylorism, serves the purposes of the master: it is the master who conceives of the slave as being like a machine" (127-128).

Sartreans and critics of what they perceive as the excesses of materialism might not be delighted to be lumped in with old-fashioned humanists, extending intuitions going back to the Cambridge Platonist defense of the soul, and 21 Vassails (1951), 315, referring to the article "MOUVEMENT." One could extend this study of the inflated vision of a mechanistic materialism to contemporary theoretical debates at the intersection of feminism and shifts in 'theory' in the humanities, sometimes under the heading of 'new materialism'; I give some indications in Wolfe (2015). subsequently human dignity. But the above analyses rely on very much the same intuitions as assertions such as "If everything is matter, I cannot see in the name of what, indeed, we might condemn Stalin's work" (this is Mauriac). 22 What do we need to condemn Stalin, then? This takes us back to the responses to La Mettrie, which reveal that the critique of the 'man-machine' idea (which is one way like another of asserting that everything is material) is really a response to moral danger. Sade will force this into the form : if everything is matter, I can commit any crime.

Nor is this opposition between a world of meaning, value, and subjectivity and a cold, dead world of matter (and/or mechanistically understood matter) restricted to a now-vanished Hegelian Marxist tradition: a prominent recent work in 'enactivist' cognitive science of recent years declares boldly that "Life is not physical in the standard materialist sense of purely external structure and function … [w]e accordingly need an expanded notion of the physical to account for the organism or living being" (Thompson 2007, 238), a point of view reflecting an enduring trend in phenomenology. 23 In sum, materialism is frequently portrayed as some combination of stupidity and wickedness -"dead matter," "mechanical, lifeless matter," "brutish matter" or -which is not the same, as we shall see -as evil itself, as in Hugo's verse which serves as the epigraph here: "evil is matter itself. Dark tree, fatal fruit," versus a varying combination of Life, Value and Freedom.

Faced with this situation, some twentieth-century thinkers sought to introduce materialism into the history of philosophy, from Althusser onwards -and one should not confuse this more sophisticated project24 with the older diktats of 'dialectical materialism' or the more dogmatic attempts to present, e.g. Helvétius or Diderot as heroes of a kind of class struggle in philosophy avant la lettre (a classic instance of which is Plekhanov 1934). Or one can seek to historicize the practice of the history of philosophy itself, in order to detect its Kantian (and otherwise idealistic) leanings. 25 Here my aim is strictly to call attention, in this combined historico-philosophical account, of several distinctive features of materialism, both in a comparative manner (when Diderot and J.J.C. Smart seek to explain mental 22 Mauriac, note of March 1953, in Mauriac (1967), 433. Thanks to Lucian Petrescu for help with this reference. 23 One thinks also of Husserl's war against positivism, his endless plans to refound each science on an eidetic basis as a science of essences, his rants against laboratories and "experimental fanatics" (Husserl 1910-1911, 304) or the "scientistic fanaticism" of our time (338). Whether nor not phenomenology can be naturalized (see Varela, Petitot and Roy eds. 1999), it will not be soluble in materialism, as Thompson himself indicates. This did not prevent such major figures of twentiethcentury materialism from searching for 'naturalized' equivalents of intentionality, e.g. David Armstrong (Armstrong 1968/1993, 57).

processes by appealing to the brain, what do they share and what is dissimilar in their arguments?) and in a specific context (what can the materialist say about phantom limb syndrome?). This is what I meant above by the need for a typology of forms of materialism.

Forms of materialism

Whether it bases itself on a critique of concepts of divinity or an internal reform of theology, on physics, on biology or neuroscience (this is not an exhaustive list!), materialism will hold (i) that everything that exists is material, or the product of interaction between or relations between material entities; a second form of materialism (ii) will focus on relations between mind and brain, although (i) and (ii) are not always separate: from responses to Locke on thinking matter to debates on animal minds in the wake of Bayle's article "Rorarius," concerns about the nature of matter and the nature of the mind could fuel one another, as in the chapter title in the free-thinker Boyer d'Argens' 1737 La philosophie du bon sens: "That the Animal Soul is a Proof that Matter can acquire the Faculty of Thought" (Boyer d'Argens 1737, ch. XIV) . Indeed, the Abbé Pluquet, in his eighteenth-century catalogue of heresies, explains that the thinkers he calls "Materialists or Materials" (Matérialistes ou Matériels, a terminology he attributes to Tertullian!) believe "that the soul is born of matter (sortait du sein de la matière)." 26 The entry on "Materialists (Atheists)" in the revolutionary-era Encyclopédie méthodique distinguished between variants of (i) and (ii), but observed that they are often collapsed: "materialists argue either that man's soul is matter, or that matter is eternal and is God; or that God is just a universal soul distributed throughout matter which moves and arranges it, either to produce beings or to create the various arrangements we see throughout the universe" (Naigeon (ed.) 1794, III, 208).

Claim (i) often took the form of a 'cosmological' thesis -i.e., concerning the constitution of the universe as a whole, as in d'Holbach's affirmation, "the universe, this vast sum of all that exists, offers us everywhere just matter and motion," in the first section of his Système de la nature (d'Holbach 1770, I, ch. I, 44). The cosmological thesis was initially framed as an attribution of basic properties such as motion to matter. Thus, in the early years of the eighteenth century, the Irish deist and freethinker John Toland rejected -perhaps the first to do so -the strong distinction between matter and motion: "Matter is but Motion under a certain Consideration" (Toland 1704, C 4). The fifth of his Letters to Serena (ibid., 163f.) is explicitly entitled Motion essential to Matter, and in it Toland states that "All the Matter in Nature, every Part and Parcel of it, has bin ever in motion, and can never be otherwise" (167), and "there's but one sort of Matter in the Universe" (174). In addition -as La Mettrie and Diderot emphasized more dramatically -matter is not just in some sort of 'intestine' motion (Toland speaks later on of its "autokinesy"), it is also fundamentally, inherently active: "Activity ought to enter into the Definition of Matter, it ought likewise to express the Essence thereof" (165), "action is essential to Matter" (160). Contrary to the common accusation that materialists reduce the world, life and mind to a heap of dead, passive matter, Toland is explicit that "Matter neither ever was nor ever can be a sluggish, dead and inactive Lump, or in a state of absolute repose" (C 3); "I deny that Matter is or ever was an inactive dead Lump in absolute Repose, a lazy and unwieldy thing" (159).

However, it is not as if materialism progresses by simply adding further and further properties to Galilean or Cartesian extension like layers in a millefeuille. Indeed, active matter, or thinking, sensing, living matter was a consequence of criticisms of the Cartesian/Malebranchian notion of inert matter and theory of mechanism that went with it. As Diderot put it, reacting to the classic mechanist metaphor of the watch or clock in his unfinished Elements of Physiology (written during the later 1760s and 1770s), "What a difference there is, between a sensing, living watch and a golden, iron, silver or copper watch!" (Diderot 1975-, XVII, 335).

The key property of living matter was organic sensitivity. Diderot sometimes suggested that "sensitivity or touch is common to all beings," or even that sensitivity was a "general property of matter" (308).

In this context, matter was not a metaphysical extensa to be assumed in theory, but instead open to experimental investigation into the particular properties of distinct types of living matter -the plasticity of the cerebellum or the regenerative properties of Trembley's polyp or, frequently appealed to by medical materialists including Mandeville (1711/1730) and La Mettrie, the particular illnesses of patients and their relation to individual constitutions -all of which served as evidence of the sorts of properties and powers possessed by matter. A virtue of these theories is that they drew on working experimental concepts and situated their arguments within experimental contexts, not solely within a theoretical account of how exemplary science works. Diderot, whose matter theory centered on epigenetic, living, sensing, self-transforming matter, stated this point as a chemically motivated critique of mathematical abstraction, in his 1770 Principes philosophiques sur la matière et le mouvement:

You can practice geometry and metaphysics as much as you like; but I, who am a physicist and a chemist, who takes bodies in nature and not in my mind, I see them as existing, various, bearing properties and actions, as agitated in the universe as they are in the laboratory where if a spark is in the proximity of three combined molecules of saltpeter, carbon and sulfur, a necessary explosion will ensue (Diderot 1975-, XVII, 34).

More broadly, he opposed the novelty and conceptual significance of the life sciences to what he (incorrectly) judged to be the historical stagnation of mathematics, including as in his Pensées sur l'interprétation de la nature (discussed in Chapter 4). 27 What is notable in this attitude is the effort to conceptualize a new ontology for the emerging life sciences as part and parcel of the reduction. This was very different from both the mechanistic models of Life and the 'animist' appeals to the soul as an explanatory or even genuine ontological principle (as in Georg-Ernest Stahl) in the late seventeenth and early eighteenth centuries, which either failed to account for specifically living, goal-directed features of organisms, or accounted for them in supernaturalistic terms.

There are several ways to describe this increasing complexity in matter theory. One reading emphasizes the shift from substance dualism to a theory in which matter takes on some of the explanatory role that 'soul' had previously (Vartanian 1982, Wright 1991). The entire story that this book seeks to tell, could be retold in terms of shifting concepts of the soul -its mortality, its corporeality, the possibility of its naturalization, tensions between Aristotelian and Epicurean models, the appearance of animal spirits on the scene, and so on. That will have to wait for another book, but I should like to make one observation, concerning the status of a 'hegemonic' entity corresponding to a centre of our personhood (self, subject, person, soul …, which Cudworth describes in his writings on morality as the to hegemonikon).

The fear that 'cosmic' materialism would lead to reductionist approaches to the mind (and thus the self, the person …) was central to early modern physicotheology and beyond, from the Boyle Lectures which Robert Boyle had endowed in his will (the title of Richard Bentley's second Boyle Lecture for 1692 is quite explicit: Matter and Motion Cannot Think) and John Ray's Wisdom of God in the 1690s to Bernard Nieuwentijt's The religious philosopher, or, The right use of contemplating the works of the Creator (1715; first English translation 1719), and William Paley's Natural Theology of 1802. This is why Isaac Newton was so adamant that gravity should not be understood as a property of matter:

It is inconceivable that inanimate brute Matter should, without the Mediation of something else, which is not material, operate upon, and affect other Matter without mutual Contact, as it must be, if Gravitation in the Sense of Epicurus, be essential and inherent in it. 28 27 Diderot (1753), § 4 in Diderot (1975-), IX, 30-31. 28 Newton to Bentley, February 25 th , 1693, letter III in Newton (1958), 302. He adds that "Gravity must be caused by an Agent acting constantly according to certain Laws; but whether this Agent be material or immaterial, I have left to the Consideration of my Readers." For a stimulating analysis of the metaphysics involved in Newton's discussion of gravity, see Schliesser (2011).

Thus he wrote to his ideological protégé the divine Richard Bentley that he "desired you would not ascribe innate Gravity to me": That Gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance thro' a Vacuum, without the Mediation of any thing else ... is to me so great an Absurdity, that I believe no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it (ibid.).

But quickly, the issue shifted from the attribution of motion or gravity to matter, to a yet more grievous attribution, shifting imperceptibly into materialist claim (ii), concerning thought. No one saw or expressed this more clearly than Fontenelle, the longtime Secretary of the Académie des Sciences, in his 1752 Théorie des tourbillons cartésiens (Theory of Cartesian Vortices), late in his long career and life. Fontenelle reflected critically on what he saw as the arbitrariness of Newtonian attraction, and added that attributing attraction to matter in terms of God's will ("wholly arbitrary") was a small step away from granting it the power to think: "If we grant this arbitrariness, we destroy any philosophical proof of the spirituality of the soul. God could just as well have granted thought to matter, as attraction" (Fontenelle 1752, § III, in Fontenelle 1829, 71, emphasis mine).

The most celebrated discussion of matter and thought in the early eighteenthcentury was the pamphlet exchange known as the Clarke-Collins correspondence. Briefly, Samuel Clarke had sought to prove in his Letter to Dodwell that consciousness cannot be the property of a system of material parts. According to Clarke, a material thing was divisible. An individual consciousness must be indivisible ("indiscerptible") and hence immaterial and immortal. Anthony Collins responded that a divisible system of matter taken as a whole may have a quality not equal to the sum of the qualities of the separate parts (Clarke 1738, III, 769): a rose is a divisible thing, yet its smell cannot be reduced to the sum of the powers of the parts29 -and thinking might be like this, too. While consciousness, thought, or the rose's smell may not be the properties of individual parts of these respective systems, they are properties of the whole.

For Clarke, if matter were conscious, then every particle of matter would have a distinct indivisible consciousness. A system of matter made up of such particles, could not have an individual consciousness, but would have to be at best a cluster or bundle of consciousnesses. Collins replied that Clarke just assumed that thinking was an individual power. For Collins, thinking was a mode of matter: "human consciousness or thinking is a mode of some generical power in matter . . . it has generation, succession and corruption like all other modes of matter" (in Clarke 1738, III, 807). Collins further insisted on a connection between the empiricist account of the origin of ideas in sensation, and the materialist account of how "ideas of sensation" originate in the process of "bodies operat[ing] upon us" (Clarke 1738, III, 863). Here, Collins added the other characteristic (and at the time quite new) materialist claim that thinking is a kind of motion in the brain (866).

If Cudworth and Bentley had feared the idea that matter could think, by the mid-eighteenth century the fear is primarily directed towards living, self-subsisting, self-organizing matter. Kant, in the 1786 Metaphysische Anfangsgründe der Naturwissenschaft (Metaphysical Foundations of Natural Science) and subsequently, argued at length against this view, which he called hylozoism, and sought to eliminate its possibility from our concepts of nature; he viewed it as "the death of all philosophy of nature."30 Consider also Goethe's reaction to d'Holbach's System of Nature:

I recollect particularly the Système de la Nature, which we laid hold of with curiosity. We could not understand how such a book could be dangerous. It seemed to us so gloomy, so Cimmerian, so deathly (so grau, so cimmerisch, so totenhaft), that we could hardly endure its presence, and shuddered before it as before an apparition. . . . But how vacant and desolate our souls grew in this sad atheistic twilight (tristen atheistischen Halbnacht)! -in which the earth vanished with all its forms of beauty, and the heaven with all its stars. Only matter remained, moved from eternity hither and thither, right and left, with no other power, on all sides producing the endless phenomena of existence (Dichtung Und Wahrheit, XI, in Goethe 1887-1919, 69-70).

Goethe is describing a reaction that was also common in Coleridge and other authors who were deeply invested in the philosophy of nature: that materialism was a dehumanizing form of reduction that stripped Nature of life and meaning; he missed the stress on living, self-organized matter and the criticisms of mechanism in French materialism (and differently, in Collins and others), which were meant to supersede the dichotomy between inert matter and active thought.

Engels, Ruyer and already Goethe articulate a powerful (at least at the level of intuition) critique of materialism: that it reduces the world of life to the world of dead matter; a sophisticated version of this critique would allow for the pertinence of certain sciences (from the search for the Urpflanz to an Aristotelian biology!) over and against other, illegitimate explanations, e.g. in terms of physics and/or mechanics. They seem blind to the presence, in Lucretius, Gassendi, La Mettrie and Diderot, and in a very different way in authors such as Dewey, Quine or Dennett, of either a specifically vital sense of matter, and/or a naturalistic openness to the fact that the description of the natural world is not, in the end, going to be a matter of pure physics. One could add, as I discuss further in the Conclusion, that they don't seem to do justice to the very active forms of anti-foundationalism in work in the theories they attack (or perhaps it is their desire for ultimate foundations which motivates some of their criticism!). As the anonymous 1738 Dissertation on the Formation of the World put it, "isn't it in vain that we seek to define the original form of matter?" (ch. II, in Stancati 2001, 96). A variety of texts, from Meslier's Mémoire (written in the 1720s but unknown until a generation later) to the Encyclopédie article "Matière," speak out against "first principles." The materialist will precisely reject the foundational character of mind, as in Cudworth's formulation that mind is "senior to the world" (Cudworth 1678, I, ch. IV, 729, 736-737; ch. V, 853).

* * * I have suggested some ways of sorting out, or through the morass of the history of materialism without thereby opting either for an overly partisan defence of a kind of eternal truth (scientific? metaphysical? atheistic?) of this doctrine, or for an antiquarian herbier of endless possible cases, sources, rhetorical forms and instantiations. Unfortunately, many episodes were left out, as I discuss again in the Conclusion -and even if this study were double the length, it would still not be about dialectical materialism or materiality. But I hope that a short historical and philosophical overview which combines Aristotle contra materialism, the problem of phantom limbs, evolution (however briefly, in Chapter 3), brains, machines and 'hylophobia' will serve some purpose. Scholars of early modern philosophy will wish for chapters on Hobbes, Spinoza, perhaps Cavendish,31 or Leibniz's critique of materialism/Epicureanism ; scholars of German Idealism will regret the absence of the problem of hylozoism, determinism, abstract matter, and so forth. This is, of course, a partial introduction, 'heavy' on the Enlightenment and its posterity.

Methodologically, the trajectory I have sought to describe indicates that there is a history, not just of materialist philosophies (e.g. Lucretius, Hobbes, Diderot, Priestley) but also of the presence of materialist 'components' or articulated wholes within philosophical systems that are not themselves materialistic: Descartes as appropriated by Regius, Malebranche as appropriated by L'âme matérielle, Spinoza and Bayle as appropriated by several generations of radical eighteenth-century thinkers -not to mention 'scientific' texts like those of Willis, Whytt or Haller, whose authors go out of their way to reject materialism, but who instantly become evidence for that view. (I don't take a position in this work on whether Spinoza was a materialist or not. 32 ) This is not just a theoretical game (whether it is described as collage, appropriation or in more systematic terms), for as noted with respect to Malebranche's psychophysiology, sometimes the texts which were criticising a view could serve as the best evidential resource for an author who was not part of an inner sanctum of experimental natural philosophy. 33

More describes the better specimen of materialist, the character Hylobares, as "a young, witty, and well moralised Materialist," in a passage well-known to historians of materialism(More 1668, 5-6). For more on the history of the term see[START_REF] Bloch | Le matérialisme[END_REF][START_REF] Bloch | Matière à histoires[END_REF],Benítez (1998), 355 (signaling an earlier usage in French, in Friedrich Spanheim's 1676 L'impie convaincu) and on the German context,[START_REF] Rumore | Materia cogitans: l'Aufklärung di fronte al materialismo[END_REF] and[START_REF] Wunderlich | Materialism in late Enlightenment Germany. A neglected tradition reconsidered[END_REF].

Observations sur Hemsterhuis, in Diderot (1975-), XXIV, 251.

Darwin, Notebook C, in Darwin (1996), 71; but this has nothing to do with the metaphysics of matter; Michael Ruse's statement that Darwinism is "the apotheosis of a materialistic theory"(Ruse 2000, 77), lacking conceptual or historical finesse, does not help us understand Darwin -or materialism -any better.

Pensées sur l'interprétation de la nature § li, inDiderot (1975-), IX, 84.

Vogt 1875, 347-348 (Vogt's 13 th Inaugural Lecture at the University of Giessen in 1845). For a similar formulation to Vogt's (thought=bile) see[START_REF] Cabanis | Rapports du physique et du moral[END_REF], 151 (the lectures forming the basis of the latter publication were given in the late 1790s).

Desgrange (1873), 15.

Quine (1981), 176.

Thus two earlier studies of the topic,[START_REF] Charbonnat | Histoire des philosophies matérialistes[END_REF] and[START_REF] Vitzthum | Materialism: An Affirmative History[END_REF], tend to overly favor the 'truth' of materialism, perhaps inadvertently mirroring the only other history of materialism, Lange's, which was intended as a careful, thorough refutation.

Cf. " L'histoire de la philosophie a toujours été l'agent de pouvoir dans la philosophie, et même dans la pensée"(Deleuze 1979, 19-21).

I am not concerned here with (a) the difference between Margaret Jacob's and Jonathan Israel's concepts of 'radical Enlightenment' or (b) the internal conceptual success and consistency of the latter (heavily debated e.g. inSecrétan et al., eds., 2007). As regards the role of Spinozism, I take it as more of a construct than a real relation to Spinoza (following[START_REF] Citton | L'Envers de la liberté. L'invention d'un imaginaire spinoziste dans la France des Lumière[END_REF], that is, a conceptual construct which need not involve first-hand acquaintance with the writings of Spinoza. For my approach to Spinozism in the Radical Enlightenment seeWolfe (2007[START_REF] Wolfe | Epigenesis as Spinozism in Diderot's biological project[END_REF].

Hegel (1989), 155;[START_REF] Hegel | Einleitung in der Geschichte der Philosophie[END_REF], 124 (where he adds, "Every philosophy is an idealism; there has never been any philosophy other than the self-knowledge of Spirit").

Schopenhauer, Die Welt als Wille und Vorstellung, II, 1 ("Zur idealistischen Grundansicht"), inSchopenhauer (1977), 11, 27.

Jean-Claude Bourdin's reading of Hegel on materialism, but also of the challenging presence of what we might call 'Radical Enlightenment' materialism within Hegel's historical presentation of philosophy, is a noteworthy attempt in this regard[START_REF] Bourdin | Hegel et les matérialistes français[END_REF]).

See Haakonssen 2006 and the papers collected in Laerke, Schliesser Smith eds. (2013), particularly those by Smith and Vermeir.

[START_REF] Pluquet | Mémoires pour servir à l'histoire des égarements de l'esprit humain par rapport à la religion chrétienne: ou Dictionnaire des hérésies, des erreurs et des schismes[END_REF][START_REF] Pluquet | Mémoires pour servir à l'histoire des égarements de l'esprit humain par rapport à la religion chrétienne: ou Dictionnaire des hérésies, des erreurs et des schismes[END_REF]), II, s.v. "Matérialistes," 300. As late as the Encyclopédie article "Matérialistes," these are presented (with an acknowledgment that it is an old definition) as those thinkers "who claim that man's soul is matter"[START_REF] Diderot | Matérialistes (Théol.)[END_REF](Diderot /1966, X, 188b), X, 188b), thus combining theses (1) and (2).

Collins, in Clarke 1738, III, 770 (this controverts Clarke, Letter to Dodwell, in Clarke 1738, III, 759).

Metaphysische Anfangsgründe,III.3, AA 4, 544. Prior to the first Critique, in a 1773 letter to Marcus Herz, Kant already insisted that in his anthropology lectures he would avoid "eternally futile inquiries as to the manner in which bodily organs are connected with thought" (AA 10, 145), as opposed to what he would call pragmatic anthropology.

See the work of Stewart Duncan, e.g.[START_REF] Duncan | Debating Materialism: Cavendish, Hobbes, and More[END_REF].

Thus Diderot, writing on physiology, could cite as evidence the Edinburgh physician Robert Whytt's 'neuropneumatological' assertion that "the soul is equally present in the extremities of the nerves through the whole body as in the brain"(Whytt 1768, 287) even though Whytt had specified this was not tantamount to materialism, since these functions of the soul were themselves dependent on what he calls an "active sentient principle," which brought together sensibility and life, and could not be a property of matter itself (ibid., 128).


George A Hagedorn

Alain Joye

Semiclassical Dynamics with Exponentially Small Error Estimates

Keywords: Semiclassical methods, quantum dynamics, exponential asymptotics

We construct approximate solutions to the time-dependent Schrödinger equation

for small values of h. If V satisfies appropriate analyticity and growth hypotheses and |t| ≤ T , these solutions agree with exact solutions up to errors whose norms are bounded by

for some C and γ > 0. Under more restrictive hypotheses, we prove that for sufficiently small T ′ , |t| ≤ T ′ | log(h)| implies the norms of the errors are bounded by

for some C ′ , γ ′ > 0, and σ > 0.

MSC: 81Q20

Introduction

In this paper, we construct exponentially accurate semiclassical approximations ψ(x, t, h) to certain normalized exact solutions Ψ(x, t, h) of the d-dimensional time-dependent Schrödinger equation

i h ∂Ψ ∂t = - h2 2 ∆ Ψ + V Ψ. (1.1)
More precisely, our main result is that for |t| ≤ T and small values of h, these approximations satisfy error estimates of the form

ψ(x, t, h) -Ψ(x, t, h) L 2 (I R d ) ≤ C exp { -γ/h } , (1.2)
where γ > 0.

Our construction of ψ(x, t, h) is technically complicated, but quite explicit. It uses a particular collection of semiclassical wave packets { ϕ j (A, B, h, a, η, • ) } that are defined in [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF], [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF], and the next section. Here A and B are d × d complex matrices that satisfy certain conditions. The quantities a and η are elements of IR d . For fixed A, B, h, a, and η, { ϕ j (A, B, h, a, η, • ) } is an orthonormal basis of L 2 (IR d ) as j ranges over all d-dimensional multi-indices.

The function ϕ j (A, B, h, a, η, • ) is concentrated near position a, and its Fourier transform is concentrated near momentum η. Its position and momentum uncertainties are proportional to √ h. The position uncertainty is determined by the matrix |A| = √ A A * , and the momentum uncertainty is determined by |B| = √ B B * . We construct ψ(x, t, h) by applying the idea of "optimal truncation" of an asymptotic expansion. For initial conditions of the form Ψ(x, 0, h) = |j|≤J c j ϕ j (A(0), B(0), h, a(0), η(0), x), (1.3) with |j|≤J |c j | 2 = 1, there exist ( [START_REF] Hagedorn | Semiclassical Quantum Mechanics III: The Large Order Asymptotics and More General States[END_REF], [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF]) approximate solutions ψ l (x, t, h) = e iS(t)/h |j|≤ J(l) c j (l, t, h) ϕ j (A(t), B(t), h, a(t), η(t), x), (1.4) that satisfy sup t∈[-T,T ]

ψ l (x, t, h) -Ψ(x, t, h) L 2 (I R d ) ≤ C(l) h l/2 (1.5)
for some constant C(l). Here J(l) = J + 3l -3, and A(t), B(t), a(t), η(t), and S(t) are solutions to the classical equations of motion ȧ(t) = η(t), η(t) = -∇V (a(t)), Ȧ(t) = i B(t), (1.6) Ḃ(t) = i V (2) (a(t)) A(t),

Ṡ(t) = η(t) 2 2 -V (a(t)),
where V (2) denotes the Hessian matrix for V , and the initial conditions A(0), B(0), a(0), η(0), and S(0) = 0 satisfy A t (0) B(0) -B t (0) A(0) = 0, (1.7) A * (0) B(0) + B * (0) A(0) = 2 I.

(1.8)

The c j (l, t, h) satisfy a linear system of ordinary differential equations that we describe in the next section.

We carefully estimate the l-dependence of C(l) in (1.5). Then, for each h, we choose l(h) to minimize the error C(l(h)) hl(h)/2 over all choices of l. It turns out that l(h) behaves like a constant times 1/h. We define ψ(x, t, h) = ψ l(h) (x, t, h) and prove that (1.2) is satisfied.

For t in a fixed compact interval, the precise statement of our results is the following:

Theorem 1.1 Suppose V is a real-valued function on IR d that is bounded below and has an analytic continuation to the set

D = { z ∈ I C d : |Im z j | < δ, j = 1, 2, • • • , d }.
(1.9)

Suppose further that there exist M > 0 and τ > 0, such that

| V (z) | ≤ M exp(τ |z| 2 ), for all z ∈ D, (1.10)
where |z| 2 = d j=1 |z j | 2 . Suppose initial conditions A(0), B(0), a(0), η(0), S(0) = 0, and c j for |j| ≤ J are specified that satisfy (1.7) and |j|≤J |c j | 2 = 1. Then for any T > 0, there exist C and γ > 0, such that the difference between the semiclassical approximation ψ(x, t, h) and the exact solution Ψ(x, t, h) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies ψ(x, t, h) -Ψ(x, t, h) L 2 (I R d ) ≤ C exp { -γ/h } , whenever |t| ≤ T .

Remarks 1. Theorem 1.1 can be generalized to allow time-dependent potentials. For example, suppose a potential V (x, t) depends smoothly on t, is bounded below, and satisfies

1 m! |D m V (x, t))| ≤ M exp(τ |x| 2 ) δ |m| ,
for |t| ≤ T for all multi-indices m. Suppose further that a classical solution (a(t), η(t)) to Newton's equations with potential V (x, t) is bounded for |t| ≤ T . Then the conclusion to Theorem 1.1 holds.

2.

Our results can also be extended to obtain weaker error estimates of the form C exp { -γ/h σ } for some σ ∈ (0, 1), when the potential belongs to a Gevrey class.

3. Theorem 1.1 is optimal in the sense that the conclusion fails if the hypotheses are relaxed slightly. For example, consider the one-dimensional potential V (x) = exp(-1/x u ), for x > 0, and V (x) = 0 for x ≤ 0, where u > 0. It is shown in [START_REF] Jung | Adiabatic Invariance and the Regularity of Perturbations[END_REF] that this potential belongs to the Gevrey class of order 1 + 1/u. For initial conditions a(0) = 0, η(0) = 0, A(0) = B(0) = 1, S(0) = 0, and c j (0) = δ j,0 , our approximation yields ϕ 0 (1 + it, 1, h, 0, 0), for all times. This function is very simple, and we can write the error term explicitly. By steepest descent analysis, we can show that there exist δ > 0 and Σ 1 > Σ 2 > 0, such that t ∈ (0, hδ ) implies

exp(-Σ 1 /h u/(1+u) ) ≤ e -itH(h)/h ϕ 0 (1, 1, h, 0, 0, •) -ϕ 0 (1 + it, 1, h, 0, 0, •) ≤ exp(-Σ 2 /h u/(1+u) ).
Note also that if we choose a(0) = -a < 0 and η(0) = η > 0, it is easy to check that the error term is O(exp(-γ/h)), for each t < a/η.

4. For all practical purposes, we can replace the c j 's by the corresponding Dyson expansion up to order l(h) (see (2.26)-(2.28)) without spoiling our exponential estimate. The normalization of the approximation, however, will be lost.

5. One cannot expect to get better agreement with the exact solution when constructing an approximation using wave packets associated with a single classical path. Such wave packets do not describe tunneling, which is an exponentially small effect in h.

Theorem 1.1 can also be generalized to allow time intervals that grow like | log(h)| as h tends to zero. However, we obtain a somewhat weaker conclusion. Our precise results are summarized by the following theorem.

Theorem 1.2 Suppose V is bounded below and analytic in

D = { z ∈ I C d : |Im z j | < δ, j = 1, 2, • • • , d }. (1.11)
Suppose further that there exist M > 0 and τ > 0, such that

| V (z) | ≤ M exp(τ |z|),
for all z ∈ D.

(1.12)

Suppose initial conditions A(0), B(0), a(0), η(0), S(0) = 0, and c j for |j| ≤ J are specified that satisfy (1.7) and |j|≤J |c j | 2 = 1, and further assume there exist N > 0 andλ > 0, such that

A(t) ≤ N exp(λ|t|). (1.13)
Then for sufficiently small T ′ > 0, there exist C ′ , γ ′ > 0, and σ > 0, such that the difference between the semiclassical approximation ψ(x, t, h) and the exact solution Ψ(x, t, h) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies

ψ(x, t, h) -Ψ(x, t, h) L 2 (I R d ) ≤ C ′ exp { -γ ′ /h σ } , whenever |t| ≤ T ′ | log(h)|.
Remark Standard existence and uniqueness theorems for systems of ODE's show that condition (1.13) is satisfied if the norm of the Hessian V (2) (a(t)) is uniformly bounded. That is the case if V is the sum of a quadratic polynomial plus an analytic function bounded on D. It is also the case if E denotes the energy of the considered trajectory and

V (2) (x)
is bounded on the connected component of the classically allowed region D E = {x ∈ IR d : V (x) ≤ E} that contains a(t). This is satisfied for all confining potentials. It is easily deduced from the proof that if τ can be chosen arbitrarily small, then we can take T ′ = ( 1 6 -ǫ) 1 λ with ǫ arbitrarily small. This yields exponential control over the same time intervals as in [START_REF] Combescure | Semiclassical Spreading of Quantum Wave Packets and Applications near Unstable Fixed Points of the Classical Flow[END_REF].

The propagation of coherent states is also considered by Combescure and Robert in [START_REF] Combescure | Semiclassical Spreading of Quantum Wave Packets and Applications near Unstable Fixed Points of the Classical Flow[END_REF], see also [START_REF] Robert | Semi-Classical Approximation in Quantum Mechanics[END_REF], using an approximation given by a linear combination of squeezed states. (The squeezed states coincide with our semiclassical wave packets, although the notation is quite different.) Their emphasis is on the long time behavior of this approximation. The bound on the error term is of the form C l (t)h l/2 , with explicit control of the time-dependence of C l (t) in terms of classical quantities. The l behavior is however not investigated.

Results of a flavor similar to ours can be found in the work of Yajima [START_REF] Yajima | Gevrey Frequency Set and Semi-classical Behavior of Wave Packets[END_REF]. They are obtained by means of the pseudo-differential techniques developed in the analytic context by Sjöstrand in [START_REF] Sjöstrand | Singularités analytiques microlocales[END_REF]. These results concern the propagation of wave packets of the form ϕ(x) = e iS(x)/h f (x), where S is analytic and f belongs to the set of compactly supported Gevrey functions of order s > 1. Assuming the potential V is analytic, Yajima constructs approximations to the actual evolution of such wave packets that are valid up to an error, whose L 2 (IR d ) norm is of order e -γ/h 1/(2s-1) , with γ > 0 (see Theorems 1.2, 1.2 and Lemma 2.5 in [START_REF] Yajima | Gevrey Frequency Set and Semi-classical Behavior of Wave Packets[END_REF]). However, it should be possible to make use of the theory [START_REF] Sjöstrand | Singularités analytiques microlocales[END_REF] to recover our results.

Similar issues have been dealt with by Bambusi, Graffi and Paul in [START_REF] Bambusi | Long Time Semiclassical Approximation of Quantum Flows: A Proof of Ehrenfest Time[END_REF]. They focus on the validity for large times of the semiclassical approximation of the Heisenberg evolution of a smooth observable, under analyticity assumptions on the hamiltonian. They prove that the semiclassical approximation remains useful for times up to order | ln(h)|, the Ehrensfest time scale. However, the Hamiltonians they can accomodate consist more or less of analytic perturbations of the harmonic oscillator that decay as x and p tend to infinity.

The paper is organized as follows: In the Section 2, we prove Theorem 1.1 under the assumption that two types of error terms satisfy certain bounds. We prove the two required bounds in Sections 3 and 4. In Section 5 we describe the proof of Theorem 1.2.

Proof of Theorem 1.1

We begin this section by presenting the definition of the semiclassical wave packets ϕ j (A, B, h, a, η, x) that is given in [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF]. A more explicit, but more complicated definition is given in [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF]. Since [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF] provides a detailed discussion of these wave packets, we do not prove all their properties here.

We adopt the standard multi-index notation. A multi-index j = (j 1 , j 2 , . . . , j d ) is a d-tuple of non-negative integers. We define

|j| = d k=1 j k , j! = (j 1 !)(j 2 !) • • • (j d !), x j = x j 1 1 x j 2 2 • • • x j d d , and D j = ∂ |j| (∂x 1 ) j 1 (∂x 2 ) j 2 • • • (∂x d ) j d .
Throughout the paper we assume a ∈ IR d , η ∈ IR d and h > 0. We also assume that A and B are d × d complex invertible matrices that satisfy

A t B -B t A = 0, ( 2.1)
A * B + B * A = 2 I. (2.2)
These conditions guarantee that both the real and imaginary parts of BA -1 are symmetric. Furthermore, Re BA -1 is strictly positive definite, and ( Re BA -1 ) -1 = A A * . We note that conditions (2.1) and (2.2) are preserved under the dynamics generated by (1.6).

Our definition of ϕ j (A, B, h, a, η, x) is based on the following raising operators that are defined for m = 1, 2, . . . , d.

A m (A, B, h, a, η) * = 1 √ 2h d n=1 B n m (x n -a n ) - d n=1 A n m (-ih ∂ ∂x n -η n ) .
Definition For the multi-index j = 0, we define the normalized complex Gaussian wave packet (modulo the sign of a square root) by

ϕ 0 (A, B, h, a, η, x) = π -d/4 h-d/4 (det(A)) -1/2 × exp -(x -a), B A -1 (x -a) /(2h) + i η, (x -a) /h . (2.3)
Then, for any non-zero multi-index j, we define

ϕ j (A, B, h, a, η, • ) = 1 √ j! ( A 1 (A, B, h, a, η) * ) j 1 ( A 2 (A, B, h, a, η) * ) j 2 • • • × ( A d (A, B, h, a, η) * ) j d ϕ 0 (A , B, h, a, η, • ).
Remarks 1. For A = B = I, h = 1, and a = η = 0, the ϕ j (A, B, h, a, η, • ) are just the standard Harmonic oscillator eigenstates with energies |j| + d/2.

2.

For each A, B, h, a, and η, the set { ϕ j (A, B, h, a, η, • ) } is an orthonormal basis for L 2 (IR d ).

3. The raising operators can also be given by another formula that was omitted from [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF] in the multi-dimensional case. If we set

g(A, B, h, a, x) = exp -(x -a), BA -1 * (x -a) /(2h) -i η, (x -a) /h , then we have ( A m (A, B, h, a, η) * ψ ) (x) = - h 2 1 g(A, B, h, a, x) d n=1 A n m ∂ ∂x n ( g(A, B, h, a, x) ψ(x) ) .
4. In [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF], the state ϕ j (A, B, h, a, η, x) is defined as a normalization factor times

H j (A; h-1/2 |A| -1 (x -a)) ϕ 0 (A, B, h, a, η, x).
Here H j (A; y) is a |j| th order polynomial in y that depends on A only through U A , where A = |A| U A is the polar decomposition of A.

5. By the argument on page 370 of [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF] or by scaling out the |A| and h dependence and using Remark 3 above, one can show that H j (A; y) e -y 2 /2 is an (unnormalized) eigenstate of the usual Harmonic oscillator with energy |j| + d/2.

6.

When the dimension d is 1, the position and momentum uncertainties of the ϕ j (A, B, h, a, η, • ) are (j + 1/2)h |A| and (j + 1/2)h |B|, respectively. In higher dimensions, they are bounded by (|j| + d/2)h A and (|j| + d/2)h B , respectively.

7. When we approximately solve the Schrödinger equation, the choice of the sign of the square root in the definition of ϕ 0 (A, B, h, a, η, • ) is determined by continuity in t after an arbitrary initial choice.

The proof of the theorem depends on the following abstract lemma, whose proof is an easy application of Duhamel's formula (see e.g. [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF]).

Lemma 2.1 Suppose H(h) is a family of self-adjoint operators for h > 0. Suppose ψ(t, h) belongs to the domain of H(h), is continuously differentiable in t, and approximately solves the Schrödinger equation in the sense that

i h ∂ψ ∂t (t, h) = H(h) ψ(t, h) + ξ(t, h), (2.4)
where ξ(t, h) satisfies

ξ(t, h) ≤ µ(t, h). (2.5)
Then, for t > 0,

e -itH(h)/h ψ(0, h) -ψ(t, h) ≤ h-1 t 0 µ(s, h) ds. (2.6)
The analogous statement holds for t < 0.

Because V is smooth and bounded below, there exist global solutions to the first two equations of the system (1.6) for any initial condition. It then follows immediately that the remaining three equations of the system (1.6) have global solutions. Furthermore, it is not difficult ( [START_REF] Hagedorn | Semiclassical Quantum Mechanics III: The Large Order Asymptotics and More General States[END_REF], [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF]) to prove that (2.1) and (2.2) are preserved by the flow.

As mentioned in the introduction, it is proved in [START_REF] Hagedorn | Semiclassical Quantum Mechanics III: The Large Order Asymptotics and More General States[END_REF] and [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF] that initial conditions of the form (1.3) give rise to approximate solutions of the form

ψ l (x, t, h) = e iS(t)/h |j|≤ J(l) c j (l, t, h) ϕ j (A(t), B(t), h, a(t), η(t), x),
with errors whose norms are of order hl/2 . Here J (l) = J + 3l -3, and A(t), B(t), a(t), η(t), and S(t) satisfy (1.6). The coefficients c j (l, t, h) satisfy the linear system

i h ċk (l, t, h) = |j|≤ J(l) K k j (l, t, h) c j (l, t, h), |k| = 0, 1, . . . , J(l), (2.7)
with initial conditions c j (l, 0, h) = c j , for |j| ≤ J in accordance with (1.3) and c j (l, 0, h) = 0 for |j| > J.

To specify the (J + 3l -2) × (J + 3l -2) matrix K(l, t, h) that appears in (2.7), we first decompose the potential as

V (x) = W a (x) + Z a (x) ≡ W a (x) + (V (x) -W a (x)), (2.8)
where W a (x) denotes the second order Taylor approximation (with the obvious abuse of notation)

W a (x) ≡ V (a) + V (1) (a) (x -a) + V (2) (a) (x -a) 2 /2, (2.9)
Next (reverting to multi-index notation), we approximate Z a (x) by its Taylor approximation of order l + 1,

Z [l] a (x) = 3≤|m|≤l+1 (D m V )(a) m! (x -a) m , (2.10)
and define the infinite matrix

K k j (l, t, h) = ϕ k (A(t), B(t), h, a(t), η(t), x), Z [l] a(t) (x) ϕ j (A(t), B(t), h, a(t), η(t), x) . (2.11)
Then, we obtain the matrix K(l, t, h) from K(l, t, h) by restricting the indices to |j| ≤ J(l) and |k| ≤ J(l). The general strategy [START_REF] Hagedorn | Semiclassical Quantum Mechanics III: The Large Order Asymptotics and More General States[END_REF], [START_REF] Hagedorn | Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension[END_REF], [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF] to show that ψ l (x, t, h) is an approximation to the actual solution Ψ(x, t, h) of (1.1) and (1.3) up to order hl/2 is as follows: From [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF], we know that for all multi-indices j, i h ∂ ∂t e iS(t)/h ϕ j (A(t), B(t), h, a(t), η(t), x)

= - h2 2 ∆ + W a(t) (x)
e iS(t)/h ϕ j (A(t), B(t), h, a(t), η(t), x) .

(2.12)

Thus, the ϕ j take into account the kinetic energy and W a(t) (x) parts of the Hamiltonian. Next, we expand the exact solution as

Ψ(x, t, h) = j b j (h, t) e iS(t)/h ϕ j (A(t), B(t), h, a(t), η(t), x),
where the b j (h, t) satisfy an infinite linear system of ordinary differential equations whose matrix is obtained from the Z a(t) (x) term in the Hamiltonian. In that system, we make a first approximation by replacing the function Z a(t) (x) by its Taylor approximation Z

[l] a(t) (x). This yields an infinite linear system whose matrix is K(l, t, h). Its entries are time dependent polynomials in h1/2 of order l -1. We make a second approximation by truncating the infinite system to obtain (2.7) that is satisfied by the c j (l, t, h). The result (1.5) is proved by using Lemma 2.1 to show that the errors generated by the Taylor approximation and the truncation approximation are of order hl/2 .

As described in the introduction, we construct the exponentially accurate approximate solution ψ(x, t, h) = ψ l(h) (x, t, h), by keeping track of the l-dependence of C(l) in (1.5) and then choosing l(h) in order to minimize the error.

In the remainder of this section, we prove Theorem 1.1 under the assumption that the following three technical lemmas are true. We prove the first lemma in Section 3. It estimates errors that arise from our replacement of Z a(t) (x) by Z

[l] a(t) (x). The second and third lemmas are proved in Section 4. They bound certain matrix elements and combinatorial quantities that arise from the truncation approximation discussed above. Lemma 2.2 Suppose V satisfies the hypotheses of Theorem 1.1, |m| = l+2, J(l) = J +3l-3, and

ψ l (x, t, h) = e iS(t)/h |j|≤ J(l) c j (l, t, h) ϕ j (A(t), B(t), h, a(t), η(t), x),
with |j|≤ J(l) |c j (l, t, h)| 2 = 1. Let ζ(x, a(t)) = (a(t) + θ x,a(t) (x -a(t))) ∈ IR d , with θ x,a(t) ∈ (0, 1)
. There exist constants g 0 and g 1 , that depend on d and J only, such that for sufficiently small h,

D m V (ζ(x, a(t))) m! (x -a(t)) m ψ l (•, t, h) ≤ g 0 M exp(4τ (δ 2 d + a(t) 2 )) g 1 h(l + 2) A(t) /δ l+2 .
(2.13) Lemma 2.3 We define the infinite matrix

X m j k (t, h) = ϕ j (A(t), B(t), h, a(t), η(t), •), (x -a(t)) m ϕ k (A(t), B(t), h, a(t), η(t), •) , (2.14)
and then define the finite matrix X m (l, t, h) from X m (t, h) by restricting its indices to |j| ≤ J(l) and |k| ≤ J(l). Then, X m j k (l, s, h) = 0 and X m j k (s, h) = 0 if |j| -|k| > |m| and, for each N > 0, there exists D(N) < ∞, such that

sup |k| ≤ J(l) J(l) + 1 ≤ |j| ≤ J(l) + l + 1 X m 0 (t 0 , h)X m 1 (l, t 1 , h)X m 2 (l, t 2 , h) • • • X mq (l, t q , h) j k ≤ D(N) h J(l) sup t∈{t 0 ,t 1 ,t 2 ,•••tq} A(t) |m 0 |+|m 1 |+|m 2 |•••+|mq| , (2.15)
for any collection m 0 , m 1 , . . . , m q of multi-indices that satisfy |m j |/ J(l) ≤ N.

Lemma 2.4 We define

F p (n, q) = 1 ≤ |m 1 |, |m 2 |, • • • , |mq| ≤ p |m 1 | + |m 2 | + • • • + |mq| = n 1 (2.16)
to be the number of distinct sets { m 1 , m 2 , . . . , m q }, where each

m j is a d-dimensional multi-index with 1 ≤ |m j | ≤ p and |m 1 | + |m 2 | + • • • + |m q | = n.
We note that F p (n, q) is zero unless q ≤ n ≤ qp. Suppose that a function L q (l) satisfies

L q (l) ≤ C q 1 hq q! q(l+1) n=3l-2 (C 2 hl) n/2 F l+1 (n, q), (2.17)
where C 1 and C 2 are constants. Let [ [ α ] ] denote the greatest integer less than or equal to α.

Then there exists g * > 0, such that for any g ∈ (0, g * ), there exist positive constants C 3 , γ 1 , and h * , that depend only on g, C 1 , and C 2 , such that

0 < h < h * , l(h) = [ [ g/h ] ], and 2 ≤ q ≤ l(h) + 2 (2.18) imply L q (l(h)) ≤ C 3 exp { -γ 1 /h } . (2.

19)

Proof of Theorem 1.1: We define ψ l (x, t, h) by (1.4), where the c j (l, t, h) are determined by the system (2.7) and initial conditions described above.

To apply Lemma 2.1 we define

ξ l (x, t, h) = i h ∂ ∂t ψ l (x, t, h) -- h2 2 ∆ + V (x) ψ l (x, t, h).
(2.20) By using (2.12), we see that this can be decomposed as a sum of two terms, ξ

l (x, t, h) = Z [l] a(t) (x) -Z a(t) (x) ψ l (x, t, h) (1)
(2)

l (x, t, h) = -P {|j|≥ J(l)} Z [l] a(t) (x) ψ l (x, t, h), (2.22)
where P {|j|≥l+2} is the orthogonal projection onto the span of the set

{ ϕ j (A(t), B(t), h, a(t), η(t), • ) : |j| ≥ l + 2 }.
By the standard Taylor series error formula,

Z a(t) (x) -Z [l] a(t) (x) = |m|=l+2 D m V (ζ(x, a(t))) m! (x -a(t)) m ,
for some ζ(x, a) = (a + θ x,a (x -a)), with θ x,a ∈ (0, 1). Thus, by the crude estimate

|m|=n 1 ≤ |m|≤n 1 ≤ d j=1 m j ≤n 1 = (n + 1) d = e d ln(n+1) ≤ e d n , (2.23)
and Lemma 2.2, we obtain

ξ (1) l (•, t, h) ≤ |m|=l+2 D m V (ζ(x, a(t))) m! (x -a(t)) m ψ l (•, t, h) ≤ g 0 M exp(4τ (δ 2 d + a(t) 2 )) g 1 h(l + 2) A(t) /δ l+2 e d l+2 .
Thus, there exist constants C 4 and C 5 , such that |t| ≤ T implies

ξ (1) l (•, t, h) ≤ C 4 C 5 h(l + 2) l+2 .
(2.24)

The quantity ξ

l (•, t, h) satisfies

ϕ j (A(t), B(t), h, a(t), η(t), x), ξ (2)
l (x, t, h) = 0, if 0 ≤ |j| ≤ J (l) or |j| > J(l) + l + 1,
and

ϕ j (A(t), B(t), h, a(t), η(t), x), ξ (2)
l (x, t, h) = |k|≤ J(l) ϕ j (A(t), B(t), h, a(t), η(t), x), Z [l] a(t) (x) ϕ k (A(t), B(t), h, a(t), η(t), x) c k (t) = ( K(t) c(t)) j , if J(l) < |j| ≤ J (l) + l + 1, (2.25)
where we denote the c j (l, t, h) collectively by the vector c(l, t, h). We easily verify these facts by using (2.7), (2.11), (2.12) and Lemma 2.3. To estimate the norm of ξ

l (•, t, h), we use the Dyson expansion with remainder to decompose c(l, t, h) = l q=0 c q (l, t, h) + r(l, t, h), (2.26) where (dropping some arguments)

c q (t) = t 0 ds 1 s 1 0 ds 2 • • • s q-1 0 ds q (ih) -q K(s 1 )K(s 2 ) • • • K(s q ) c(0), (2.27)
and

r(t) = t 0 ds 1 s 1 0 ds 2 • • • s l 0 ds l+1 (ih) -(l+1) K(s 1 )K(s 2 ) • • • K(s l+1 ) c(s l+1 ).
(2.28) Using (2.11), (2.10), and (2.14), we see that each c q (l, t, h) is of order hq/2 and that r(l, t, h) is of order h(l+1)/2 .

To estimate the norm of ξ

l (•, t, h) we study the j th component of K(t)c(l, t, h), with |j| > J(l). Because of (2.26), this coefficient is a sum of two types of terms: those that arise from c q (l, t, h) and those that arise from r(l, t, h). Using (2.10), we expand K(l, t, h) in (2.27) and (2.28) and K(l, t, h) in terms of X m (l, t, h) and X m (t, h), to obtain

K(t)c q (t) = (ih) -q (l+1)(q+1) n=3(q+1) m 0 , m 1 , m 2 , • • • , mq |m 0 | + |m 1 | + • • • + |mq| = n 3 ≤ |m j | ≤ l + 1 t 0 ds 1 s 1 0 ds 2 • • • s q-1 0 ds q D m 0 V (a(t))D m 1 V (a(s 1 )) • • • D mq V (a(s q )) m 0 !m 1 !m 2 ! • • • m q ! × X m 0 (t)X m 1 (s 1 )X m 2 (s 2 ) • • • X mq (s q ) c(0), (2.29)
and

K(t)r(t) = (ih) -(l+1) (l+1)(l+2) n=3(l+2) m 0 , m 1 , m 2 , • • • , m l+1 |m 0 | + |m 1 | + • • • + |m l+1 | = n 3 ≤ |m j | ≤ l + 1 t 0 ds 1 s 1 0 ds 2 • • • s l 0 ds l+1 D m 0 V (a(t))D m 1 V (a(s 1 )) • • • D m l+1 V (a(s l+1 )) m 0 !m 1 !m 2 ! • • • m l+1 ! × X m 0 (t)X m 1 (s 1 )X m 2 (s 2 ) • • • X m l+1 (s l+1 ) c(s l+1 ). (2.30)
The values of m j that occur in both (2.29) and (2.30) satisfy

|m j |/ J(l) ≤ (l + 1)/(3l -3) ≤ 1, (2.31)
as long as l ≥ 3. So, we can apply Lemma 2.3 with N = 1.

Recall that X m j k (l, s, h) = 0 and X m j k (s, h) = 0 if |j| -|k| > |m|. Since c k (0) is non-zero only for |k| ≤ J, and we need only consider the j th coefficient of K(t)c(t) for |j| > J(l), the only relevant values of n in (2.29) must satisfy

n ≥ 3l -2.
(2.32)

This condition is also satisfied for all values of n in (2.30), since the sum begins with n = 3l+6.

To use the analyticity assumptions to get estimates on the derivatives of V , we define

C δ (x) = {z ∈ I C d : z j = x j + δe iθ j , θ j ∈ [0, 2π), j = 1, 2, • • • , d}. If z ∈ C δ (a(t)), then, for all j = 1, 2, • • • , d, |z j | ≤ δ + |a j (t)|.
Hence, writing 1 m! D m V (a(t)) as a d-dimensional Cauchy integral, we get the bound

1 m! |D m V (a(t))| ≤ sup z∈C δ (a(t)) |V (z)| δ |m| ≡ v(t) δ |m| (2.33) where v(t) ≤ M exp(4τ (δ 2 d + a 2 (t))).
Furthermore, since A(t) depends continuously on t, there exists w(T ), such that sup t∈[-T,T ] A(t) ≤ w(T ).

The norm of the vector c(t) is 1 since K(l, t, h) is self-adjoint. Thus, the non-zero entries of c(0) are each bounded by 1, and by a crude estimate, there are at most (J + 1) d of them. Similarly, c(t) has at most ( J(l) + 1) d non-zero entries, each of which is bounded by 1. Thus, for l ≥ 3, (2.11), (2.10), (2.14), (2.29), (2.30), (2.32), (2.31), and Lemma 2.3 imply the following two estimates when j satisfies J(l) + 1 ≤ |j| ≤ J(l) + l + 1:

( K(t) c q (t)) j ≤ v(t) T 0 v(s) ds q hq q! (l+1)(q+1) n=3l-2 (D(1) w(T )/δ) n hn/2 J(l) n/2 F l+1 (n, q + 1) (J + 1) d = v(t)h(q + 1)(J + 1) d T 0 v(s)ds T 0 v(s)ds q+1 hq+1 (q + 1)! (l+1)(q+1) n=3l-2 (D(1)w(T ) h J(l)/δ) n F l+1 (n, q + 1) (2.34) and ( K(t)r(t)) j ≤ v(t)h(l + 2)( J(l) + 1) d T 0 v(s)ds T 0 v(s)ds l+2 hl+2 (l + 2)! (l+1)(l+2) n=3l-2 (D(1)w(T ) h J(l)/δ) n F l+1 (n, l + 2), (2.35)
where F p (n, q) is defined by (2.16). By Lemma 2.4, (2.34), and (2.35), both ( K(t) c q (t)) j and ( K(t)r(t)) j are bounded by C 3 exp { -γ 1 /h } for an appropriate choice of l(h) = [ [ g/h ] ] and sufficiently small h. Each of the l + 2 terms in (2.26) contributes a term of this type to ξ

(2) l (x, t, h), so h-1 ξ (2) l (x, t, h) ≤ C 3 h-1 (l(h) + 2) exp { -γ 1 /h } ≤ C 3 exp { -γ 2 /h } ,
for any γ 2 < γ 1 when h is sufficiently small. We shrink g if necessary to make

C 2 5 g < 1 (2.36)
and set l = l(h) in (2.24). This yields a similar estimate

h-1 ξ (1) l (•, t, h) ≤ C 6 exp { -γ 3 /h } , for some γ 3 > 0.
We combine these two estimates and apply Lemma 2.1 to obtain (1.2) with γ = min {γ 2 , γ 3 }. This proves the theorem.

Proof of Lemma 2.2

For simplicity, we drop the t dependence in the notation throughout this section.

To prove Lemma 2.2, we use Hölder's inequality to see that |j|≤ J(l) |c j (l, h)| 2 = 1 implies

|j|≤ J(l) |c j (l, h)| ≤    |j|≤ J(l) |c j (l, h)| 2    1/2    |j|≤ J(l) 1    1/2 ≤ J(l) + 1 d/2 .
Thus, it is sufficient to prove

D m V (ζ(x, a)) m! (x -a) m ϕ j (A, B, h, a, η, x) ≤ g 3 M exp(4τ (δ 2 d + a 2 )) g 4 h(l + 2) A /δ l+2 , (3.1)
for some g 3 and g 4 , whenever |j| ≤ J(l).

We mimic the proof of 2.33 to obtain a bound on

D m V (ζ(x, a))/m!. If z ∈ C δ (ζ(x, a)), then, for all j = 1, 2, • • • , d, |z j | ≤ δ + |ζ j (x, a)| ≤ δ + |a j | + |x j -a j |.

Using this and applying (b+c

) 2 ≤ 2(b 2 +c 2 ) several times, we see that z ∈ C δ (ζ(x, a)) implies | V (z) | ≤ M exp(2τ (x -a) 2 ) exp(4τ (δ 2 d + a 2 )).
I = IR d (x -a) 2m exp(4τ (x -a) 2 ) |ϕ j (A, B, h, a, η, x)| 2 dx. (3.3)
Performing the change of variables x → y = |A| -1 (x-a)/h 1/2 , and using the explicit formula for ϕ j , we see that

I = h|m| 2 |j| j! π d/2 IR d (|A|y) 2m exp(-y 2 + 4τ h(|A|y) 2 ) |H j (A; y)| 2 dy, (3.4)
where H l (A; y) is the polynomial described in Remarks 4 and 5 that immediately follow the definition of ϕ j (A, B, h, a, η, x) in Section 2. We assume henceforth that h is sufficiently small that

4 τ h A 2 ≤ 1/2. (3.5)
The estimate (|A|y

) 2 ≤ A 2 y 2 implies (|A|y) 2 k ≤ A 2 y 2 , for k = 1, 2, • • • , d. From this, we conclude I ≤ (h A 2 ) |m| 2 |j| j! π d/2 y 2|m| exp(-y 2 /2) |H j (A; y)| 2 dy. (3.6)
We next need an estimate on |H j (A; y)|. By Remark 5 after the definition in Section 2,

-∆ + y 2 H j (A; y) e -y 2 /2 = (2|j| + d) H j (A; y) e -y 2 /2 . (3.7)
This equation states that H j (A; y) e -y 2 /2 is a eigenfunction of -∆ + y 2 corresponding to the eigenvalue 2|j| + d. Introducing normalization factors, we conclude that

2 -|j|/2 (j!) -1/2 π -d/4 H j (A; y) e -y 2 /2 = |k|=|j| b k ϕ k 1 (y 1 )ϕ k 2 (y 2 ) • • • ϕ k d (y d ), (3.8)
where

ϕ k (y) = 2 -k/2 (k!) -1/2 π -1/4 H k (y) e -y 2 /2
is the normalized eigenfunction of -∂ 2 ∂y 2 + y 2 corresponding to the eigenvalue 2k + 1, and the coefficients b k satisfy |k|=|j| |b k | 2 = 1. We can thus deduce an estimate of H j (A; y) from an estimate for usual Hermite polynomials H k (y):

|H j (A; y)| 2 = | |k|=|j| b k H k 1 (y 1 )H k 2 (y 2 ) • • • H k d (y d )| 2 ≤ |k|=|j| |H k 1 (y 1 )H k 2 (y 2 ) • • • H k d (y d )| 2 .
(3.9)

The Hermite polynomials, in turn, satisfy the following bounds.

Lemma 3.1 If |y| > √ 2k + 1, then |H k (y)| ≤ 2 k |y| k , (3.10)
and there exists a numerical constant κ > 0, such that for all y ∈ IR,

|H k (y)| ≤ κ 2 k/2 √ k! e y 2 /2 . (3.

11)

Proof: The second statement is well known. See [START_REF] Abromowitz | Handbook of Mathematical Functions[END_REF], formula 22.14.17 or [START_REF] Gradsteyhn | Table of Integrals, Series, and Products[END_REF], formula 8.954.2. By symmetry, it is sufficient to prove (3.10) for y > 0. It is well known that the k th eigenfuction of the harmonic oscillator is non-zero in the classically forbidden region |y| > √ 2k + 1. It is also well known that H k (y) = 2 k y k + P k (y) where P k (y) is a polynomial of degree k -2. Thus, we have

H k (y)/(2 k y k ) = 1 + B k (y), where B k satisfies B k (y) = O(1/y 2 )
for large y, and

B k (y) > -1 if y > √ 2k + 1.
Using standard properties of the Hermite polynomials, we see that

B ′ k (y) = (y H ′ k (y) -k H k (y))/(2 k y k+1 ) = (2 k y H k-1 (y) -k H k (y))/(2 k y k+1 ) = k (2 y H k-1 (y) -(2 y H k-1 (y) -2 (k -1) H k-2 (y)))/(2 k y k+1 ) = k (k -1) H k-2 (y)/(2 k-1 y k+1 ). So, if y > √ 2k + 1, we see that B ′ k (y) > 0. Thus, y > √ 2k + 1 implies -1 < B k (y) < 0, which implies (3.10).
In (3.6), we use the multinomial expansion

(y 2 ) |m| = |n|=|m| |m| n 1 • • • n d y 2n 1 1 y 2n 2 2 • • • y 2n d d to obtain I ≤ (h A 2 ) |m| 2 |j| j! π d/2 |n|=|m| |k|=|j| |m| n 1 • • • n d d r=1 y 2nr r |H kr (y r )| 2 exp(-y 2 r /2)dy r .
Using Lemma 3.1, the one dimensional integrals in the final factor here can then be estimated as follows (without indices):

y 2n |H k (y)| 2 exp(-y 2 /2) dy ≤ y 2 ≤2k+1
κ 2 2 k k! y 2n exp(y 2 /2) dy + 4 k y 2(n+k) exp(-y 2 /2) dy

≤ κ 2 2 k k! 2 2n + 1 (2k + 1) n+1/2 exp(k + 1/2) + 4 k 2 (k+n+1/2) z 2(k+n) e -z 2 dz = 2 k+1 2n + 1 κ 2 e k+1/2 k!(2k + 1) n+1/2 + 4 k 2 (k+n+1/2) π 1/2 (2(k + n) -1)! 2 2(n+k)-1 (k + n -1)! = 2 k+1 2n + 1 κ 2 e k+1/2 k!(2k + 1) n+1/2 + 2 k-n+3/2 π 1/2 (2(k + n) -1)! (k + n -1)! . (3.12)
Here we have used

z 2(k+n) e -z 2 dz = π 1/2 1 • 3 • 5 • • • (2(k + n) -1) 2 k+n .
Now, Stirling's formula guarantees the existence of a > 0, such that

a n n n ≤ n! ≤ n n , (3.13)
for all integers n ≥ 1. Using this in (3.12), we obtain

y 2n |H k (y)| 2 exp(-y 2 /2) dy ≤ f 0 f k+n 1 (k + n)!,
for some constants f 0 and f 1 . From this we conclude (restoring the indices)

I ≤ (h A 2 ) |m| 2 |j| j! π d/2 |n|=|m| |k|=|j| f d 0 f |m|+|j| 1 |m| n 1 • • • n d (k + n)! (3.14)
In this expression, we have

(k + n)! = (|k| + |n|)! |k| + |n| k 1 + n 1 • • • k d + n d -1 = (|m| + |j|)! |m| + |j| k 1 + n 1 • • • k d + n d -1 ≤ (|m| + |j|)! (3.15)
We further use

|n|=|m| |m| n 1 • • • n d = d |m| and (2.23) in (3.14) to obtain I ≤ (h A 2 ) |m| f 0 √ π d (f 1 d) |m| e d f 1 2 |j| (|m| + |j|)! j! = (h A 2 ) |m| f 0 √ π d (f 1 d) |m| e d f 1 2 |j| |j| j 1 • • • j d |m| + |j| |j| |m|! ≤ (h A 2 ) |m| f 0 √ π d (2 f 1 d) |m| (e d f 1 ) |j| |j| j 1 • • • j d |m|! ≡ (h A 2 d) |m| f d 2 f |m| 3 f |j| 4 |j| j 1 • • • j d |m|!, (3.16)
where f 2 , f 3 , f 4 are numerical constants. Estimates (3.2) and (3.16) imply

D m V (ζ(x, a)) m! (x -a) m ϕ j (A, B, h, a, η, x) ≤ M exp(4τ (δ 2 d + a 2 ))(h A 2 (d/δ 2 )f 3 ) |m|/2 f d/2 2 f |j|/2 4 |m|! |j| j 1 • • • j d .
So, by the Schwarz inequality, (1.4), (3.9), and (3.18),

D m V (ζ(x, a)) m! (x -a) m ψ l (x, t, h) ≤ J(l) n=0 |j|=n D m V (ζ(x, a)) m! (x -a) m ϕ j (A, B, h, a, η, x) 2 ≤ M 2 exp(8τ (δ 2 d + a 2 ))(h A 2 (d/δ 2 )f 3 ) |m| f d 2 |m|! J(l) n=0 |j|=n f |j| 4 |j| j 1 • • • j d = M 2 exp(8τ (δ 2 d + a 2 )) (h A 2 (d/δ 2 )f 3 ) |m| f d 2 |m|! J(l) n=0 f n 4 d n = M 2 exp(8τ (δ 2 d + a 2 )) (h A 2 (d/δ 2 )f 3 ) |m| f d 2 |m|!   (f 4 d) J(l)+1 -1 f 4 d -1   ≤ M 2 exp(8τ (δ 2 d + a 2 )) (h A 2 (d/δ 2 )f 3 ) |m| f d 2 |m|! 2 (f 4 d) J(l) . (3.17)
The last step depends on the following fact: We can assume without loss that q = f 4 d ≥ 2, so that for any positive integer p, we have -1 ≤ q p+1 -2q p , and hence, q p+1 -1 ≤ 2q p+1 -2q p . Thus, (q p+1 -1)/(q -1) ≤ 2q p .

(3.18)

We use the hypothesis |m| = l + 2 and note that |m|! ≤ (l + 2) l+2 . We then conclude that there exist constants g 0 and g 1 , that depend only on d and J, such that (2.13) holds. This implies the lemma.

Proofs of Lemmas 2.3 and 2.4

The proof of Lemma 2.3 relies on two preliminary lemmas.

| ϕ j (A, B, h, a, η, x), (x -a) m ϕ k (A, B, h, a, η, x) | ≤ h|m|/2 ( √ 2d) |m| A |m| (|k| + 1)(|k| + 2) • • • (|k| + |m|), (4.1)
and

ϕ j (A, B, h, a, η, x), (x -a) m ϕ k (A, B, h, a, η, x) = 0, if |j| -|k| > |m|.
Proof: [START_REF] Hagedorn | Raising and Lowering Operators for Semiclassical Wave Packets[END_REF] to express (x i -a i ) in terms of raising and lowering operators. Doing so, we obtain

For i = 1, 2, • • • , d, we can use equation (3.28) of
(x i -a i ) ϕ k (A, B, h, a, η, x) = h/2 d p=1 A ip k p + 1 ϕ k ′ (p) (A, B, h, a, η, x) + h/2 d p=1 Āip k p ϕ k ′′ (p) (A, B, h, a, η, x),
where k ′ (p) and k ′′ (p) are constructed from k by replacing the component k p by k p + 1 and k p -1, respectively. Thus, (x i -a i ) ϕ k (A, B, h, a, η, x) may be written as a sum of 2d terms, each of which has the form b q ϕ q (A, B, h, a, η, x), where |q| ≤ |k| + 1 and |b q | ≤ h/2 A |k| + 1. From this and a simple induction, (x -a) m ϕ k (A, B, h, a, η, x) may be written as a sum of (2d) |m| terms, each of which has the form b q ϕ q (A, B, h, a, η, x), where |k| -|m| ≤ |q| ≤ |k| + |m| and

|b q | ≤ (h/2) |m| A |m| (|k| + 1)(|k| + 2) • • • (|k| + |m|).
This implies the lemma.

In the next lemma we use the shorthand X m (t) to denote X m (l, t, h) of Lemma 2.3.

Lemma 4.2 For each N > 0, there exists D(N) < ∞, such that

sup {j,k : |j|, |k|≤ J(l)} | (X m 1 (t 1 ) X m 2 (t 2 ) • • • X mq (t q )) j k | ≤ D(N) √ h sup t∈{t 1 ,t 2 ,•••tq} A(t) |m 1 |+|m 2 |•••+|mq| J(l) (|m 1 |+|m 2 |+•••+|mq|)/2 , (4.2)
for any collection m 1 , m 2 , . . . , m q of multi-indices that satisfy |m j |/ J(l) ≤ N.

Proof: The matrix X m is obtained from X m by restricting its indices to |j|, |k| ≤ J(l). Using Lemma 4.1, we see that for such values of the indices,

X m j k (t) ≤ ( √ 2h d A(t) ) |m| J (l) |m|/2 (1 + 1/ J(l))(1 + 2/ J(l))) • • • (1 + |m|/ J(l)).
Thus, |m|/ J(l) ≤ N implies

X m j k (t) ≤ ( √ 2h d A(t) ) |m| (1 + N) |m|/2 J (l) |m|/2 .
We estimate the absolute value of the j, k matrix element of the product

| (X m 1 (t 1 ) X m 2 (t 2 ) • • • X mq (t q )) j k | ≤ |p i | ≤ J(l), i = 1, • • • , q -1 |p 1 -j| ≤ |m 1 |, |p 2 -p 1 | ≤ |m 2 |, • • • , |k -p q-1 | ≤ |mq| X m 1 j p 1 (t 1 ) X m 2 p 1 p 2 (t 2 ) • • • X mq p q-1 k (t q ) (4.3)
by bounding the absolute values of each of the matrix elements X m i p i-1 p i (t i ) by

X m i p i-1 p i (t i ) ≤ ( √ 2h d A(t i ) ) |m i | (1 + N) |m i |/2 J(l) |m i |/2 , (4.4)
and multiplying by a bound on the number of terms.

To estimate the number of terms, we first note that for any multi-index r, the number of

i | ≤ |m| for i = 1, 2, • • • , d, which is (2|m| + 1) d .
Thus, the number of terms is bounded by

|p i | ≤ J(l), i = 1, • • • , q -1 |p 1 -j| ≤ |m 1 |, |p 2 -p 1 | ≤ |m 2 |, • • • , |k -p q-1 | ≤ |mq| 1 ≤ (2|m 1 | + 1) d (2|m 2 | + 1) d • • • (2|m q | + 1) d ≤ (e 2 ) d(|m 1 |+|m 2 |+•••+|mq|) . (4.5)
The final inequality follows because 1 + 2|m j | ≤ e 2|m j | . We obtain the lemma by using (4.4) and (4.

A(t) -m ( X m 0 (t 0 ) X m 1 (t 1 ) X m 2 (t 2 ) • • • X mq (t q )) j k = √ h sup t∈{t 0 ,t 1 ,•••,tq} A(t) -m |r|≤ J(l) X m 0 j r (t 0 ) | (X m 1 (t 1 ) X m 2 (t 2 ) • • • X mq (t q )) r k | ≤ |r| ≤ J(l) |j -r| ≤ |m 0 | 2 |m 0 |/2 (|r| + 1)(|r| + 2) • • • (|r| + |m 0 |) × D(N) |m 1 |+|m 2 |+...+|mq| J(l) (|m 1 |+|m 2 |+...+|mq|)/2 ≤ |r| ≤ J(l) |j -r| ≤ |m 0 | 2 |m 0 |/2 J(l) |m 0 |/2 (1 + 1/ J(l))(1 + 2/ J(l)) . . . (1 + |m 0 |/ J(l)) × D(N) |m 1 |+|m 2 |+...+|mq| J(l) (|m 1 |+|m 2 |+...+|mq|)/2 ≤ |r| ≤ J(l) |j -r| ≤ |m 0 | 2 |m 0 |/2 (1 + N) |m 0 |/2 D(N) |m 1 |+|m 2 |+...+|mq| J (l) (|m 1 |+|m 2 |+...+|mq|)/2 ≤ 2 |m 0 |/2 (1 + N) |m 0 |/2 D(N) |m 1 |+|m 2 |+...+|mq| J(l) (|m 1 |+|m 2 |+...+|mq|)/2 |j-r|≤|m 0 | 1.
Since the sum in the last line is (1+2|m 0 |) d ≤ e 2d|m 0 | , we obtain the desired estimate.

We now turn to the proof of Lemma 2.4. Our first step is to study the combinatorial factor F p (n, q) in the one dimensional case.

(4.6)

This quantity is zero if n < q or n > qp. Otherwise, it satisfies

G p (n, q) ≤        n -1 q -1 if q ≤ n ≤ [q(p + 1)/2] q(p + 1) -n -1 q -1 if [q(p + 1)/2] + 1 ≤ n ≤ qp. (4.7)
Proof: If n < q or n > qp, the sum in (4.6) contains no terms, so G p (n, q) = 0. The quantity G p (n, q) is the number of ways that the number n can be decomposed as n = m 1 + m 2 + . . . + m q with each m j satisfying 1 ≤ m j ≤ p. We uniquely associate to each such decomposition of n, a corresponding decomposition of n ′ = q(p + 1)

-n = m ′ 1 + m ′ 2 + . . . + m ′ q by setting m ′ j = p + 1 -m j .
This association is a one-to-one correspondence, so we see that G p (n, q) satisfies the symmetry relation G p (n, q) = G p (q(p + 1) -n, q). (4.8)

As a consequence, the second inequality in (4.7) (with [q(p + 1)/2] + 1 ≤ n ≤ qp) follows from the first (with q ≤ n ≤ [q(p + 1)/2]).

To prove the first inequality in (4.7) we drop the condition m j ≤ p and exactly calculate the resulting function. Dropping the upper bound on the m j , we clearly have G p (n, q) ≤ G(n, q), where G(n, q) = 1.

So, the lemma will be proved once we establish

G(n, q) = n -1 q -1 . (4.9)
To prove this, we use induction. Formula (4.9) is trivial to verify for all n ≥ 1 when q = 1 and for all n ≥ 2 when q = 2. Assume (4.9) has been verified whenever n ≥ q -1, and suppose n ≥ q. We have

G(n, q) = n-q+1 m 1 =1 G(n -m 1 , q -1) = n-q+1 m 1 =1 n -m 1 -1 q -2 ,
so the induction step is complete, since it is known (see [START_REF] Vilenkin | Combinatorics[END_REF], p.36) that

n-q+1 m 1 =1 n -m 1 -1 q -2 = n -1 q -1 . (4.10)
This ends the proof of the lemma.

Our next step is to obtain an estimate for G p (n, q) that depends only on n.

Lemma 4.4 Let G p (n, q) be defined by (4.6). There exists a constant C 7 , such that p ≥ 1, q ≥ 1, and n ≥ 1 imply

G p (n, q)) ≤ (C 7 ) n . ( 4

.11)

Proof: We note that G p (n, q) is zero unless q ≤ n ≤ qp. Furthermore, G p (q, q) = G p (qp, q) = G 2 (2, 1) = 1 and G 1 (n, q) = δ n q . So, we need only prove existence of C 7 , such that p ≥ 2, q ≥ 1, (p, q) = (2, 1) imply sup q<n<pq (G p (n, q)) 1/n ≤ C 7 .

(4.12)

To prove this, we first study the case where q < n ≤ [q(p + 1)/2]. Since q ≤ n, we have

q 1/n = exp(ln(q)/n) ≤ exp(ln(n)/n) ≤ e 1/e .
Thus, by (4.7), the condition q < n ≤ [q(p + 1)/2], and (3.13), we have

(G p (n, q)) 1/n ≤ (n -1)! (q -1)! (n -q)! 1/n ≤ q 1/n (n -1)! q! (n -q)! 1/n ≤ e 1/e
n aq q/n (n -q) (n-q)/n . (4.13)

An explicit computation shows that ∂ ∂n n q q/n (n -q) (n-q)/n = n q q/n (n -q) (n-q)/n q n 2 ( ln(q) -ln(n -q) ) .

From this, we deduce that the right hand side of (4.13) attains its maximum at n = 2q. Evaluating that maximum, we obtain sup q<n≤[q(p+1)/2] (G p (n, q)) 1/n ≤ 2 a e 1/e . (4.14)

For [q(p + 1)/2] < n < pq, the number n ′ = q(p + 1) -n satisifes q < n ′ ≤ [q(p + 1)/2] and n ′ /n ≤ 1. Thus, by (4.8) and (4.14), we obtain We now generalize Lemma 4.4 to the multi-dimensional case.

(G p (n, q)) 1/n = (G p (n ′ , q)) 1/n ≤ 2 a e 1/e n ′ /n ≤ 2 a e 1/e . Thus, sup [q(p+1)/2]<n<pq (G p (n, q))
Lemma 4.5 Let F p (n, q) be defined by (2.16). For all p ≥ 1, q ≥ 1, and n ≥ 1, we have

F p (n, q) ≤ C (n+qd) 7 , (4.16)
where C 7 is the constant of Lemma 4.4.

Proof: With m j temporarily denoting numbers instead of multi-indices, we define

Γ p (n, q) = 0 ≤ m 1 , m 2 , • • • , mq ≤ p m 1 + m 2 + • • • + mq = n 1,
which is zero unless 0 ≤ n ≤ pq.

By defining m ′ j = m j +1, we see that every decomposition of n as n = m 1 +m 2 +• • •+m q , with 0 ≤ m j ≤ p, corresponds uniquely to a decomposition of n + q as n + q

= m ′ 1 + m ′ 2 + • • • + m ′ q , with 1 ≤ m ′ j ≤ p + 1.
Therefore, we have the identity Γ p (n, q) = G p+1 (n + q, q). (4.17)

We now let the m j denote multi-indices and let m j (k) denote the k th component of m j . Then, using (4.17) and Lemma 4.4, we easily obtain

F p (n, q) ≤ 0 ≤ m j (k) ≤ p q j=1 d k=1 m j (k) = n 1 = Γ p (n, qd) ≤ C (qd+n) 7 .
Proof of Lemma 2.4: Suppose 2 ≤ q ≤ l + 2, and let L q (l) satisfy (2.17), i.e.,

L q (l) ≤ C q 1 hq q! q(l+1) n=3l-2 (C 2 h l) n/2 F l+1 (n, q).
By Lemma 4.5 and (3.13), there exist C 8 and C 9 , such that

L q (l) ≤ (C 1 C d 7 ) q (a h q) q q(l+1) n=3l-2 (C 2 h l) n/2 C n 7 ≤ C q 9 (h q) q q(l+1) n=3l-2 C 8 h1/2 l 1/2 n . ( 4

.18)

Note that we can take

C 8 = max { 1, C 2 C 7 }, (4.19)
C 9 = max { 1, C 1 C d 7 }, (4.20)
because we can assume C 8 ≥ 1 and C 9 ≥ 1 without loss of generality.

We arbitrarily choose two positive numbers α < 1 and β < 1. We then define By summing a geometric series in (4.18), we see that

g = 1 C 2
L q (l) ≤ C q 9 (1 -α) (h q) q C 8 h1/2 l 1/2 3l-2 = C q 9 C 2q 8 (1 -α) (h q C 2 8 ) q C 8 h1/2 l 1/2 3l-2 . (4.23)
As a function of q, l q q = exp ( q ln(l/q) ) is maximized at q = l/e, where it has the value e l/e . Thus, l q q ≤ e l/e . This, q ≤ l + 2, and h l C 2 8 ≤ α 2 < 1 imply 1 (h q C 2 8 ) q = 1 (h l C 2 8 ) q l q q ≤ e l/e (h l C Note that the factor 2 3 can be replaced by (1 + ǫ), with ǫ arbitrarily small, by taking h sufficiently small.

Remark We can choose α and β to satisfy β e 1/e ≤ α <

Proof of Theorem 1.2

To prove Theorem 1.2, we revisit the proof of Theorem 1.1 and make the T dependence of all constants explicit. We then allow T to grow with h with the restriction that our approximation remain close to the actual solution as h → 0. Since V is bounded below and energy is conserved, |a(t)| grows at most linearly with time. Thus, there exist v 1 > 0 and v 0 > 0, such that v(t) defined by (2.33) satisifies v(t) ≤ v 0 exp(v 1 t).

(5.1)

From assumption (1.13), it follows that w(T ) = sup 0≤|t|≤T A(t) ≤ N exp(λT ).

(5.2)

In the sequel, we denote all innessential constants that do not depend on T by the same symbol c.

Consider Lemma 2.2. For z ∈ C δ (ζ(x, a(t))), we can prove the existence of ρ > 0 such that To arrive at these conclusions, we imposed various conditions. Those conditions were (3.5), that now requires h ≤ c exp(-2λT ), (2.36), than now requires c exp(2λT ) exp(-νT ) < 1, and h ≤ g(T )/2 used in (4.25), that now requires h ≤ c exp(-νT ).

D m V (ζ
These are all satisfied, provided we take

T (h) ≤ T ′ | ln(h)|,
with T ′ > 0 sufficiently small. Using this in our estimate for the error term ξ(t, h), we see that for sufficiently small T ′ , |t| ≤ T ′ | ln(h)| implies

ψ(x, t, h) -Ψ(x, t, h) L 2 (I R d ) ≤ C ′ exp(-γ ′ /h σ ),
for some γ ′ > 0, σ > 0, and C ′ > 0.

  m V (ζ(x, a)) as a d-dimensional Cauchy integral, we easily obtain the bound1 m! |D m V (ζ(x, a))| ≤ M exp(4τ (δ 2 d + a 2 )) δ |m| exp(2τ (x -a) 2 ). (3.2)Thus, estimate (3.1) follows from the corresponding estimate on the integral

Lemma 4 . 1

 41 The matrix elements of (x -a) m satisfy

2 8 ) 9 C 2 l 1 /2 l- 6 = e 1 /e C 9 C 3 8 h1/2 l 1/2 l C 2 9 ( 1 8 . 2 9( 1 - 8 ≤ 2 3 e C 2 9 ( 1 -α) g 3 C 2 8 e

 89216189182182918 l+2 . From this, C 8 ≥ 1, and C 9 ≥ 1, we concludeL q (l) ≤ C l+2 -α) (h l) 3 C 2 see from (4.24) that if h ≤ g/2, then we have L q (l) ≤ e -| ln(β)|l C α) (g -h) 3 C 2 -| ln(β) |g/h . (4.25)

Lemmas 4 . 3 , 4 . 4 ,

 4344 and 4.5 have no time dependence, and the time dependence of Lemmas 2.1, 2.2, 2.3, 3.1, 4.1, and 4.2 has been made explicit in their conclusions.

  (x, a(t))) m! (x -a(t)) m ψ l (x, t, h) ≤ c exp(ρ|t|) c h (l + 2) exp(λ|t|) our bound on |V (z)| we haveD m V (ζ(x, a(t))) m! ≤ c exp(ρ|t|) exp(c|x -a(t)|)instead of(3.2). With this and the estimate exp(cy) ≤ exp(cy 2 ) + exp(c) for all c > 0 and y > 0, we see that the estimates in the proof of Lemma 2.2 are still valid. This yields (5.3). Consequently, the constants C 4 and C 5 appearing in (2.24) satisfyC 4 (T ) = c exp(ρT ) C 5 (T ) = c exp(λT ),where ρ > 0. By using estimates (5.1) and (5.2) in (2.34) and (2.35), we see that when we apply Lemma 2.4, the constants C 1 and C 2 satisfyC 1 (T ) = c exp(ρT ) C 2 (T ) = c exp(2λT ).We still must determine the T dependence of C and γ of Theorem 1.1. We do this by determining the T dependence of g, γ 1 and C 3 in Lemma 2.4. Equations (4.19) and (4.20) yield C 8 and C 9 , in terms of which the above constants are determined. We find that C 8 (T ) = c exp(λT ) C 9 (T ) = c exp(ρT ). Consequently, from (4.26), (4.27) ,(4.28), we have g(T ) = c exp(-νT ), ν = 2ρ + 6λ γ 1 (T ) = c exp(-νT ), C 3 (T ) = c exp(µT ), µ = 8ρ + 10λ.

  multi-indices indices p that satisfy |p| -|r| ≤ |m| is equal to the number of vectors v with integer components, such that | d i=1 v i | ≤ |m|. This is bounded by the number of vectors with integer components, such that |v

  5) to bound (4.3). We mimic the proof of Lemma 4.2. Let m = |m 0 | + |m 1 | + |m 2 | + . . . + |m q |. By Lemmas 4.1 and 4.2,

	Proof of Lemma 2.3 √ h sup
	t∈{t 0 ,t 1 ,•••,tq}

  1 e 1/e . Then, since β 2 C 2 9 C 4 8 e 2/e ≤ β 2 e 2/e ≤ α 2 , we obtain the conclusion of Lemma 2.4 with -e -1/e ) β 6 , (4.28) where C 8 and C 9 are related to C 1 , C 2 , and C 7 by (4.19) and (4.20). Note that C 7 is purely combinatorial and has no time dependence.

	g =	β 2 9 C 6 C 2 8 e 2/e ,	(4.26)
	γ 1 =	| ln(β)| β 2 C 2 9 C 6 8 e 2/e	(4.27)
	C 3 =	2 3 e 1+6/e C 8 9 C 10 8 (1

≤ m 1 , m

, • • • , mq ≤ p m 1 + m 2 + • • • + mq = n

≤ m 1 , m

, • • • , mq m 1 + m 2 + • • • + mq = n

* Partially Supported by National Science Foundation Grant DMS-9703751. † Partially Supported by Fonds National Suisse de la Recherche Scientifique, Grant 8220-037200.

1