Laurent Nottale: Scale relativity and fractal space-time

World Scientific/Imperial College Press, 2011, 764 pp, GBP105.00, ISBN 978-1-84816-650-9

Patrick Peter

I have heard about scale relativity for many years, but never had the opportunity to actually get into it until I was asked to read Laurent Nottale's "Scale relativity and fractal space-time". The book consists in two parts, the first one, Chapters 1-8, in which the theory is exposed, and the second where applications are discussed, in Chapters 9-14.

The basic principle out of which the entire theory is developed in this book is that the particle trajectories need not be differentiable, following the original Feynman's idea whose path integral formulation of quantum mechanics can be recast in terms of continuous but nowhere differentiable trajectories with fractal dimension D F = 2. This propagator-based formulation is taken here seriously by assuming that particles do follow such trajectories (although later on in the course of the book, it is said that the particles do not, in fact, follow such trajectories, those being a mere representation of a geodesic flow leading to probabilities). This is the first idea developed in great length in Chapters 1-3.

The second idea, related to the first, consists in assuming that since the trajectories are fractals, their properties, as seen by an observer, are depending on the scale at which we observe them, i.e., on the accuracy of the measurement that are performed on them. Then, following the example of the invariance under Lorentz transformations in special relativity, the author proposes, in Chapter 3, a similar invariance with respect to scale transformations. This leads to a new kind of covariant derivative as well as to the existence of some special length scales, equivalent to the velocity of light in special relativity, which are identified to the Planck and cosmological constant scales, depending on the context. P. Peter (B) Institut d'Astrophysique de Paris, UMR7095 CNRS, Université Pierre and Marie Curie, 98bis boulevard Arago, 75014 Paris, France e-mail: peter@iap.fr With the two ideas previously developed, it is argued in Chapter 5 that for a given particle whose trajectory is now a non-differentiable geodesic, the velocity field should be two-valued (left and right derivatives with respect to time being different) and recast into a single complex variable, later somehow to be identified with a wave function (although at this stage, trajectories are no longer part of the description). It is then claimed that it can also be implemented by means of coordinates having a classical part as usual and a stochastic (with vanishing mean) part, whose mean square provides a new constant, later to be related with the Planck constant: then, the trajectories are geodesics of a non-differentiable space. The classical geodesic (Newton-like trajectory) equation in these coordinates is then put in a form equivalent, under some arguable hypothesis, to a Schrödinger equation. It is then argued that the probability of finding a particle at a given position had to be proportional to the geodesic flow, hence leading to the Born rule.

Moving on to special relativistic trajectories in Chapter 6, the Klein-Gordon equation is obtained in a similar way, and after some considerations of algebra doubling, again based on the two-valuedness of derivatives, the Dirac equation is introduced. This aims at showing a geometrically-based theory of spin. Finally, the scale covariant derivative is used to produce gauge vectors, thus completing the set of known objects and forming the basis for a particle physics model (Chapter 7). Chapter 8 provides some ideas as to how to generalize what precedes by extending to fractal dimensions other than the usual quantum mechanical D F = 2, including scaling laws in a varying fractal dimension model.

The second part of the book is introduced in Chapter 9 and consists in applications ranging from ordinary laboratory experiments to astrophysics/cosmology and high energy particle physics, with a final chapter dedicated to biology and other humanrelated sciences. It begins in Chapter 10 with probably the most interesting issue, namely the transition between quantum and classical behaviors, which is said here to have a purely "geometric interpretation" (transition between fractal and non-fractal geodesics). An interesting example is a calculation of many possible trajectories in a 2-slit Young experiment or the treatment of the harmonic oscillator. All through this chapter, use is made of the quantum potential, defined in exactly the same way as in the ontological formulation of quantum mechanics.

Then comes Chapter 11, devoted to particle physics and the running of the coupling constants. The characteristic dilation-invariant scale is then taken to be the Planck length, which serves the same purpose that the usual renormalization scale normally assumes. Amusingly, this chapter also contains a prediction for the Higgs boson that should have been observed at m H ≃ 113.7GeV...it would appear, according to the book itself, that the theory it describes would be already ruled out by LHC data! When it comes to cosmology in Chapter 12, the theory now takes as its fundamental the cosmological constant length scale, although the Planck scale still plays a role close to the now reinterpreted primeval singularity. The new theory is said to solve the horizon puzzle without invoking a phase of inflation. Bizarrely, a practicing cosmologist will hardly recognize any equation in this chapter, which also does not mention primordial perturbations ... in the time of Planck ultimate observation of these fluctuations, this is disturbing.

Chapter 13 presents a fractal-based extended Newtonian-like theory of gravitation, leading to specific (and, as far as I know, not verified) new astrophysical predictions, ranging from the solar system (planet trajectories) to the formation of large scale structures in cosmology or star formation in galaxies. It is argued that the scale relativity approach permits to solve the dark matter question.

Last but not least, Chapter 14 discusses all other possible applications of scale relativity and fractal mathematics in biology (evolution theory for instance, but a new approach to the origin of life is presented, not invoking biological phenomena), history and sociology. Having no serious knowledge on these topics, I cannot pretend to argue in one way or another, but the presentation suggests some reformulation, possibly even useful, of data analysis.

Generally speaking, I found the book not clearly written, and it is always quite time consuming to understand almost any point which is raised. Out of 555 references covering all the topics discussed in the book, i.e. ranging from particle physics and cosmology to neuroscience (and with mention of historical papers such as 1905 and 1916 Einstein's), 95 are signed or co-signed by the author himself, including numerous proceedings: that amounts to a staggering almost 20 % self-references! Apart from the fact that I have some doubts regarding the mathematical rigor of the discussion, I have found in many places either misleading or possibly erroneous statements concerning in particular the quantum mechanical treatment of particle physics, at least according to the widely accepted view shared by the vast majority of living physicists. This would not, by itself, be a problem for a monograph dealing with a highly controversial topic, as the book actually is, but then comes what is, in my view, the most important point: contrary to what the topic discussed should imply, it is clearly aimed at being a textbook, with over 700 pages, often repeating many times the same thing (which is definitely a good pedagogical method) and including exercises and problems to be worked out by students ... My personal opinion is that the topic is probably interesting (the quantum-toclassical transition treatment for instance is original and could lead to some yet-to-do developments), and could perhaps deserve more attention than it had until now. Basically, there is only a handful of people involved in these researches, and I cannot imagine the present book can in any way change this fact. Indeed, it is based on a highly controversial set of ideas that have not reached any level of acceptance by the physics community (as far as I can tell), and should thus be presented with much more caution than is done in the present work; a 150 page monograph at most might have been useful for practicing researchers. In particular, I would not suggest to any physics student to read this book.


Shinji Mukoyama
email: shinji.mukohyama@ipmu.jp

Jean-Philippe Uzan
email: uzan@iap.fr

EMERGENCE OF THE LORENTZIAN STRUCTURE IN CLASSICAL FIELD THEORY *

Keywords: Field theory, gravitation, Lorentz signature, emergence PACS Number(s): 03.50.-z, 04.20.Cv, 04.50.Kd

The Lorentzian metric structure allows one to implement the relativistic notion of causality in any field theory and to define a notion of time dimension. We propose that at the microscopic level the metric is Riemannian and that the Lorentzian structure, usually thought as fundamental, is in fact an effective property, that emerges in some regions of a 4-dimensional space with a positive definite metric. We argue that a decent classical field theory for scalars, vectors and spinors in flat spacetime can be constructed, and that gravity can be included under the form of a covariant Galileon theory instead of general relativity.

Introduction

Part of the art of theoretical physics is to find mathematical structures that allow to formalize and "decomplexify" the laws of nature. These structures include the description of spacetime (dimension, topology, etc.), of the matter and interactions (fields, symmetries, etc.). While there is a large freedom in the choice of these mathematical structures, the developments in theoretical physics taught us that some of them are better-suited to describe some classes of phenomena. However, these choices are only validated by the mathematical consistency of the theory and, in the end, by the agreement of their consequences with experiments. Some structures, considered as fundamental in the domain of validity of a theory, can be replaced by other structures in a new description, e.g. the spacetime of Newtonian physics can be formally seen as the limit of the Minkowski structure in regimes, where v ≪ c. In this respect, the role of fundamental constants as concept synthesizers or as limiting quantities is central and signal the need for new mathematical structures in a new regime. 1,2 At each step, some properties, such as the topology of space, the number of spatial dimensions or the numerical values of the free parameters that are the fundamental constants, may remain ap r i o r ifree in one framework, or imposed in another framework (e.g. the number of space dimensions is fixed in string theory 3 ). It may even be that different structures can reproduce what we know about physics and one has to rely on less-defined criteria, such as simplicity and economy, to choose between them.

Among all these structures, and in the framework of metric theories of gravitation, the signature of the metric is in principle arbitrary. Indeed, it seems that on the scales that have been probed so far, there is a need for only one time dimension and three spatial dimensions. It is also now universally accepted that the relativistic structure is a central ingredient of the construction of any realistic field theory, in particular as the cleanest way to implement the notion of causality. Spacetime enjoys a local Minkowski structure and, when gravity is included, the equivalence principle implies (this is not a theoretical requirement, but an experimental fact, required at a given accuracy) that all the fields are universally coupled to the same Lorentzian metric. Thus, we now take for granted that the spacetime is 4-dimensional manifold endowed with a metric of signature (-, +, +, +).

While the existence of two time directions may lead to confusion, 4,5 several models for the birth of the universe [6][7][8][9] are based on a change of signature via an instanton in which a Riemannian and a Lorentzian manifolds are joined across a hypersurface which may be thought of as the origin of time. While there is no time in the Euclidean region, where the signature is (+, +, +, +), it flips to (-, +, +, +). Eddington even suggested 10 that it can flip across some surface to (-, -, +, +) and signature flips also arise in brane or loop quantum cosmology. [START_REF] Gibbons | [END_REF][12][START_REF] Mielczarek | Signature change in loop quantum cosmology[END_REF] It is legitimate to investigate whether the signature of the metric is only a convenient way to implement causality or whether it is just a property of an effective description of a microscopic theory in which there is no such notion. What does it take to construct a realistic field theory in a positively definite metric? If one considers a scalar field χ and assumes that its Lagrangian is a scalar and leads to second-order equations, the only terms that can be included are a "kinetic term", 1 2 δ µν ∂ µ χ∂ ν χ, and a potential term V (χ). As a consequence its field equation is elliptic, determining a static configuration.

In order to make dynamics emerge locally, we introduce a scalar field φ such that (1) its derivative has a nonvanishing vacuum expectation value (VEV) in region M 0 of the Riemannian space (see Fig. 1) and ( 2) that it couples to all other fields.

The first condition is implemented by assuming that ∂ µ φ =c o n s t . =0i nM 0 . We can thus set ∂ µ φ = M 2 n µ in M 0 with n µ a unit constant vector. Its norm

X E ≡ δ µν ∂ µ φ∂ ν φ = M 4 is constant and satisfies X E > 0, in M 0 .
(2.1)

Fig. 1. Example of a spatial configuration of the clock field. Locally, one can define regions such as M 0 , M ′ 0 and M ′′ 0 , in each of which a time direction emerges. Indeed, this direction does not preexist at the microscopic level and can be different from patches to patches. Now, under this assumption, one of the coordinates of the Euclidean space can be chosen as dt = n µ dx µ , i.e.

t ≡ φ M 2 , (2.2)
up to a constant that can be set to zero without loss of generality and we introduce three independent coordinates x i (i =1,...,3) on the hypersurfaces Σ t normal to n µ . The second condition is implemented by considering that χ couples only to ∂ µ φ, so that one can consider the action

S χ = d 4 x - 1 2 δ µν ∂ µ χ∂ ν χ -V (χ)+ 1 M 4 (δ µν ∂ µ φ∂ ν χ) 2 . (2.3)
On M 0 where the condition (2.2) holds, it is straightforward to deduce that

S χ → dtd 3 x 1 2 (∂ t χ) 2 - 1 2 δ ij ∂ i χ∂ j χ -V ≡ dtd 3 x - 1 2 η µν ∂ µ χ∂ ν χ -V . (2.4)
On M 0 , the action (2.3) thus describes the dynamics of a scalar field propagating in an effective 4-dimensional Minkowski spacetime with metric η µν = diag(-1, +1, +1, +1). The apparent Lorentzian dynamics, with a preferred time direction, results from the coupling to φ. As a consequence, φ is related to what we usually call "time", so that we shall call it a clock field.

Field theory in flat spacetime

In our example, the clock field allows for the emergence of an effective Lorentzian dynamics because the scalar field is actually coupled to the metric ĝµν = δ µν -2 M 4 δ µα δ νβ ∂ α φ∂ β φ that reduces M 0 to η µν . Let us now show 14 that it can be generalized to fields of any spin.

First, consider a vector field A µ with Faraday tensor

F µν = ∂ µ A ν -∂ ν A
S A = 1 4 d 4 x -F µν F µν E + 4 M 4 F µρ E F ν Eρ ∂ µ φ∂ ν φ →- 1 4 dtd 3 xη µα η νβ F µα F νβ . (2.5)
Because of the coupling to the clock field, the vector field propagates effectively in a Minkowski metric and we recover the standard Maxwell action. The generalizations to a non-Abelian group and to charged scalar fields are straightforward. [START_REF] Mukohyama | [END_REF] Again, as for the scalar field, the action is equivalent to a minimal coupling to the effective metric ĝµν . The case of spinors is more subtle because one needs to define the Clifford algebra compatible with the Euclidean metric (see Ref. 14 for details of the construction), that is the matrices γ µ E satisfying {γ µ E ,γ ν E } = -2δ µν .T h e yc a nb e obtained from the standard γ-matrices as γ 0 E ≡ iγ 5 and γ i E ≡ γ i . The matrices

S µν E ≡ i 4 [γ µ E ,γ ν E ]
then satisfy the algebra of SO(4) rotation generators. Hence, the SO(4) rotation for Dirac field ψ is

ψ → Λ E, 1 2 ψ, Λ E, 1 2 =exp - i 2 ω E µν S µν E , (2.6)
where ω E µν are real numbers. Since all S µν E are Hermitian, Λ E, 1 2 is unitary. This implies that ψ → ψΛ -1

E, 1 2 and ψ † → ψ † Λ -1 E, 1 2
so that both ψψ and ψγ 5 E ψ (= ψ † ψ) are scalars under a SO(4).

The dynamics is obtained by starting from the action for a Euclidean Dirac spinor, ψ( i

2 γ µ E ← → ∂ µ -m)ψ.
Assuming that the clock field φ has derivative couplings to ψ, there are six Hermitian bilinear coupling terms that transform as scalar under SO(4) and that do not include more than one derivative acting on spinors. Among them, it is sufficient to consider δ µν (i ψγ 5

E ← → ∂ µ ψ)∂ ν φ and (i ψγ ρ E ← → ∂ µ ψ)∂ ρ φ∂ ν φ.I ti s then easily shown that on M 0 S ψ = dx 4 ψ i 2 γ µ E ← → ∂ µ -m ψ + δ µν 2M 2 [(i ψγ 5 E ← → ∂ µ ψ) -(i ψγ ρ E ← → ∂ µ ψ)∂ ρ φ]∂ ν φ → dx 4 ψ i 2 γ 0 ← → ∂ 0 + i 2 γ i ← → ∂ i -m ψ.
(2.7)

The coupling to the clock field implies that ψ effectively propagates in an effective Lorentzian metric and we recover the standard Minkowskian Dirac action with the algebra for the γ-matrices, {γ µ ,γ ν } = -2η µν . Note that the interpretation in terms of the effective metric ĝµν cannot be extended to spinors.

Gravity and physics in curved spacetime

It is possible to extend this construction to gravity. For this purpose, we now consider a general 4-dimensional Riemannian manifold M with a positive definite metric g E µν . To minimize the number of degrees of freedom, we demand that the equation of motion for φ is second-order. Hence, its action is restricted to the Riemannian version of the Horndeski theory 15 with shift symmetry. Equivalently, it is given by the shift-symmetric generalized Galileon. 16,17 For the effective equations, i.e. once the Lorentzian structure has emerged, we would like to ensure that the system is invariant not only under time translation but also under charge, parity and time reversal (CPT). For this reason, we require that besides the shift symmetry (φ → φ +const.) the theory also enjoys a Z 2 symmetry (φ →-φ) for the clock field action. With these symmetries, the Riemannian action reduces to

S g = dx 4 √ g E {G 4 (X E )R E + K(X E ) -2G ′ 4 (X E )[(∇ 2 E φ) 2 -(∇ E µ ∇ E ν φ) 2 }, (2.8)
where

X E ≡ g µν E ∂ µ φ∂ ν φ. As demonstrated in Ref. 14, it reduces on M 0 to S g = dx 4 √ -g{f (X)R +2f ′ (X)[(∇ 2 φ) 2 -(∇ µ ∇ ν φ)(∇ µ ∇ ν φ)] + P (X)}, (2.9)
in terms of X = -g µν ∂ µ φ∂ ν φ,w h e r eg µν is the emergent Lorentzian metric and where f and P are two functions related to G 4 and K.T h i sac t i on( 2.9) is a special case of the covariant Galileon 16 and the equations of motion are second-order.

It can then be shown 14 that the action for scalar and vector fields can easily be generalized to curved spacetime, while the case of spinor is still an open question.

Discussion

By introducing couplings of the standard fields to a clock field, we have shown that a Lorentzian dynamics, including gravity, can emerge on a patch M 0 of a Riemannian space. This goes far beyond earlier attempts. 18 We thus propose that not only the Lorentz invariance 19 but also the Lorentzian signature may be an emergent phenomena at long distances. At short distances, on the other hand, higher derivative corrections compatible with the Riemannian diffeomorphism become important and the theory is genuinely Riemannian.

In flat spacetime, we can describe scalars, vectors and spinors and hence the whole standard model of particle physics. It is important to emphasize that: (1) it is limited to classical field theories; (2) when restricted to M 0 all fields propagate in the same effective Minkowski metric so that the equivalence principle is safe in first approximation; (3) indeed, the couplings to the clock field have been tuned for that purpose since the action (2.3) could have been chosen as

S χ = d 4 x - κ χ 2 δ µν ∂ µ χ∂ ν χ -V (χ)+ α χ 2M 4 (δ µν ∂ µ φ∂ ν χ) 2
and a Lorentzian signature is recovered, only if α χ >κ χ > 0. In the case where these constants are not tuned, different fields can have different lightcones; (4) In the bosonic sector, since the theory is invariant under the Euclidean parity (x µ →-x µ ) and field parity (φ →-φ), both P and T invariances in the Lorentzian theory are ensured. This explains why we have included only quadratic terms in ∂ µ φ in the actions for scalars and vectors; (5) In the fermionic sector, one of the terms in the Euclidean Dirac action and one of the coupling terms are not CPT invariant after the clock field has a VEV, unless they cancel each other; and (6) The configuration of the clock field is not arbitrary, but determined by solving its equation of motion. Since its action enjoys a shift symmetry, it shall take the form of a current conservation.

In curved spacetime, gravity takes the form of a covariant Galileon theory, that depends on two free functions constrained 14 by the stability of Friedmann-Lemaître background with respect to linear scalar and tensor perturbations. For the matter sector, the actions for scalar and vector fields are easily generalized and each depends on two free parameters (κ, α) that are allowed to be functions of X E in general but may as well be assumed constant. Besides, there is an environmental parameter which characterizes the clock field configuration on the patch M 0 . The emergent model has the following properties: (1) it induces two components which respectively behave as dark matter and dark energy when considering the dynamics of a homogeneous cosmology, similar to ghost condensates. 20 This sets two constraints for the cosmology to be consistent with standard cosmology, at least at the background level; (2) In general, scalars and vectors propagate in two different effective metrics. In order for the weak equivalence principle to hold, we have to impose that these two metrics coincide. In the simplest situation in which the coefficients (κ, α) are assumed to be constant, one only requires a tuning on the parameters of the Lagrangians, but then it is satisfied whatever the configuration of the clock field may be. In this sense, the tuning is not worse than the usual assumption that all the fields propagate in the same metric; (3) In general, the parameters entering our effective Lorentzian actions are environmentally determined. This means that if X E is not strictly constant on M 0 , fundamental constants may be spacetime dependent, which is strongly constrained 1,2 ; and (4) The speeds of light and of graviton may not coincide, which is constrained by the observations of cosmic rays [21][22][23] because particles propagating faster than gravity waves emit gravi-Cerenkov radiation.

To conclude, from a theoretical point of view, our construction gives a new insight into the need for Lorentzian metric as a fundamental structure. As we have shown, this is not a mandatory requirement and a decent field theory, at least at the classical level, it can be constructed from a Riemannian metric. Such a formalism may be fruitful in the debate on the emergence of time and, speculating, for the development of quantum gravity.

It also opens up a series of questions and possibilities. We can list: (1) the construction of Majorana and Weyl spinors; (2) the development of a quantum theory; (3) the possibility from the classical viewpoint that singularities in our local Lorentzian region may be related to singularities in the clock field (e.g. similar to topological defects) and not in the metric of the Euclidean theory; and (4) the possibility that a de Sitter spacetime may be an "illusion" in an anti-de Sitter Riemannian space. It then follows that a Euclidean AdS/CFT correspondence at the microscopic level would reveal itself as a dS/CFT correspondence in our effective Lorentzian universe. These are, for now, speculations but they illustrate that this framework may be fruitful for extending our current field theories, including general relativity.

2 .

 2 Emergence of the Lorentz Signature 2.1. Clock field Let us start by physics in flat spacetime and consider a 4-dimensional Riemannian manifold M with a positive definite Euclidean metric g E µν = δ µν in Cartesian coordinates. The theories on this manifold have thus no natural concept of time.

  µ and its standard Maxwell action -F µν F µν E /4 (where the subscript E indicates that the indices are raised with the Euclidean metric δ µν ) and assume that it couples only as[START_REF] Bars | [END_REF] M 4 F µρ E F ν Eρ ∂ µ φ∂ ν φ.I ti se a s i l ys h o w nt h a to nM 0


A Fedotov
email: a.fedotov@spbu.ru

F Klopp

Complex WKB method for difference equations with meromorphic coefficients

We discuss one-dimensional difference Schrödinger equations in the complex plane. To study their solutions in the quasiclassical limit, V. Buslaev, A. Fedotov and E. Shchetka developed an analog of the complex WKB method. We consider the equation with a potential having a simple pole and study the asymptotics of its solutions in a neighborhood of this pole.

Introduction

We consider the difference equation

ψ(z + h)+ψ(z -h)+v(z)ψ(z)=0, ( 1
)
where z is the complex variable, v is a given analytic or meromorphic function, and h>0i st h e translation parameter. One encounters such equations, for example, in solid state physics when studying, say, an electron in a crystal in a weak magnetic field. The translation parameter h is proportional to the magnetic flux through the periodicity cell. In solid state physics, one has v(z)=w(z) -E, where the function w is called the potential and the parameter E is called the spectral parameter. The equation with w(z) = cos(z) is the famous Harper equation, see, e.g., [START_REF] Wilkinson | Critical properties of electron eigenstates in incommensurate systems[END_REF]; the equation with w(z)=tan(z) is a close relative of the well-known Maryland equation introduced by D. Grempel, S. Fishman and R. Prange in [START_REF] Grempel | Localization in an incommensurate potential: An exactly solvable model[END_REF].

Difference equations with the small translation parameters arise also when studying the scattering of waves by wedge-shaped domains in the framework of the Sommerfeld-Malyuzhinets method. The translation parameters appear to be proportional to the angles of the wedges, see [START_REF] Babich | Diffraction theory: the Sommerfeld-Malyuzhinets Technique[END_REF].

We study the asymptotics of solutions to (1) as h → 0. Since formally exp h d dz Ψ(z)=Ψ(z + h), the parameter h in (1) can be regarded as a small parameter in front of the derivative and, thus, appears to be a standard quasiclassical parameter.

To study the one-dimensional differential equations in the quasiclassical limit, one uses the classical complex WKB method, see [START_REF] Fedoryuk | Asymptotic Analysis. Linear Ordinary Differential Equations[END_REF]. To study onedimensional difference equations with analytic coefficients, in papers [START_REF] Buslaev | The complex WKB method for Harper's equation[END_REF][START_REF] Fedotov | Complex WKB method for the difference Schrödinger equations with the potentials being trigonometric polynomials[END_REF] their authors developed an analog of the complex WKB method.

In this paper, we consider the case of meromorphic v. To be more precise, we assume that B 0 is a neighborhood of z =0( h e r ea n db e l o wa neighborhood of a point is an open disc centered at this point), and that v is analytic in B 0 \{0} and has a simple pole at zero. Let ψ be a solution to [START_REF] Wilkinson | Critical properties of electron eigenstates in incommensurate systems[END_REF] analytic in B 0 \ R + (R + denotes the set of positive real numbers). Equation [START_REF] Wilkinson | Critical properties of electron eigenstates in incommensurate systems[END_REF] implies that ψ(z)=-ψ(z -2h) -v(z -h)ψ(z -h). Therefore, for sufficiently small h, ψ can be meromorphically continued into B 0 . After that, it can have poles at the points z = h, 2h, 3h... When h is small, these points become close one to another. We describe the quasiclassical asymptotics in B 0 of solutions to (1) having poles at z = h, 2h, 3h...

A brief introduction to the complex WKB method

Let us briefly describe the main constructions of the complex WKB method for the difference equation (1) with an analytic coefficient v. We note that formally this equation can be written in the form

(2 cos p + v(z))ψ(z)=0, p = -ih d dz . (2)
We define the complex momentum p by the formula 2cos p(z)+v(z)=0.

It is an analytic multivalued function. Its branch points satisfy the relations ±2+v(z) = 0. We call a subset D of the domain of analyticity of v regular, if v(z) = ±2i nD.

Let D be a simply connected regular domain, and let p be a branch of the complex momentum analytic in D. All the other branches of p that are analytic in D are of the form ±p(z)+2πm, m ∈ Z.

The complex momentum is the main analytic object of the complex WKB method. In terms of the complex momentum, one defines the canonical domains that are the main geometric objects of the method. The definition of the canonical domains can be found in [START_REF] Buslaev | The complex WKB method for Harper's equation[END_REF][START_REF] Fedotov | Complex WKB method for the difference Schrödinger equations with the potentials being trigonometric polynomials[END_REF]. Here, we note that the canonical domains are regular and simply connected, and that one has the following assertion.

Theorem 1 Any regular point is contained in a canonical domain.

This statement is an analog of Lemma 5.3 from [START_REF] Fedotov | On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit[END_REF]. The principle result of the method is Theorem 2 ([5]) Let K ⊂ C be a canonical domain, let z 0 ∈ K,a n dl e tp be a branch of the complex momentum analytic in K. For sufficiently small h,t h e r ee x i s t sψ, a solution to (1), analytic in K and such that in K as h → 0

ψ(z)= e i h z z0 p(z) dz + o(1) sin(p(z)) . ( 3
)
The asymptotics is locally uniform in z.

A continuation principle

Let z 0 be a regular point, and let V 0 be its regular neighborhood. Assume that there is ψ,as o l u t i o n to (1) that is analytic in V 0 and admits in V 0 the uniform asymptotic representation (3). There exist general statements (Continuation principles) allowing to describe the asymptotics of ψ outside of V 0 .O n eo ft h e mi s Theorem 3 (The rectangle lemma)

Let z 1 ∈ V 0 . Consider the straight line L = {z ∈ C : Im z =I m z 1 }.L e t z 2 ∈ L, Re z 2 > Re z 1 ,a n d let the segment [z 1 ,z 2 ]={z ∈ L :R e z 1 ≤ Re z ≤ Re z 2 } be regular. If Im p(z) < 0 along [z 1 ,z 2 ], then the asymptotic representation (3) is valid and uniform in an independent of h regular neighborhood of [z 1 ,z 2 ].
This theorem is an analog of Lemma 5.1 from [START_REF] Fedotov | On the absolutely continuous spectrum of one dimensional quasi-periodic Schrödinger operators in the adiabatic limit[END_REF]. It roughly says that the asymptotic formula (3) stays

1: B 0 , V 0 ,t h eδ-neighborhood of R + (the δ-neighborhood is shaded in).
valid along a horizontal line as long as ψ exponentially grows.

Typical quasiclassical formulation of the problem

For the sake of simplicity, we additionally assume that v(z) ∈ R for z ∈ R. In this case, if B 0 is chosen sufficiently small, then: (1) in B 0 , v(z) ∈ R only for z ∈ R; (2) the set B 0 \{0} is regular. We assume that B 0 has these two properties. Let p b eab r a n c ho ft h ec o m p l e xm o m e n t u m analytic in B 0 \ R + .F o rB 0 under consideration, Im p(z)d oe sn o tv a n i s hi nB 0 \ R + . For the sake of definiteness, we assume that it is negative.

In B 0 we pick z 0 < 0, see Fig. 1. In view of Theorem 1, there is a regular neighborhood V 0 ⊂ B 0 of z 0 such that there exists a solution ψ to equation ( 1) that is analytic in V 0 and admits representation (3) there.

Let δ>0 be sufficiently small. Let B δ be the domain B 0 without the δ-neighborhood of R + . As Im p(z) < 0i nB 0 \ R + , then by Theorem 3 asymptotics (3) is valid and uniform in B δ to the right from V 0 . The problem is to describe ψ in the δ-neighborhood of R + .

The main result

Below, instead of saying that a function can be analytically continued into a domain, we say that it is analytic in this domain.

The complex momentum p has a logarithmic branch point at zero. One can check the following assertion.

Lemma 1 In B 0 \ R + we fix analytic branches of ln and p. The function z → p(z) -i ln z is analytic in B 0 . The function z → z sin p(z) is analytic and does not vanish in B 0 .

For z ∈ B 0 \ R + , we set U 0 (z)= h -2πz sin p(z) × exp z h ln 1 h + i h z 0 (p(z) -i ln(-z)) dz .
Here p, √ sin p, z → ln(-z)andz → √ -z are functions analytic in B 0 \ R + .I n V 0 , p and √ sin p coincide with p and √ sin p in (3). The functions z → ln(-z)a n dz → √ -z satisfy the conditions ln(-z)| z=-1 = 0 and √ -z| z=-1 = 1. By Lemma 1, U 0 is analytic in B 0 .

Our main result is Theorem 4 Let δ>0 be sufficiently small. In the δ-neighborhood of R + ,a sh → 0, the solution ψ admits the uniform asymptotic representation

ψ(z)=Γ 1 - z h U 0 (z) e i h 0 z0 pdz + o(1) . (4)
So, the special function describing the asymptotic behavior of ψ near the poles generated by the pole of the potential at z = 0 is the Euler Γ-function.

Near the point z = 0, the formula (4) can not be simplified. For large values of |z/h|, the Γ-function in (4) can be replaced with its asymptotics. To be more precise, let us pick ǫ>0. By means of the asymptotic formula is analytic in a neighborhood of zero independent of h, say, a disc of radius δ>0. Therefore, to prove (4), due to the maximum principle, it suffices to check that f (z)=1+o(1) for, say, |z| = δ/2. This is done by means of a rather standard asymptotic computation made using the complex WKB method for difference equations.

Γ(1 + ζ)= 2πζ e ζ(ln ζ-1)+o(1) , |ζ|→∞, (5

Figure

  Figure 1: B 0 , V 0 ,t h eδ-neighborhood of R + (the δ-neighborhood is shaded in).

  ) that is uniform in the sector | arg ζ|≤π -ǫ,o n e checks that if | arg z -π|≤π -ǫ and, say, |z|≥δ/2, representation (4) turns into (3). Now let us discuss the case where |z|≥δ/2a n d | arg z|≤ǫ. In this case, to simplify (4), first we use the relation Γ(1z) dz + o(1) (1 -e -2πiz/h ) sin(p(z)) ,h → 0, where p, z → z 0 p(z) dz and sin(p)a r eo b t a i n e d by analytic continuation in the anticlockwise direction from B 0 \ R + into the sector under consideration. 6 The ideas of the proof Let us note that the function f (z)=ψ(z)/(Γ(1 -z/h)U 0 (z))

Acknowledgments

This work was supported by the CNRS and the Russian Foundation of Basic Research under the French-Russian grant 17-51-150008.