periodic limit of inverse scattering i t is well known that a p periodic potential q x can be reconstructed from spectral data of the corresponding hill operator d2 dx2 q x in terms of a riemann function we regard the periodic potential q x as the pointwise limit of a scattering potential qn c x defined to equal q x when np x np to equal zero when x np as n and c2 the scattering potential qn c x can be recovered from the scattering data of the corresponding schrodinger operator in terms of a dyson determinant according to a well known theory we derive the riemann function corresponding to the periodic potential q x by taking the above limit of the dyson determinant for the scattering potential we first calculate the scattering data of the potential qn c x through recursive formulas in terms of the left transmission and reflection coefficients t and r of the potential which is equal to q x when 0 x p and equal to zero otherwise we use these data to express the dyson determinant of qn c x we then expand the dyson determinant into a fredholm series and compute the main contributions to the expansion in the asymptotic limit n and c2 using a method developed by lax levermore and venakides in their study of the small dispersion limit of the initial value problem of korteweg de vries equation the computation of the leading order contributions reduces to a quadratic functional maximization problem constrained by a positivity condition and by a mass quantization condition the solutions to this maximization problem constitute the differentials on a riemann surface the main ingredients for the riemann function corresponding to the periodic potential the limit of the dyson determinant for qn c x as n and c2 is shown to equal the exact riemann function corresponding to the periodic potential q x times an exponential function with exponent being a quadratic polynomial in x our calculation includes the correct phase shifts of the function 1993 john wiley sons inc

discriminant transmission coefficient and stability bands of hill s equation the discriminant k2 of hill s equation is shown to be related to the transmission coefficient t k ei k of one period of the potential by k2 2 t k cos k k this result is used to find the boundaries of the stability bands

the non abelian specker group is free abstract the paper investigates a group h which can be constructed as a subgroup of the inverse limit of the finitely generated free groups fn by taking only those elements g lim f n for which there exists some bound b g such that none of the generators of any of the groups fn occurs more often than b g times in any of the entries of at least one sequence of fn elements describing g by using a similar condition h can also be described as a subgroup of the fundamental group of the hawaiian earrings despite not having any obvious candidate for a free basis h is proven to be free by a non constructive basis selection method since h is related to lim f n the same way as the classical specker group is related to z n we decided to call h the non abelian specker group

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

the number of extensions of a number field with fixed degree and bounded discriminant we give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant x these bounds improve on work of schmidt we also prove various related results such as lower bounds for the number of extensions and upper bounds for galois extensions

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

on geometric phases for soliton equations this paper develops a new complex hamiltonian structure forn soliton solutions for a class of integrable equations such as the nonlinear schrodinger sine gordon and korteweg de vries hierarchies of equations that yields amongst other things geometric phases in the sense of hannay and berry for example one of the possible soliton geometric phases is manifested by the well known phase shift that occurs for interacting solitons the main new tools are complex angle representations that linearize the corresponding hamiltonian flows on associated noncompact jacobi varieties this new structure is obtained by taking appropriate limits of the differential equations describing the class of quasi periodic solutions a method of asymptotic reduction of the angle representations is introduced for investigating soliton geometric phases that are related to the presence of monodromy at singularities in the space of parameters in particular the phase shift of interacting solitons can be expressed as an integral over a cycle on an associated riemann surface in this setting soliton geometric asymptotics are constructed for studying geometric phases in the quantum case the general approach is worked out in detail for the three specific hierarchies of equations mentioned some links with functions the braid group and geometric quantization are pointed out as well

geometry of kdv 4 abel sums jacobi variety and theta function in the scattering case the study of the general projective curve x defined by the vanishing of an irreducible polynomial p x y originated in the discovery of addition formulas for integrals v r x y dx of rational functions on such a curve

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

counting primitive points of bounded height let k be a number field and k a finite extension of k we count points of bounded height in projective space over the field k generating the extension k k as the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension k k in a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over k we also introduce the notion of an adelic lipschitz height generalizing that of masser and vaaler this will lead to further applications involving points of fixed degree on linear varieties and algebraic numbers of fixed degree satisfying certain subfield conditions

counting maximal arithmetic subgroups we study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semisimple lie group using an extension of the method developed by borel and prasad

the infinite period limit of the inverse formalism for periodic potentials soit l operateur de hill h p d 2 dx 2 v p x quand v p x est une fonction p periodique de x on montre que les formules pour la solution du probleme inverse ont une limite si on prend v p comme une repetition periodique d une fonction a support compact v x et que l on fait tendre p vers l infini