scattering amplitude and algorithm for solving the inverse scattering problem for a class of non convex obstacles abstract consider a class r1 of non convex smooth obstacles b such that there exist a number of directions sj sj j 1 2 n uniformly distributed on the unit sphere with the following property the rays incident in the directions sj are reflected by the obstacle once according to geometrical optics laws it is also assumed that the obstacles in r1 satisfy a non trapping condition for b r1 the high frequency asymptotics of the scattering amplitude f s s0 k is studied and analytical formulas are derived for determining the surface of the obstacle from the scattering amplitude known at a fixed large wave number k and for the directions s0 and s of the incident and scattered waves in solid angles around the directions sj j 1 n the investigation of the scattering amplitude and the solution to the inverse scattering problem are based on approximate formulas for f derived from the kirchhoff approximation and these formulas are justified for non convex obstacles of class r1

phase space lattices with threefold symmetry this paper presents a semiclassical analysis of the spectrum of the hamiltonian ir cos p e cos ci v 3 5 cos f p a 5 which is a model for bloch electrons in a magnetic field the energy levels are determined to a first approximation by bohr sommerfeld quantisation because of the translational symmetry in phase space this leads to a lattice of infinitely degenerate states this degeneracy is lifted by tunnelling effects which can be described by an effective hamiltonian of the same form as that above but with different values of 0 and h the calculation of the effective hamiltonian therefore defines a renormalisation group transformation and it predicts that the spectrum has a complex recursive structure which is confirmed by numerical experiments the results complement earlier work on lattices with fourfold symmetry in which a similar type of spectrum occurs the greater complexity of the problem in lattices with threefold symmetry necessitated developing simpler canonically invariant methods of analysis

an exact renormalisation group for bloch electrons in a magnetic field bloch electrons in a magnetic field can be modelled by a one dimensional hamiltonian h h x p which is periodic in x and p when beta the ratio of planck s constant h to the area of a unit cell is a rational number p q the spectrum is a set of q bloch bands if beta is perturbed away from this rational value the spectrum becomes a cantor set but it is still possible to define a set of generalised bloch waves which form a complete basis set for a given band projecting these states into the band and taking matrix elements of the hamiltonian leads to a new effective hamiltonian h1v x p and planck constant h v describing the spectrum of the vth band since degrees of freedom have been eliminated by projecting into a band this is a renormalisation group rg transformation the rg transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a liouville number the measure of the spectrum vanishes if h x p has centres of threefold or fourfold symmetry in the phase plane and the rg transformation explains the hierarchical structure of the spectrum observed in these cases

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

on semiclassical dispersion relations of harper like operators we describe some semiclassical spectral properties of harper like operators i e of one dimensional quantum hamiltonians periodic in both momentum and position the spectral region corresponding to the separatrices of the classical hamiltonian is studied for the case of integer flux we derive asymptotic formulae for the dispersion relations the width of bands and gaps and show how geometric characteristics and the absence of symmetries of the hamiltonian influence the form of the energy bands

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

hierarchical clustering in the spectra of incommensurate systems using degenerate perturbation theory to all orders the authors derive the hofstadter cluster rules 1976 for the spectrum of harper s equation 1955 they also extend their method to calculate the fractal dimension df of the spectrum and speculate that df 1 2 for typical values of the incommensurability parameter phi the clustering rules are also shown to be valid for a one dimensional model of quasicrystals

an exact effective hamiltonian for a perturbed landau level considers the effect of a scalar potential v x y on a landau level in two dimensions an exact effective hamiltonian is derived which describes the effect of the potential on a single landau level expressed as a power series in v ec where ec is the cyclotron energy the effective hamiltonian can be represented as a function h x p in a one dimensional phase space the function h x p resembles the potential v x y when the area of a flux quantum is much smaller than the square of the characteristic length scale of v then h approximately v also h x p retains the translational and rotational symmetries of v x y exactly but reflection symmetries are not retained beyond the lowest order of the perturbation expansion

on the numerical approximation of high frequency acoustic multiple scattering problems by circular cylinders the aim of this paper is to propose a numerical strategy for computing the solution of two dimensional time harmonic acoustic multiple scattering problems at high frequency the scatterers are assumed to be circular leading therefore to semi analytical representation formulae of the scattered field through the solution of a large linear system of equations taking advantage of the special block toeplitz structure of the matrix of the linear system a fast iterative and preconditioned numerical method yielding large memory savings is proposed several numerical experiments for general configurations are presented to show the efficiency of the numerical method

general quantization of canonical maps on a two torus canonical maps on a two torus in phase space are quantized under most general conditions recent results by keating et al 1999 nonlinearity 12 579 are thus fully extended in two directions a the translational component of a general canonical map is included in the quantization b all values of planck s constant consistent with the toral boundary conditions bcs are considered generically these values are rational numbers whose numerator must satisfy a number theoretical condition besides the condition on planck s constant the quantization is possible only for particular allowed bcs on the torus the general equation determining these bcs is derived allowed bcs may not exist in some cases representative examples are the irrational skew translations and kronecker maps exact versions of egorov s theorem are shown to hold under some conditions composition and representation properties of the quantization scheme are studied

on the green function of the almost mathieu operator the square tight binding model in a magnetic field leads to the almost mathieu operator which for rational fields reduces to a q q matrix depending on the components of the wavevector in the magnetic brillouin zone we calculate the corresponding green function without explicit knowledge of eigenvalues and eigenfunctions and obtain analytical expressions for the diagonal and the first off diagonal elements the results which are consistent with the zero magnetic field case can be used to calculate several quantities of physical interest e g the density of states over the entire spectrum impurity levels in a magnetic field

coupling of dirichlet to neumann boundary condition and finite difference methods in curvilinear coordinates for multiple scattering the applicability of the dirichlet to neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape the original boundary value problem bvp for the multiple scattering problem is reformulated as an interface bvp a heterogenous medium with variable physical properties in the vicinity of the obstacles is considered a rigorous proof of the equivalence between these two problems for smooth interfaces in two and three dimensions for any finite number of obstacles is given the problem is written in terms of generalized curvilinear coordinates inside the computational region then novel elliptic grids conforming to complex geometrical configurations of several two dimensional obstacles are constructed and approximations of the scattered field supported by them are obtained the numerical method developed is validated by comparing the approximate and exact far field patterns for the scattering from two circular obstacles in this case for a second order finite difference scheme a second order convergence of the numerical solution to the exact solution is easily verified

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

on the multifractal spectrum of the fibonacci chain we consider the multifractal properties of a quasiperiodic tight binding hamiltonian where the hopping elements are arranged according to the fibonacci chain by using the trace map approach and an assumption relating the cycles of the trace map to the integrated density of states it is shown that the maximal scaling of the spectrum does not always occur at the edges or the centre of the spectrum which is confirmed by numerical simulations a good description of the multifractal spectrum is obtained by applying the 3z model which was recently developed for the harper model

zero hausdorff dimension spectrum for the almost mathieu operator we study the almost mathieu operator at critical coupling we prove that there exists a dense g delta set of frequencies for which the spectrum is of zero hausdorff dimension

spectrum and bandwidth of an exactly soluble incommensurate model the structure of the energy spectrum of the exactly soluble maryland model is examined in detail for both incommensurate and commensurate modulations the study is motivated by a formal similarity with harper s equation describing crystal electrons in a uniform magnetic field an exact expression is derived for the characteristic polynomial which determines the energy eigenvalues allowing explicit verification of the transition from band structure for commensurate modulations to a point spectrum in the commensurate limit an application is the calculation of the total width of the bands a semiclassical argument is used to assess the relevance of the results to models of physical interest

an improved surface radiation condition for high frequency acoustic scattering problems a new artificial boundary condition for two dimensional acoustic scattering problems is constructed in this paper it is designed from the principal symbol of the dirichlet to neumann map so that both the propagating and evanescent fields are properly modeled in the high frequency regime the new artificial boundary condition can be used as an on surface radiation condition for efficient approximation of the wave field the accuracy and efficiency of our method are demonstrated in numerical examples

on surface radiation condition for multiple scattering of waves the formulation of the on surface radiation condition osrc is extended to handle wave scattering problems in the presence of multiple obstacles the new multiple osrc simultaneously accounts for the outgoing behavior of the wave fields as well as the multiple wave reflections between the obstacles like boundary integral equations bie this method leads to a reduction in dimensionality from volume to surface of the discretization region however as opposed to bie the proposed technique leads to boundary integral equations with smooth kernels hence these fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities moreover under weak scattering conditions this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration inherited from the original osrc the proposed multiple osrc is generally a crude approximate method if accuracy is not satisfactory this approach may serve as a good initial guess or as an inexpensive pre conditioner for krylov iterative solutions of bie

on the inverse diffraction problem abstract it is proved that for a convex reflecting scatterer the support function is uniquely determined by the scattering amplitude in particular by the backscattering amplitude the support function determines the position orientation and the shape of the scatterer uniquely stability of the solution is discussed

generalized wannier function and renormalization of harper s equation harper s equation a model for bloch electrons in a magnetic field has a band spectrum when the dimensionless magnetic field beta is a rational number p q this paper considers the definition of generalized wannier functions which can be used to represent the bloch bands of the spectrum of the rational harper equation by means of a von neumann lattice one representation of these bloch bands can be extended to the irrational case and taking matrix elements in this basis leads to a renormalization group transformation acting on the hamiltonian the results in the present paper considerably extend a previous analysis of this renormalization group transformation in that the formalism is suitable for systematic calculations of the renormalized hamiltonian and that the transformation preserves the rotational symmetry of the harper hamiltonian in phase space

adaptive perfectly matched layer method for multiple scattering problems a uniaxial perfectly matched layer upml method is proposed for solving three dimensional helmholtz equation with multiple scatterers the exterior domain is truncated to enclose each scatterer by a bounded domain individually based on reliable a posteriori error estimate an efficient adaptive finite element algorithm is proposed to solve the multiple scattering problem the efficiency of the adaptive pml method is demonstrated by extensive numerical experiments

total bandwidth for the harper equation iii corrections to scaling earlier work on the spectral measure for the harper equation discrete mathieu equation showed that if there is a large common period p for the lattice and the sinusoidal potential the spectral measure scaled as c p where c is a constant independent of the number of oscillations q of the potential in p lattice spacings the corrections to this scaling law are found for q 1 and q 2 for q 1 it is shown that the corrections to scaling are logarithmic functions of p while for q 2 the leading corrections are reduced by a factor p 2 analytical and numerical support is given for the assertion that this difference between odd and even values of q persists for higher values of q provided they are substantially less than p

formulation and accuracy of on surface radiation conditions for acoustic multiple scattering problems the aim of this paper is 1 to address an extension of the on surface radiation condition osrc method to multiple scattering by a cluster of convex obstacles and 2 to analyze the numerical accuracy of the formulation for some model problems first the acoustic scattering problem is considered and materials about integral equations and osrcs are recalled then an osrc integral equation formulation is developed for the dirichlet multiple scattering problems finally an extensive numerical study of the method is proposed in the special case of the scattering by circular cylinders even if this situation is not completely general it provides an assessment of the accuracy of the approximation with respect to various parameters including thanks to the choice of the osrc operators

dirichlet to neumann boundary conditions for multiple scattering problems a dirichlet to neumann dtn condition is derived for the numerical solution of time harmonic multiple scattering problems where the scatterer consists of several disjoint components it is obtained by combining contributions from multiple purely outgoing wave fields the dtn condition yields an exact non reflecting boundary condition for the situation where the computational domain and its exterior artificial boundary consist of several disjoint components because each sub scatterer can be enclosed by a separate artificial boundary the computational effort is greatly reduced and becomes independent of the relative distances between the different sub domains the dtn condition naturally fits into a variational formulation of the boundary value problem for use with the finite element method moreover it immediately yields as a by product an exact formula for the far field pattern of the scattered field numerical examples show that the dtn condition for multiple scattering is as accurate as the well known dtn condition for single scattering problems j comput phys 82 1989 172 numerical methods for problems in infinite domains elsevier amsterdam 1992 while being more efficient due to the reduced size of the computational domain

von neumann lattices of wannier functions for bloch electrons in a magnetic field the problem of bloch electrons in a magnetic field in two dimensions can be reduced to a one dimensional problem with a hamiltonian h that is a periodic function of x and p wannier functions can be defined for the sub bands of the spectrum of this effective hamiltonian when the chern class quantized hall conductance integer of the sub band is zero the weyl wigner formalism can be used to represent these wannier functions by a von neumann lattice it is shown how this von neumann lattice of wannier functions can be defined for irrational as well as rational magnetic fields an important benefit from using the weyl wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective hamiltonian in phase space it is shown how the wannier functions can be defined so that their wigner functions have the point symmetries of the effective hamiltonian an example of how these results can prove useful is given if we take matrix elements of the hamiltonian between the wannier states of a sub band we derive a new effective hamiltonian describing this sub band which is again a periodic function of coordinate and momentum operators since by projecting onto a sub band we have also reduced the number of degrees of freedom this operation is a renormalization group transformation it is shown that the symmetry of the new effective hamiltonian in phase space is the same as that of the original one this preservation of symmetry helps to explain some unusual properties of the spectrum when the hamiltonian has fourfold symmetry

estimates of derivatives of random functions i abstract approximate formulas for the derivatives of functions of several variables contaminated by noise are given error estimates are derived the formulas are optimal in some sense

the ten martini problem we prove the conjecture known as the ten martini problem after kac and simon that the spectrum of the almost mathieu operator is a cantor set for all nonzero values of the coupling and all irrational frequencies

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

hofstadter rules and generalized dimensions of the spectrum of harper s equation we consider the harper model which describes two dimensional bloch electrons in a magnetic field for irrational flux through the unit cell the corresponding energy spectrum is known to be a cantor set with multifractal properties in order to relate the maximal and minimal fractal dimension of the spectrum of harper s equation to the irrational number involved we combine a refined version of the hofstadter rules with results from semiclassical analysis and tunnelling in phase space for quadratic irrationals with continued fraction expansion the maximal fractal dimension exhibits oscillatory behaviour as a function of n which can be explained by the structure of the renormalization flow the asymptotic behaviour of the minimal fractal dimension is given by as the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket our results imply that the time evolution of high order moments is sensible to the parity of n

fredholm indices and the phase diagram of quantum hall systems the quantized hall conductance in a plateau is related to the index of a fredholm operator in this paper we describe the generic phase diagram of fredholm indices associated with bounded and toeplitz operators we discuss the possible relevance of our results to the phase diagram of disordered integer quantum hall systems

on the total bandwidth for the rational harper s equation in the last years several contributions have been done around the total bandwidth of the spectrum for the harper s operator in particular an interesting conjecture has been proposed by thouless which gives also strong convincing arguments for the proof in special cases on the other hand in the study of the cantor structure of the spectrum b helffer and j sjostrand have justified an heuristic semiclassical approach proposed by m wilkinson the aim of this article is to explain how one can use the first step of this approach to give a rigorous semi classical proof of the thouless formula in some of the simplest cases we shall also indicate how one can hope with more effort to prove rigorously recent results of last and wilkinson on the same conjecture

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

scaling for the discrete mathieu equation the scaling of the total width of the band for the discrete mathieu equation is studied in the critical region near the transition between localized and extended states for the special case in which there is one period of the modulation forp lattice spacings a general expression for the bandwidthw in the critical region is found at the critical point an analytic expression forpw is found which agrees to one part in 108 with the result deduced from numerical work

the dtn nonreflecting boundary condition for multiple scattering problems in the half plane abstract the multiple dirichlet to neumann multiple dtn non reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half plane the multiple dtn map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries as opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half plane the adapted multiple dtn condition is simultaneously imposed in each of the artificial circular boundaries as a result the computational effort is significantly reduced a computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary fitted grids approximate solutions to problems involving two scatterers of arbitrary geometry are presented the proposed numerical method is validated by comparing the approximate and exact far field patterns for the scattering from a single and from two circular obstacles in the half plane

wkb analysis of energy band structure of modulated systems a discrete wkb approximation is applied to harper s equation describing the motion of an electron in a square lattice subject to a magnetic field the treatment is valid when the modulation parameter proportional to the magnetic field is a rational number m n in the limit of large n a uniform asymptotic expression is derived for the floquet wavenumber which characterizes the dynamics and spectral properties the case of energies near the classical separatrix crucial to the properties of the spectrum near the transition between localized and extended states is included an application is a detailed calculation of the total width of the bands in the critical region for m 1 and 2 the results are consistent with thouless conjecture that the bandwidth obeys a universal scaling form independent of m

symmetry properties of exact energy solutions to the harper equation and related q normalizations proofs are given that exact solutions to the wavefunctions of the harper equation can be established in an explicit manner by resorting to three term recurrence relations implied by a q calculus approach proposed previously the q normalization of wavefunctions resulting in the appearance of peaks is discussed the exact q 6 energy solution has also been derived and analysed in some more detail there are reasons to say that fractal structures concerning band and gap distributions have to be accounted for if q 6

energy spectra for one dimensional quasiperiodic potentials bandwidth scaling mapping and relation with local isomorphism energy spectra for one dimensional tight binding models with two types of quasiperiodic potentials are studied for which the incommensurability is characterised by quadratic irrationals one is the step potential model for which the structure is a generalised fibonacci chain for special structures scaling properties of the spectrum are found numerically a critical index delta for the total bandwidth is determined after deriving recursion relations for the trace of transfer matrices it is shown that generalised fibonacci chains have the same energy spectrum if and only if they are locally isomorphic the other potential is sinusoidal for which the critical index delta is determined at the critical point

an example of phase holonomy in wkb theory discusses the application of wkb theory to harper s equation psi n 1 psi n 1 2 alpha cos 2 pi beta n delta psi n e psi m in the case in which beta is very close to a rational number p q the wkb wavefunction for this system is a vector valued quantity proportional to an eigenvector u of a matrix h x p which is parametrised by the phase space coordinates x and p the complex phase of u is determined by a non holonomic connection rule when transported around a cycle and in phase space u is multiplied by a phase factor ei gamma c this phase change manifests itself as a modification of the bohr sommerfeld quantisation conditions

hierarchical band clustering and fractal spectra in incommensurate systems an analytic derivation is given of rules exactly describing the hierarchical band clustering in the spectrum of harper s equation for various incommensurate systems these rules involving transformations of the incommensurability parameter phi are shown to apply exactly for arbitrary coupling strength the approach also provides energy scale factors and hence the fractal dimension df 0 50 or 0 01 of the self similar spectrum occurring when phi takes a fixed point value phi g the golden mean

on spectral properties of harper like models we study spectral properties of harper like models by algebraic and combinatorial methods and derive sufficient conditions for the existence of spectral gaps with qualitative estimates for this class the chambers relation holds and we obtain an analytic expression for the representation dependent part models corresponding to the rectangular and triangular lattice are studied in the second case we show that one class of spectral gaps is open for magnetic fields with rational magnetic flux per unit cell a quantitative estimate for the gap widths is given for the anisotropic case and for irrational magnetic flux fulfilling some liouville condition the spectrum is a cantor set

vortex structure and characterization of quasiperiodic functions quasiperiodic functions qpfs are characterized by their full vortex structure in one unit cell this characterization is much finer and more sensitive than the topological one given by the total vorticity per unit cell the chern index it is shown that qpfs with an arbitrarily prescribed vortex structure exist by constructing explicitly such a standard qpf two qpfs with the same vortex structure are equivalent in the sense that their ratio is a function which is strictly periodic nonvanishing and at least continuous a general qpf can then be approximately reconstructed from its vortex structure on the basis of the standard qpf and the equivalence concept as another application of this concept a simple method is proposed for calculating the quasiperiodic eigenvectors of periodic matrices possible applications to the quantum chaos problem on a phase space torus are briefly discussed

first order correction to the renormalization of the phase space lattice hamiltonian we develop a renormalization group analysis for a hamiltonian which is a periodic function of and this is a model for bloch electrons in a magnetic field and includes harper s model as a special case the eigenfunctions are bloch waves when the ratio of to the area of the unit cell is a rational number p q the renormalization procedure produces an effective hamiltonian for a subset of the spectrum which can be expanded in the zeroth order term has been obtained in earlier papers here we obtain the first order term of this expansion in terms of the q dimensional vectors which define the rational bloch states k and are bloch wavevectors the effective hamiltonian is not invariant under gauge transformations multiplying the vector we show that the infinitesimal gauge transformation is equivalent to evolving under a gauge hamiltonian obtained to lowest order in by quantizing

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

total bandwidth for harper equation correction to renormalization analysis this paper studies the correction to the effective hamiltonian of the harper equation in respect of the total bandwidth the effective hamiltonian describes the further splitting from the spectrum of a lower order fraction when the spectrum of a higher order fraction is considered the total bandwidth for a split sub band is given by the thouless scaling law modified by the curvature of the effective hamiltonian at its saddle points in this paper it is shown that the leading term of the effective hamiltonian is exactly solvable moreover the correction to the curvature of the effective hamiltonian is obtained numerically and the explicit analytical expression is also found this gives the second sum rule which is of importance in deriving the thouless scaling law for a generic fraction

bandwidth statistics from the eigenvalue moments for the harper hofstadter problem a method for studying the product of bandwidths for the harper hofstader model is proposed which requires knowledge of the moments of the midband energies a general formula for these moments is conjectured and the asymptotic representation for the product of bandwidths computed in the limit of a weak magnetic flux using szeg o s theorem for hankel matrices then a first approximation for the edge of the butterfly spectrum is given and its connection with l evy s formula for brownian motion discussed

high order surface radiation conditions for time harmonic waves in exterior domains abstract we formulate a new family of high order on surface radiation conditions to approximate the outgoing solution to the helmholtz equation in exterior domains motivated by the pseudo differential expansion of the dirichlet to neumann operator developed by antoine et al 1999 we design a systematic procedure to apply pseudo differential symbols of arbitrarily high order numerical results are presented to illustrate the performance of the proposed method for solving both the dirichlet and the neumann boundary value problems possible improvements and extensions are also discussed

positive hausdorff dimensional spectrum for the critical almost mathieu operator we show that there exists a dense set of frequencies with positive hausdorff dimension for which the hausdorff dimension of the spectrum of the critical almost mathieu operator is positive

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

bayliss turkel like radiation conditions on surfaces of arbitrary shape this paper addresses the extension of the bayliss turkel second order radiation condition to an arbitrarily shaped surface the derivation is based mainly on the pseudo differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition the radiation condition then ranges among the most accurate of those of order two as a by product of the derivation almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared

justification of geometrical optics for non convex obstacles using new results about the propagation of singularities we shall show that it suffices to approximate the exact solution of the helmholtz equation locally for justification of geometrical optics the methods which we describe allow us to show that the geometrical optics approximation converges to the exact solution of the helmholtz equation for a large class of obstacles apart from some technical assumptions concerning the behaviour of the reflected rays we only need that solutions of the wave equation decay exponentially outside of the obstacle this is true if rays are not trapped cf s in this paper however we shall not treat the most general case but approximate the solution of the helmholtz equation only at points x satisfying the additional conditions a2 and a3 given below a more general case will be considered in a sequel to this paper let b be a bounded open set with c boundary in r3 and q r3 b the solution of au k2u 0 1 1

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

a six fold rotation operator for the wannier functions of the phase space lattice hamiltonian we consider a set of localized wannier functions which can be used to describe the bloch bands of one dimensional hamiltonians whose associated weyl function is periodic under a hexagonal lattice of translations in phase space these hamiltonians can be used to describe electrons on two dimensional hexagonal lattices penetrated by a magnetic field the effect on the wannier functions of a rotation of the phase plane is considered the resulting transformation is found to be surprisingly complicated when the quantized hall conductance integer for the bloch band is non zero

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

level statistics in a quasiperiodic system the authors examine the level statistics of harper s equation a simple model of a one dimensional quasiperiodic system through the distribution of the spacings between adjacent eigenvalues each normalised by the average local density of states they present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential showing it has a simple form in each of these limits they show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side and also that at the critical point the distribution of the normalised spacings has a distinct but simple form