effective masses and conformal mappings letgn n n denote the set of gaps of the hill operator we solve the following problems 1 find the effective massesmn 2 compare the effective massmn with the length of the gapgn and with the height of the corresponding slit on the quasimomentum plane both with fixed numbern and their sums 3 consider the problems 1 2 for more general cases the dirac operator with periodic coefficients the schrodinger operator with a limit periodic potential to obtain 1 3 we use a conformal mapping corresponding to the quasimomentum of the hill operator or the dirac operator

marchenko ostrovki mapping for periodic zakharov shabat systems abstract consider the zakharov shabat or dirac operator t in the hilbert space l2 r l2 r with real periodic vector potential q q1 q2 h l2 t l2 t the spectrum of t is absolutely continuous and consists of intervals separated by gaps gn z n z n n z from the dirichlet eigenvalue mn n z of the zakharov shabat equation with dirichlet boundary conditions at 0 1 the square of the height of vertical slits on the quasimomentum domain and the points on these slits we construct the marchenko ostrovki vertical slits mapping for the periodic zakharov shabat systems h h l2 l2 using nonlinear functional analysis in hilbert spaces we show that this mapping is a real analytic isomorphism in the second part of our paper we prove a new identity for the effective masses

on the convergence of the series expansion analysis for self avoiding walks attached to a surface extensions of exact enumeration data for self avoiding walks attached to the surfaces of face centered cubic and triangular lattices are reported the convergence of the estimates for the critical points pc and exponent gamma 1 obtained by the baker hunter 1973 confluent singularity analysis is discussed and comparison made with estimates of pc from the corresponding bulk series

the estimates of periodic potentials in terms of effective masses letgn an an n 1 denote the gaps mn be the effective masses and n an 1 an a0 0 be the spectral bands of the hill operatort d2 dx2 v x inl2 r wherev is a 1 periodic real potential froml2 0 1 let the length gapln gn hn be the height of the corresponding slit on the quasimomentum domain and n 2 2n 1 n 0 be the band reduction let l n sqrt a n sqrt a n n geqq 1 n 1 denote the gap length for the operator sqrt t geqq 0 introduce the sequencesl ln h hn l ln n m mn and the norms left f right m 2 sum nolimits n 0 left 2 pi n right 2m f n 2 m geqq 0 m 0 the following results are obtained i the estimates of v l h 1 l 1 in terms of m 2 ii identities for the dirichlet integral of quasimomentum and integral of potentials and so on iii the generation of i ii for more general potentials

stability analysis of a vlasov wave system describing particles interacting with their environment we study a kinetic equation of the vlasov wave type which arises in the description of the behaviour of a large number of particles interacting weakly with an environment composed of an infinite collection of local vibrational degrees of freedom modeled by wave equations we use variational techniques to establish the existence of large families of stationary states for this system and analyze their stability

crossover exponent for piecewise directed walk adsorption on sierpinski fractals we study the problem of critical adsorption of piecewise directed random walks on a boundary of fractal lattices that belong to the sierpinski gasket family by applying the exact real space renormalization group method we calculate the crossover exponent associated with the number of adsorbed steps for the complete fractal family we demonstrate that our results are very close to the results obtained for ordinary self avoiding walk and discuss the asymptotic behaviour of at the fractal to euclidean lattice crossover

chaotic dynamics of a free particle interacting linearly with a harmonic oscillator we study the closed hamiltonian dynamics of a free particle moving on a ring over one section of which it interacts linearly with a single harmonic oscillator on the basis of numerical and analytical evidence we conjecture that at small positive energies the phase space of our model is completely chaotic except for a single region of complete integrability with a smooth sharp boundary showing no kam type structures of any kind this results in the cleanest mixed phase space structure possible in which motions in the integrable region and in the chaotic region are clearly separated and independent of one another for certain system parameters this mixed phase space structure can be tuned to make either of the two components disappear leaving a completely integrable or completely chaotic phase space for other values of the system parameters additional structures appear such as kam like elliptic islands and one parameter families of parabolic periodic orbits embedded in the chaotic sea the latter are analogous to bouncing ball orbits seen in the stadium billiard the analytical part of our study proceeds from a geometric description of the dynamics and shows it to be equivalent to a linked twist map on the union of two intersecting disks

periodic weighted operators abstract consider the periodic weighted operator ty 2 2 y in l 2 r x 2 dx where the real function is 1 periodic positive and let q l 2 0 1 the spectrum of t consists of intervals separated by gaps n n 1 with the lengths n let h n be a height of the corresponding slit in the quasimomentum domain and let g n n 1 be the gap with the length g n of the operator t 0 the following results are obtained i the quasimomentum k for the weighted operator t is constructed and the basic properties of k are studied ii two sided estimates of l 2 norms of the sequences n n 1 h n 1 g n 1 in terms of 0 1 q x 2 dx iii the asymptotics of the gap length n as n are determined the proofs are based on the analysis of the quasimomentum as a conformal mapping embedding theorems and the identities between the quasimomentum and the potential in order to prove these results the asymptotics of the fundamental solutions and the lyapunov function are obtained at high energy

effective masses for laplacians on periodic graphs abstract we consider laplacians on both periodic discrete and periodic metric equilateral graphs their spectrum consists of an absolutely continuous part which is a union of non degenerate spectral bands and flat bands i e eigenvalues of infinite multiplicity we estimate effective masses associated with the ends of each spectral band in terms of geometric parameters of the graphs moreover in the case of the bottom of the spectrum we determine two sided estimates on the effective mass in terms of geometric parameters of the graphs the proof is based on floquet theory factorization of fiber operators perturbation theory and the relation between effective masses for laplacians on discrete and metric graphs obtained in our paper

spectral statistics of a system with sharply divided phase space we construct a family of non kolmogorov arnold moser non kam piecewise linear continuous 2d area preserving maps which have sharply divided phase space with regions of regular elliptic and chaotic hyperbolic motion for such systems the shape of the islands of regular motion is either a solid ellipse or a solid polygon depending on the ir rationality of the frequency and thus the total area of the regular region of phase space can be computed analytically or at least rigorously estimated from below we analyse the spectral statistics for a few examples of the quantization of our maps and show that they provide a convenient playground for testing and confirming the key assumptions required for the validity of berry robnik formulae

critical behaviour at an edge for the saw and ising model the authors have studied the behaviour of two and three dimensional self avoiding walks confined to a wedge of wedge angle alpha series have been obtained and analysed for the angular dependent critical exponents characterising various edge susceptibilities in terms of a general scaling form for the edge free energy fe approximately t d 2 nu psi e ht y0 nu h1t y1 nu h2t y2 nu they find for the two dimensional case the following scaling indices y0 91 48 y1 3 8 y2 alpha 5 pi 8 alpha they argue that these results are exact from which follow all exponents for the bulk surface and edge problem in three dimensions the authors obtain y0 approximately 2 488 y1 0 65 or 0 02 y2 alpha alpha b pi alpha where a 0 51 or 0 04 b 0 847 or 0 017 which for y2 is precisely of the functional form given by mean field theory y2 alpha 1 2 pi 2 they argue that a 1 2 for all three dimensional o n models this simple angular dependence of y2 is different from that suggested by cardy s one loop epsilon expansion 1983 for the square lattice they have also studied the case in which the wedge is rotated through an angle of pi 4 and find that the various exponents are unchanged for the three dimensional ising model in a wedge analogy with the saw results plus mean field results in conjunction with rg and series work yield y0 approximately 2 485 y1 0 71 or 0 02 and y2 a b pi alpha with a 1 2 and b 0 79 or 0 02

inverse problem and the trace formula for the hill operator ii consider the hill operator t d 2 dx 2 q x on l 2 mathbb r where q in l 2 0 1 is a 1 periodic real potential and int 0 1q x dx 0 the spectrum of t is absolutely continuous and consists of intervals separated by gaps gamma n a n a n n geq 1 let mu n n geq 1 be the dirichlet eigenvalue of the equation y qy ly on the interval 0 1 introduce the vector g n g cn g sn in mathbb r 2 with components g cn a n a n 2 mu n and g sn gamma n 2 4 g cn 2 s n where the sign s n or s n for all n geq 1 using nonlinear functional analysis in hilbert spaces we show that the mapping g q to g q g n 1 iy in ell 2 oplus ell 2 is a real analytic isomorphism in the second part a trace formula for q in l 2 0 1 is proved

equivariant singularity theory with distinguished parameters two case studies of resonant hamiltonian systems we consider hamiltonian systems near equilibrium that can be formally reduced to one degree of freedom spatiotemporal symmetries play a key role the planar reduction is studied by equivariant singularity theory with distinguished parameters the method is illustrated on the conservative spring pendulum system near resonance where it leads to integrable approximations of the iso energetic poincare map the novelty of our approach is that we obtain information on the whole dynamics regarding the quasi periodic solutions the global configuration of their invariant manifolds and bifurcations of these

estimates for the hill operator ii consider the hill operator ty y q t y in l2 r where the real potential q is 1 periodic and q q l2 0 1 the spectrum of t consists of spectral bands separated by gaps n n 1 with length n 0 we obtain two sided estimates of the gap lengths n2 n 2 in terms of 01q t 2dt moreover we obtain the similar two sided estimates for spectral data the height of the corresponding slit on the quasimomentum domain action variables for the kdv equation and so on in order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the hill operator that makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory then the proof is based on the analysis of the conformal mapping the embedding theorems and the identities furthermore we obtain the similar two sided estimates for potentials which have p 2 derivatives

dynamics of self trapping in the discrete nonlinear schro dinger equation abstract we study dynamical aspects of the discrete nonlinear schrodinger equation dnls for chains of different sizes with periodic and open boundary conditions we focus on the occurrence of a self trapping transition in the different geometries the initial condition used is that which places the particle or power on one lattice site or nonlinear waveguide and the quantity studied is the time averaged probability for the particle to remain in that site we show that the self trapping transition in long chains occurs for parameter values not very different from that for very small clusters

estimates for periodic zakharov shabat operators abstract we consider the periodic zakharov shabat operators on the real line the spectrum of this operator consists of intervals separated by gaps with the lengths g n 0 n z let n be the corresponding effective masses and let h n be heights of the corresponding slits in the quasi momentum domain we obtain a priori estimates of sequences g g n n z n n z h h n n z in terms of weighted l p norms at p 1 the proof is based on the analysis of the quasi momentum as the conformal mapping

self avoiding walks in wedges the authors consider the number of self avoiding walks confined to a subset zd f of the d dimensional hypercubic lattice zd such that the coordinates x1 x2 xd of each vertex in the walk satisfy x1 or 0 and 0 or xk or fk x1 for k 2 3 d they show that if fk x to infinity as x to infinity the connective constant of walks in zd f is identical to the convective constant of walks in zd they also explore conditions on fk which lead to a smaller connective constant for walks in zd f and in particular consider walks between two parallel d 1 dimensional hyperplanes finally they contrast some of these results with recent work by grimmett on percolation on subsets of the square lattice

self avoiding walks attached to triangular and face centred cubic lattice surfaces extended exact enumeration data the analysis of exact enumeration data of self avoiding walks attached to the surfaces of triangular and face centred cubic lattices is consistent with gamma 1 0 718 or 0 008 136 delta pc where delta pc pc 0 0995 in three dimensions and 0 9530 or gamma 1 or 0 9548 in two dimensions where pc for the triangular lattice is assumed to lie in the range 0 240 915 or pc or 0 240 93

inverse problem for periodic weighted operators define the periodic weighted operator ty 2 2y in l2 r x 2 dx suppose a function w21 r z is 1 periodic real positive 0 1 and let q l2 0 1 the spectrum of t consists of intervals n n 1 n separated by gaps n n n n 1 with the lengths n and 0 0 let m2n n 1 be the dirichlet eigenvalue of the equation y 2qy z2y y 0 y 1 0 where mn 0 introduce the lyapunov function z q for t and note that z zn q 0 for some zn n n let x z q be the solution of the equation 2q z2 z c satisfying 0 z q 0 0 z q 1 introduce the vector hn hcn hsn r2 with components hcn log 1 n 1 mn q hsn hn 2 hcn 2 1 2 sign zn mn where hn is defined by the equation cosh hn 1 n zn q 1 and coincides with the euclidien norm of the vector hn using nonlinear functional analysis in hilbert space we prove that the mapping h q h q hn 1 is a real analytic isomorphism

the lyapunov function for schrodinger operators with a periodic 2 2 matrix potential abstract we consider the schrodinger operator on the real line with a 2 2 matrix valued 1 periodic potential the spectrum of this operator is absolutely continuous and consists of intervals separated by gaps we define a lyapunov function which is analytic on a two sheeted riemann surface on each sheet the lyapunov function has the same properties as in the scalar case but it has branch points which we call resonances we prove the existence of real as well as non real resonances for specific potentials we determine the asymptotics of the periodic and the anti periodic spectrum and of the resonances at high energy we show that there exist two type of gaps 1 stable gaps where the endpoints are the periodic and the anti periodic eigenvalues 2 unstable resonance gaps where the endpoints are resonances i e real branch points of the lyapunov function we also show that periodic and anti periodic spectrum together determine the spectrum of the matrix hill operator

a hamiltonian model for linear friction in a homogeneous medium we introduce and study rigorously a hamiltonian model of a classical particle moving through a homogeneous dissipative medium at zero temperature in such a way that it experiences an effective linear friction force proportional to its velocity at small speeds the medium consists at each point in a space of a vibration field modelling an obstacle with which the particle exchanges energy and momentum in such a way that total energy and momentum are conserved we show that in the presence of a constant not too large external force the particle reaches an asymptotic velocity proportional to this force in a potential well on the other hand the particle comes exponentially fast to rest in the bottom of the well the exponential rate is in both cases an explicit function of the model parameters and independent of the potential

the marchenko ostrovski mapping and the trace formula for the camassa holm equation we consider the periodic weighted operator ty 2 2y 14 4 in l2 r 2dx where is a 1 periodic positive function satisfying q l2 0 1 the spectrum of t consists of intervals separated by gaps in the first part of the paper we construct the marchenko ostrovski mapping q h q and solve the corresponding inverse problem for our approach it is essential that the mapping h has the factorization h q h0 v q where q v q is a certain nonlinear mapping and v h0 v is the marchenko ostrovski mapping for the hill operator moreover we solve the inverse problem for the gap length mapping in the second part of this paper we derive the trace formula for t

exact enumeration results for self avoiding walks on the honeycomb lattice attached to a surface we consider self avoiding walks on the honeycomb lattice interacting with a surface with different energies associated between sites in contact with a linear boundary to the left of the origin and those in contact with the right of the boundary we numerically confirm recent exact results for the polymer adsorption transition and corresponding critical exponents with mixed ordinary and special boundary conditions the phase diagram is elucidated with the aid of some rigorous arguments

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out