space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

cyclic boson algebra and q boson realizations of cyclic representations of the quantum algebra slq 3 the authors introduce the concept of cyclic boson algebra and study its representations using this algebra to realize slq 3 they construct the cyclic representation of the quantum universal enveloping algebra uq sl 3 identical to slq 3 on the q fock space restricting this representation to the subalgebra slq 2 slq 3 they naturally obtain the irreducible cyclic representations of slq 2

a variational iterative technique applied to quantum mechanical problems a technique for solving nonlinear operator equations iteratively is applied to quantum mechanical calculations it is shown that alternative methods of applying the technique lead to another way of approaching variational calculations and generalisations of perturbation theory respectively

a simple iterative method for resonance calculation a recently developed procedure for solving the stationary schrodinger equation in matrix representation form is proved to be useful in calculating quasibound energy levels results for a simple quantum mechanical model with potential energy function v r 1 lambda r lambda 0 are shown to be very accurate

indecomposable generalisations of the standard angular momentum representations of so 3 indecomposable representations of the lie algebra of the rotation group so 3 are studied on the heisenberg weyl basis which is the basis for a subspace of a quotient space of the space of the universal enveloping algebra of the heisenberg weyl algebra the author obtains an infinite dimensional indecomposable representation which induces the finite dimensional representation on certain quotient spaces and subduces the infinite dimensional indecomposable representations on invariant subspaces the standard angular momentum representations of so 3 are given as special cases

generalised meshes for quantum mechanical problems a new method to discretise schrodinger equations on a mesh is described this method is based on an accurate approximation of a variational calculation the regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail it is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form the applications to some simple potential problems show that the method is very accurate as well as very simple applications to many body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations

numerical methods for the eigenvalue determination of second order ordinary differential equations an accurate method for the numerical solution of the eigenvalue problem of second order ordinary differential equation using the shooting method is presented the method has three steps firstly initial values for the eigenvalue and eigenfunction at both ends are obtained by using the discretized matrix eigenvalue method secondly the initial value problem is solved using new highly accurate formulas of the linear multistep method thirdly the eigenvalue is properly corrected at the matching point the efficiency of the proposed methods is demonstrated by their applications to bound states for the one dimensional harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential in quantum mechanics

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

cayley differential unitary space time codes one method for communicating with multiple antennas is to encode the transmitted data differentially using unitary matrices at the transmitter and to decode differentially without knowing the channel coefficients at the receiver since channel knowledge is not required at the receiver differential schemes are ideal for use on wireless links where channel tracking is undesirable or infeasible either because of rapid changes in the channel characteristics or because of limited system resources although this basic principle is well understood it is not known how to generate good performing constellations of unitary matrices for any number of transmit and receive antennas and for any rate this is especially true at high rates where the constellations must be rapidly encoded and decoded we propose a class of cayley codes that works with any number of antennas and has efficient encoding and decoding at any rate the codes are named for their use of the cayley transform which maps the highly nonlinear stiefel manifold of unitary matrices to the linear space of skew hermitian matrices this transformation leads to a simple linear constellation structure in the cayley transform domain and to an information theoretic design criterion based on emulating a cauchy random matrix moreover the resulting cayley codes allow polynomial time near maximum likelihood ml decoding based on either successive nulling canceling or sphere decoding simulations show that the cayley codes allow efficient and effective high rate data transmission in multiantenna communication systems without knowing the channel

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

bound states for truncated coulomb potentials the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros bound states energy eigenvalues for two truncated coulombic potentials are calculated using pslet in contrast with the shifted large n expansion technique pslet results compare excellently with those from a direct numerical integration

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

inhomogeneous boson realization of indecomposable representations of lie algebras by making use of the differential realization of lie algebras in the space of inhomogeneous polynomials of a certain number of variables the corresponding inhomogeneous boson realization of lie algebras is given a new kind of indecomposable representations of lie algebras are studied on the universal enveloping algebra of heisenberg weyl algebra its subspaces and its quotient spaces the finite dimensional representations are naturally obtained on the subspaces of fock space as an example the indecomposable and irreducible representations of the lie algebra su 2 are discussed in detail

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

energy levels for a double well potential in three dimensional system using hill determinant approach abstract energy levels of the schrodinger equation for the double well potential v x y z z 2 1 2 z 2 x 2 y 2 z 2 1 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system are calculated using the hill determinant approach for several eigenstates and large values of the perturbation parameters and z2 emphasis is specially placed on the larger values of z2 because the eigenvalues for different states have almost degenerate eigenvalues some of the results calculated by the hill determinant approach are compared with the results produced by other techniques

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

orthogonal designs with maximal rates orthogonal designs have been used as space time block codes for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have a very simple decoupled maximum likelihood decoding algorithm the rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block which reflects the bandwidth efficiency of the employed space time block code constructed from the orthogonal design this paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates which may be real or complex and square or rectangular matrices we first provide several representations of orthogonal designs and their characterizations in terms of hurwitz radon matrix equations next we observe that the real orthogonal designs square or rectangular and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s however unfortunately it is not the case for the complex rectangular orthogonal designs with rates as high as possible we then construct a class of complex orthogonal designs for any number of transmit antennas the proposed complex orthogonal designs for the number of transmit antennas n 2m 1 and 2m have the same rate m 1 and 2m where m is any natural number finally we demonstrate that for n 2m 1 and 2m with any given natural number m the value m 1 and 2m is the maximum possible rate that the complex orthogonal designs square or rectangular with n transmit antennas can achieve

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

application of the eigenvalue moment method to important one dimensional quantum systems the authors study the effectiveness of the eigenvalue moment method as applied to various one dimensional quantum problems which have appeared in the literature of particular interest are the radial potential problems including angular momentum effects ze2 r beta and r2 lambda r2 1 gr2 studied by de meyer and vanden berghe 1990 and more recently by fernandez 1991 and witwit 1991 respectively they also examine the potentials 1 2x2 or gx4 1 alpha gx2 exhaustively studied by auberson and boissiere 1983 finally they examine the one dimensional regulated bohr atom with potential z mod x mod alpha well studied by loudon 1959 and haines and roberts 1969

a simple iterative solution of the schrodinger equation in matrix representation form a simple procedure for solving the schrodinger equation is presented it is based upon an iterative solution of the secular equation a large enough convergence rate is obtained by using a basis set of properly scaled functions the effect of the scaling parameter on the convergence rate is studied in order to improve the calculation the method is applied to simple though non trivial quantum mechanical models such as the quartic sextic and octic anharmonic oscillators a double well potential and the linear confining potential highly accurate eigenvalues for all values of the coupling parameter are obtained

two generalized complex orthogonal space time block codes of rates 7 11 and 3 5 for 5 and 6 transmit antennas space time block codes from orthogonal designs have two advantages namely fast maximum likelihood ml decoding and full diversity rate 1 real pulse amplitude modulation pam space time codes real orthogonal designs for multiple transmit antennas have been constructed from the real hurwitz radon families which also provides the rate 1 2 complex quadrature amplitude modulation qam space time codes complex orthogonal designs for any number of transmit antennas rate 3 4 complex orthogonal designs space time codes for three and four transmit antennas have existed in the literature but no high rate 1 2 complex orthogonal designs for other numbers of transmit antennas exist we present rate 7 11 and rate 3 5 generalized complex orthogonal designs for five and six transmit antennas respectively

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

the cyclic boson operators and new representations of the quantum algebra slq 2 for q a root of unity the concept of the cyclic boson operator is introduced for the explicit construction of new representations of the quantum algebra slq 2 when q is a root of unity the method used can easily be generalized to other quantum algebras or quantum superalgebras

representations of ueres sl2 via restricted q fock spaces two restricted c q q 1 forms of the well known q boson algebra are introduced and the corresponding restricted q fock spaces defined all of the irreducible highest weight representations including the infinite dimensional ones of ueres sl2 of type 1 are constructed through the restricted q fock spaces

on the nonexistence of rate one generalized complex orthogonal designs orthogonal space time block coding proposed recently by alamouti 1998 and tarokh et al 1999 is a promising scheme for information transmission over rayleigh fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum likelihood ml decoding algorithm tarokh et al extended the theory of classical orthogonal designs to the theory of generalized real or complex linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space time block codes stbc with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas it has been known that the stbc constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation contrary to this in this correspondence we prove that there does not exist rate one stbc from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation

analytical bound eigenstates and eigenvalues of a truncated coulomb potential the author obtains exact eigenfunctions and eigenvalues of the schrodinger equation with the potential v r ze2 r beta beta 0 for particular values of the cut off parameter beta each eigenfunction is a finite polynomial times an exponential and the eigenvalues are exactly those for the pure coulomb problem i e beta 0

tight bounds to the schrodinger equation eigenvalues increasing tight upper and lower bounds to the schrodinger equation eigenvalues are obtained from the riccati equation for the logarithmic derivative of the wavefunction the solution of this non linear equation is written as a pade approximant and the bounds are given by the roots of a sequence of determinants numerical results are shown for the anharmonic and quartic oscillators

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

exact solution to a supersymmetric gaudin model we investigate a special case of the gaudin model related to the superalgebra osp 1 2 we present an exact solution to that system diagonalizing a complete set of commuting observables and providing the corresponding eigenvectors and eigenvalues the approach used in this paper is based on the co algebra symmetry of the model already known from the calogero gaudin system

converging bounds for the eigenvalues of multiminima potentials in two dimensional space the eigenvalue problem in of schrodinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls it is shown that the dirichlet and the von neumann problems in generate upper and lower bounds respectively to the eigenvalues of the unbounded system to illustrate the method rapidly converging two sided bounds for the energy levels of two and three well oscillators are presented using simple trigonometric basis functions

new inhomogeneous boson realizations and inhomogeneous differential realizations of lie algebras the inhomogeneous boson realizations ihbr and the corresponding inhomogeneous differential realizations ihdr of lie algebras which play an important role in the search of quasi exactly solvable problems qesp of quantum mechanics are studied all possible ihdr of semisimple lie algebras can be obtained in this way as examples the ihbr and the corresponding ihdr of lie algebras su 2 and su 3 are studied in detail

unitary signal constellations for differential space time modulation with two transmit antennas parametric codes optimal designs and bounds we focus on the design of unitary signal constellations for differential space time modulation with double transmit antennas by using the parametric form of a two by two unitary matrix we present a class of unitary space time codes called parametric codes and show that this class of unitary space time codes leads to a five signal constellation with the largest possible diversity product and a 16 signal constellation with the largest known diversity product although the parametric code of size 16 is not a group by itself we show that it is a subset of a group of order 32 furthermore the unitary signal constellations of sizes 32 64 128 and 256 obtained by taking the subsets of the parametric codes of sizes 37 75 135 and 273 respectively have the largest known diversity products we also use large diversity sum of unitary space time signal constellations as another significant property for the signal constellations to have good performance in low snr scenarios the newly introduced unitary space time codes can lead to signal constellations with sizes of 5 and 9 through 16 that have the largest possible diversity sums subsequently we construct a few sporadic unitary signal constellations with the largest possible diversity product or diversity sum a four signal constellation which has both the largest possible diversity product and the largest possible diversity sum and three unitary signal constellations with the largest possible diversity sums for sizes of 6 7 and 8 are constructed respectively furthermore by making use of the existing results in sphere packing and spherical codes we provide several upper and lower bounds on the largest possible diversity product and the largest possible diversity sum that unitary signal constellations of any size can achieve

generalized psk in space time coding a wireless communication system using multiple antennas promises reliable transmission under rayleigh flat fading assumptions design criteria and practical schemes have been presented for both coherent and noncoherent communication channels in this paper we generalize one dimensional 1 d phase shift keying psk signals and introduce space time constellations from generalized psk gpsk signals based on the complex and real orthogonal designs the resulting space time constellations reallocate the energy for each transmitting antenna and feature good diversity products consequently their performances are better than some of the existing comparable codes moreover since the maximum likelihood ml decoding of our proposed codes can be decomposed to 1 d psk signal demodulation the ml decoding of our codes can be implemented in a very efficient way

differential unitary space time modulation we present a framework for differential modulation with multiple antennas across a continuously fading channel where neither the transmitter nor the receiver knows the fading coefficients the framework can be seen as a natural extension of standard differential phase shift keying commonly used in single antenna unknown channel systems we show how our differential framework links the unknown channel system with a known channel system and we develop performance design criteria as a special ease we introduce a class of diagonal signals where only one antenna is active at any time and demonstrate how these signals may be used to achieve full transmitter diversity and low probability of error

solving the anharmonic oscillator tuning the boundary condition we outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x2m potential we solve the schrodinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified borel resummation technique to obtain the large x behaviour the process allows us to calculate energy eigenvalues to an arbitrary level of accuracy high degrees of precision are achieved even with modest computing power our technique extends to all levels of excitation and produces the correct solution to the double well oscillators even though they are dominated by non perturbative effects

energy levels of double well potentials in a three dimensional system the renormalized inner product perturbation technique is used to compute the energy levels for a double well potential v x y z z2 z2 x2 y2 z2 axxx4 ayyy4 azzz4 2axyx2y2 2axzx2z2 2ayzy2z2 in the three dimensional system results are produced for a wide range of parameters z2 and of state numbers nx ny nz comparison is made with results obtained by other means

upper and lower bounds of the ground state energy of anharmonic oscillators using renormalized inner projection renormalization is a method often used to approximate the eigenvalues of a hamiltonian that cannot be solved exactly it consists of splitting the hamiltonian into a solvable part and a remainder which is then minimized the inner projection technique first introduced by lowdin in the sixties was developed to bracket the eigenvalues between lower and upper bounds combining renormalization and lowdin s inner projection yielded the so called renormalized inner projection technique in this study this method will be applied to the quartic sextic and octic anharmonic oscillators lower and upper energy bounds are obtained for finite values of the coupling constant as well as for the infinite case the relation between the renormalized inner projection and perturbation theory will also be discussed another feature of this study is the importance of symbolic computation in allowing us to manipulate expressions with unevaluated parameters and to perform calculations in rational arithmetics or high decimal precision thus lowdin s rational approximants can be expressed explicitly as rational fractions in terms of the coupling constant and values for the limit constant can be obtained with amazing high accuracy namely 62 33 and 21 decimal places for the quartic sextic and octic oscillator respectively

quantum theory of a double well potential energy levels for symmetric and nonsymmetric double well potentials in a three dimensional system abstract energy eigenvalues are obtained for a double well potential v x y z z x 2 z y 2 z z 2 1 2 z x 2 x 2 z y 2 y 2 z z 2 z 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system using matrix diagonalization for various values of parameters zx2 zy2 zz2 and several energy levels enx ny nz comparison is made with results obtained by other means

exact bound state solutions of the potential v r ze2 r β high accuracy approximations of the bound state energies of the potential v r ze2 r are obtained by means of the dynamical group method certain bound state solutions of the schrodinger problem are constructed in analytic closed form