commuting periodic operators and the periodic wigner function commuting periodic operators cpo depending on the coordinate and the momentum operators are defined the cpo are functions of the two basic commuting operators and with a being an arbitrary constant a periodic wigner function pwf w x p is defined and it is shown that it is applicable in a normal expectation value calculation to the cpo as done in the original wigner paper moreover this pwf is non negative everywhere and it can therefore be interpreted as an actual probability distribution the pwf w x p is shown to be given as an expectation value of the periodic dirac delta function in the phase plane

accuracy barriers of two time level difference schemes for hyperbolic equations for two time level schemes for u 1 cu x we give for each noninteger courant number mu a bound for the order of a stable scheme if this bound is attained we give a lower bound for the error constant and give the scheme which attains these bounds all these barriers to the accuracy of stable schemes are expressed in a physical fashion in terms of the number of upwind and downwind points this notation reveals a certain symmetry with respect to a characteristic the proofs are carried out by comparing a general scheme to the optimal one using order stars

a proof of the order barrier for upwind schemes by dahlquist s second barrier demonstration d un resultat d engquist osher sur la limitation de l ordre des semidiscretisations stables de u t u x a l aide de la seconde barriere de dahlquist

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

dynamic interaction between multiple cracks and a circular hole swept by sh waves abstract the focus of this contribution is to develop a hypersingular integral equation method hiem to solve the dynamic interaction between multiple cracks and a circular hole illuminated by plane sh waves the cracks can be in arbitrary positions in particular they can terminate at the boundary of the circular hole in order to solve the proposed problem the green s function of a point harmonic force applied at an infinite plane with a circular hole is obtained by the graf formula and the wave function expansion method in order to construct the hypersingular integral equations for the cracks the green s function is separated into a closed form singular part and a regular part on the basis of the reciprocity principle as well as the obtained green s function the hypersingular integral equations for the cracks are obtained numerical solution of the hypersingular integral equations yields the dynamic stress intensity factors dsif at the crack tips comparison of present results with known results verifies the proposed method some numerical examples are presented in the paper

recursive t matrix algorithms for the solution of electromagnetic scattering from strip and patch geometries two recursive t matrix algorithms are presented and their reduced computational complexities and reduced memory requirements are demonstrated these algorithms are applied to the problem of electromagnetic scattering from conducting strips and patches with canonical geometries the geometries are reminiscent of finite sized frequency selective surfaces computational complexities of o n sup 2 and o n sup 7 3 and memory requirements of o n and o n sup 4 3 are shown to be feasible for two dimensional and three dimensional geometries respectively the formulation uses only two components of the electric field therefore the vector electromagnetic problem of scattering from three dimensional patch geometries can be solved using scalar addition theorems for spherical harmonic wave functions for a two dimensional strip problem both tm and te polarizations can be solved simultaneously using this formulation numerical scattering results in the form of radar cross sections rcs are validated by comparison with the method of moments

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

stability of semidiscretizations of hyperbolic problems semidiscretization of hyperbolic systems yields systems of ordinary differential equations of first or second order in both cases we investigate the stability of a general class of methods namely the m stage k step methods when applied to linear test equations a map sigma is introduced which allows us at least for constant linear systems to investigate whether a second order system should be integrated directly or should first be reduced to a first order system

water wave propagation in unbounded domains part ii numerical methods for fractional pdes this paper concerns numerical solutions for a fractional partial differential equation arising as a nonreflecting boundary condition in water wave propagation the fractional derivative operator is written as divergence of a singular integrable convolution which allows the equation to be viewed as a conservation law with a linear nonlocal flux a semi discrete finite volume scheme is presented using conservative piecewise polynomial reconstruction of the solution the convolution with the singular kernel is then integrated exactly time integration uses runge kutta schemes of matching order stability is discussed convergence is established and numerical examples are presented

finite volume transport schemes we analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in l 1 and l for odd order schemes in dimension one it gives sharp fractional order estimates of convergence for bv solutions it shows odd order finite volume advection schemes are better than even order finite volume schemes therefore the gibbs phenomena is controlled for odd order finite volume schemes numerical experiments sustain the theoretical analysis in particular the oscillations of the lax wendroff scheme for small courant numbers are correlated with its non stability in l 1 a scheme of order three is proved to be stable in l 1 and l

wilson like functions on von neumann lattices an orthonormal set of functions is defined on a von neumann lattice in phase space there is one function assigned to each unit cell of area h the functions are of the wilson type in that in the x direction they are obtained from one another by uniform translations while in the p direction they are double peaked and cannot be obtained by translations a similar construction is also carried out with the x and p axes interchanged the results apply to any dimension an explicit example is worked out for the ground state of a harmonic oscillator and a relation to coherent states is pointed out

on the operator bases underlying wigner s kirkwood s and glauber s phase space functions the hermitian operators behind wigner s phase space function 1932 are recognised to be simple ordered exponentials of the dynamical variables this operator basis is highly symmetric it supplies a perfectly unbiased formulation of operator equations in terms of phase space functions which is as close to classical physics as it possibly can be the author demonstrates how the symmetry properties of the basis can be exploited to simplify the computation of wigner functions enormously the ordered operator methods are also applicable to the phase space functions of kirkwood 1933 and glauber 1963 type both kinds are briefly discussed with emphasis on how they differ from wigner s description

accuracy bounds for semidiscretizations of hyperbolic problems bounds are given for the error constant of stable finite difference methods for first order hyperbolic equations in one space dimension which use r downwind and s upwind points in the discretization of the space derivatives and which are of optimal order p min r s 2 r 2 2 s it is known that this order can be obtained by interpolatory methods examples show however that their error constants can be improved

stability and accuracy for implicit semidiscretizations of hyperbolic problems in this paper we give bounds for the error constants of certain classes of stable implicit finite difference methods for first order hyperbolic equations in one space dimension we consider classes of methods that user downwind ands upwind points in the explicit part andr downwind ands upwind points in the implicit part respectively and that are of optimal orderp min r r s s 2 r r 1 2 s s

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

stability and accuracy of difference schemes for hyperbolic problems abstract bounds for the error order and error constant of full and semidiscretizations of hyperbolic problems are given in order to be able to compare fulldiscretizations a scaling is introduced this scaling reflects the amount of work needed to perform one time step using the semidiscretization associated with the fulldiscretization lower bounds for these scaled error constants are given these results are derived using the order star technique

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

von neumann s hypothesis concerning coherent states an orthonormal basis of modified coherent states is constructed each member of the basis is an infinite sum of coherent states on a von neumann lattice a single state is assigned to each unit cell of area h planck constant in the phase plane the uncertainties of the coordinate x and the square of the momentum p2 for these states are shown to be similar to those for the usual coherent states expansions in the newly established set are discussed and it is shown that any function in the kq representation can be written as a sum of two fixed kq functions approximate commuting operators for x and p2 are defined on a lattice in phase plane according to von neumann s prescription

uniform asymptotic stability of strang s explicit compact schemes for linear advection we consider a family of explicit compact schemes for advection in one dimension the order is arbitrarily high these stencils may be called strang s stencils after the seminal work of strang j math phys 41 1962 pp 147 154 we prove that odd order schemes are stable in all l q under cfl one the strategy of the proof is similar to the one of thomee j differential equations 1 1965 pp 273 292 with a careful verification that all sharp estimates on the amplification factor are independent of the cfl number this is possible based on a general representation formula for the amplification factor numerical results in one dimension confirm the analysis

stability analysis of numerical methods for systems of functional differential and functional equations abstract this paper is concerned with the numerical solution of functional differential and functional equations which include functional differential equations of neutral type as special cases the adaptation of linear multistep methods one leg methods and runge kutta methods is considered the emphasis is on the linear stability of numerical methods it is proved that a stable methods can inherit the asymptotic stability of underlying linear systems some general results of stability on explicit and implicit methods are also given

gpg stability of runge kutta methods for generalized delay differential systems abstract the gp g stability of runge kutta methods for the numerical solutions of the systems of delay differential equations is considered the stability behaviour of implicit runge kutta methods irk is analyzed for the solution of the system of linear test equations with multiple delay terms after an establishment of a sufficient condition for asymptotic stability of the solutions of the system a criterion of numerical stability of irk with the lagrange interpolation process is given for any stepsize of the method

periodic and discrete zak bases weyl s unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals planck s constant then their common eigenstates constitute the zak basis with each state specified by two phase parameters accordingly the transformation function from the position basis to the zak basis maps the hilbert space on the line onto the hilbert space on the torus this mapping is one to one provided that the zak basis states are periodic functions of their phase parameters but then the mapping cannot be continuous on the whole torus with the periodicity of the zak basis enforced the basis has a double fourier series the fourier coefficients identify a discrete basis which complements the periodic zak basis to form a pair of mutually unbiased bases the discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators these partner operators are of angular momentum type with integer eigenvalues and generate the fundamental rotations of the torus conversely the displacement operators are the ladder operators for their partners for each consistent phase convention for the periodic zak basis and thus for the line onto torus mapping there is a corresponding discrete zak basis and a corresponding pair of partner operators examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters the latter emphasizes the heisenberg weyl symmetry between position and momentum we discuss briefly the relation between the zak bases and aharonov s modular operators finally as an application of the zak operators for the torus we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled

rational approximants and numerical methods for initial value problems abstract we survey three instances of an interaction between numerical analysis of initial value problems and rational approximation theory rational functions can be used to design numerical algorithms for differential equations assist in their implementation and analyse their behaviour and our three examples have been designed to highlight all these aspects they are a order stars and their application to the derivation of bounds on the order of stable methods for linear hyperbolic differential equations b dimensional splitting and its implementation to solve efficiently nonlinear parabolic differential equations c estimates and bounds on the truncation error when the matrix exponential is approximated by rational functions

transition matrix in acoustic scattering by a strip the transition matrix describing the scalar scattering by a strip has been obtained using a boundary integral equation technique it avoids the procedure of the extended boundary condition taking the strip as the limit of an elliptic cylinder our final results confirm the known results obtained from the extended boundary condition technique the method presented in this paper seems to be very attractive to similar problems in elastodynamic scattering

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

the optimal accuracy of difference schemes we consider difference approximations to the model hyperbolic equation ut ux which compute each new value u x t at as a combination of the known values u x ra x t u x s a x a t for such schemes we find the optimal order of accuracy stability is possible for small a t a x if and only if p min r s 2 r 2 2s a similar bound is established for implicit methods in this case the most accurate schemes are based on pade approximations p z q z to za near z 1 and we find an expression for the difference i q 12 p 12 this allows us to test the von neumann condition i p q i 1 we also determine the number of zeros of q in the unit circle which decides whether the implicit part is uniformly invertible