Showing papers in "Journal of Mathematical Physics in 1993"
TL;DR: In this paper, the Schrodinger equation with quartic anharmonic and symmetric double-well potentials of the form V(A,B)=Ax2/2+Bx4(B≥0) was studied.
Abstract: Rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the Schrodinger equation with quartic anharmonic and symmetric double‐well potentials of the form V(A,B)=Ax2/2+Bx4(B≥0) are presented. This procedure exploits some exactly soluble model potentials and appears to be of quite general utility.
277 citations
TL;DR: In this paper, a nonperturbative approximate analytical solution for the Lane-Emden equation using the Adomian decomposition method is derived. The solution is in the form of a power series with easily computable coefficients.
Abstract: A nonperturbative approximate analytical solution is derived for the Lane–Emden equation using the Adomian decomposition method. The solution is in the form of a power series with easily computable coefficients. The Pade approximants method is used to accelerate the convergence of the power series. Comparison with some known exact and numerical solutions shows that the present solution is highly accurate.
224 citations
TL;DR: In this article, the structure of the q-Poincare group with braid statistics was studied and the abstract structure of q-Lorentz group was also studied.
Abstract: The q‐Poincare group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B is a braided‐quantum group structure on the q‐Minkowski space of four‐momentum with braided‐coproduct Δ_p=p⊗1+1⊗p. Here the necessary B is not a usual kind of quantum group, but one with braid statistics. Similar braided vectors and covectors V(R’), V*(R’) exist for a general R‐matrix. The abstract structure of the q‐Lorentz group is also studied.
170 citations
TL;DR: In this article, it was shown that every Lie algebra can be represented as a bivector algebra and every Lie group can be expressed as a spin group, which is a spin version of the general linear group, and an invariant method for constructing real spin representations of other classical groups is developed.
Abstract: It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
168 citations
TL;DR: In this paper, it was shown that the q-Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R • matrix.
Abstract: Braided differential operators ∂i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang–Baxter matrix R. The quantum eigenfunctions expR(x‖v) of the ∂i (braided‐plane waves) are introduced in the free case where the position components xi are totally noncommuting. A braided R‐binomial theorem and a braided Taylor theorem expR(a‖∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R‐matrix (and hence to n dimensions) the well‐known properties of the usual one‐dimensional q‐differential and q‐exponential. As a related application, it is shown that the q‐Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[ p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R‐ matrix.
142 citations
TL;DR: In this paper, the relation between x-ray spectra and abstract dynamical systems for structural studies is discussed. And the relation of the two is shown in Section 2.1.
Abstract: Crystallographers use x‐ray spectra to understand the structure of solids and ergodic theorists use spectra of abstract dynamical systems for structural studies; the relation of the two is shown here.
137 citations
TL;DR: In this paper, two different methods of finding Lie point symmetries of differential-difference equations are presented and applied to the two-dimensional Toda lattice, in particular to differential equations of the delay type.
Abstract: Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations, in particular, to differential equations of the delay type. The concept of conditional symmetries is extended from purely differential to differential‐difference equations and shown to incorporate Backlund transformations.
127 citations
TL;DR: In this paper, finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction.
Abstract: Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite‐dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.
122 citations
TL;DR: In this paper, it was shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one-form in three dimensions.
Abstract: It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one‐form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two‐monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)‐valued connection and belongs to a nontrivial Godbillon–Vey class. On the other hand, for the Euler top and a special case of three‐species Lotka–Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi‐Hamiltonian structures. The globally integrable bi‐Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one‐form in 3+1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one‐form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack–McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka–Volterra, May–Leonard, and Maxwell–Bloch systems admit globally integrable bi‐Hamiltonian structure.
111 citations
TL;DR: In this article, it was shown that the renormalized perturbation expansion can be computed by Pade approximants, and that Levin's sequence transformation diverges and is not able to sum all the perturbations.
Abstract: 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. Cižek, 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.
106 citations
TL;DR: A detailed analysis of the constant quantum Yang-Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk 2k3j2j3 Rk 1l2k 1k2 in two dimensions is presented in this article.
Abstract: A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Grobner‐basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well‐known C, P, and T reflections, and rotations and scalings R=κ(Q⊗Q)R(Q⊗Q)−1.
TL;DR: In this article, it is shown that if V(R) and V* (R) are endowed with the necessary braid statistics Ψ then their braided tensor product V(r)⊗_V*(R), which is a realization of the braided matrices B(r), leads to an invariant quantum trace.
Abstract: Quantum matrices A(R) are known for every R matrix obeying the quantum Yang–Baxter equations. It is also known that these act on ‘‘vectors’’ given by the corresponding Zamalodchikov algebra. This interpretation is developed in detail, distinguishing between two forms of this algebra, V(R) (vectors) and V*(R) (covectors). A(R)→V(R21)⊗V*(R) is an algebra homomorphism (i.e., quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. It is shown that if V(R) and V*(R) are endowed with the necessary braid statistics Ψ then their braided tensor‐product V(R)⊗_V*(R) is a realization of the braided matrices B(R) introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from B(R) act on themselves by conjugation in a way impossible...
TL;DR: In this article, a generalization of a class of infinitesimal Backlund transformations originally introduced in a gas-dynamics context by Loewner in 1952 leads to a linear representation for a novel class of 2+1-dimensional nonlinear equations.
Abstract: A reinterpretation and generalization of a class of infinitesimal Backlund transformations originally introduced in a gas‐dynamics context by Loewner in 1952 leads to a linear representation for a novel class of 2+1‐dimensional nonlinear equations. The latter may be parametrized in terms of a triad of eigenfunctions. Moreover, like the well‐known integrable Davey–Stewartson and Nizhnik–Novikov–Veselov equations, the nonlinear systems typically contain the two spatial variables on an equal footing. The ∂‐dressing method is outlined for these generalized Loewner systems. It is noted that basic reductions lead to 2+1‐dimensional versions of the principal chiral‐fields model, Toda lattice system, and notably the classical sine–Gordon equation. Loewner systems on Grassmannian, the projective CPn and RPn manifolds, are considered. In particular, the 2+1‐dimensional integrable sine–Gordon system in which x and y occur in a symmetric manner arises naturally in this context. A gauge‐equivalent system likewise emerges out of a special reduction of a 2+1‐dimensional Toda lattice scheme constructed here via the Loewner formalism.
TL;DR: In this article, an algebraic description of finite Lorentz transformations of vectors in ten-dimensional Minkowski space is given by means of a parametrization in terms of the octonions.
Abstract: An explicit algebraic description of finite Lorentz transformations of vectors in ten‐dimensional Minkowski space is given by means of a parametrization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way automorphisms of the two highest dimensional normed division algebras, namely, the quaternions and the octonions, are described in terms of conjugation maps. Similar techniques are used to define SO(3) and SO(7) via conjugation, SO(4) via symmetric multiplication, and SO(8) via both symmetric multiplication and one‐sided multiplication. The noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions.
TL;DR: In this paper, a non-local square root of the Klein-Gordon equation is proposed, which is a special relativistic equation for a scalar field of first order in the time derivative.
Abstract: A nonlocal square root of the Klein–Gordon equation is proposed. This nonlocal equation is a special relativistic equation for a scalar field of first order in the time derivative. Its space derivative part is described by a pseudodifferential operator. The usual quantum mechanical formalism can be set up. The nonrelativistic limit and the classical limit in the form of plane wave solutions and the Ehrenfest theorem are correctly included. The nonlocality of the wave equation does not disturb the light cone structure, and the relativity principle of special relativity is fulfilled. Uniqueness and existence of solutions of the Cauchy problem for this equation can be proved. The second quantized version of this theory turns out to be macrocausal.
TL;DR: In this article, the eigenvalue correlations of generalized Gaussian and Laguerre random matrix ensembles are calculated exactly and the fluctuations are shown to be nonuniversal in certain intervals of the spectrum.
Abstract: The eigenvalue correlations of the generalized Gaussian and Laguerre random matrix ensembles are calculated exactly. The fluctuations are shown to be nonuniversal in certain intervals of the spectrum. A physical example from quantum transport where such nonuniversal effects occur is discussed.
TL;DR: In this article, the authors considered nonclassical symmetry solutions of physically relevant partial differential equations via the reduction methods of Bluman and Cole and Clarkson and Kruskal, and provided consistent conditions to show that these two methods are equivalent in the sense that they lead to the same symmetry solutions.
Abstract: Nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of Bluman and Cole and Clarkson and Kruskal. Consistency conditions will be provided to show that, if satisfied, these two methods are equivalent in the sense that they lead to the same symmetry solutions. The Boussinesq equation and Burgers’ equation are used as illustrative examples. Exact solutions, one of which is new, will be presented for Burgers’ equation obtained from the Bluman and Cole method, yet not obtainable by Clarkson and Kruskal’s method.
TL;DR: A generalization of the standard class of solutions in the Kaluza-Klein (4+1) gravity, wherein the spherically symmetric metric depends not only on the radius but also on the extra coordinate, is considered in this paper.
Abstract: A generalization of the standard class of solutions in the Kaluza–Klein (4+1) gravity, wherein the spherically symmetric metric depends not only on the radius but also on the extra coordinate, is considered. Two new classes of exact solutions of the empty Kaluza–Klein field equations are given. However, it is known that apparently empty solutions of the (4+1) Kaluza–Klein equations can be interpreted as solutions with effective matter properties of the (3+1) Einstein equations. The physical importance of the new solutions is that in this approach the dependency on the extra coordinate allows us to obtain more general equations of state than before, including ones for radiation, dust, vacuum, and stiff matter.
TL;DR: In this paper, the Fairbanks theorem was used to find Lax pairs for each of the three integrable Hamiltonian systems under consideration, including the one with genus one and genus two theta functions.
Abstract: The Hamiltonian system corresponding to the (generalized) Henon–Heiles Hamiltonian H= 1/2(px2+py2)+1/2Ax2+1/2By2+x2y+ey3 is known to be integrable in the following three cases: (A=B, e=1/3); (e=2); (B=16A, e=16/3). In the first two the system has been integrated by making use of genus one and genus two theta functions. We show that in the third case the system can also be integrated by making use of elliptic functions. Finally, using the Fairbanks theorem, we find Lax pairs for each of the three integrable systems under consideration.
TL;DR: In this paper, simple eigenvalue tests are given to ascertain that a given real 4×4 matrix transforms the four-vector of Stokes parameters of a beam of light into the fourvector of the Stokes parameter of another beam, and whether a given 4 × 4 matrix is a weighted sum of pure Mueller matrices.
Abstract: Simple eigenvalue tests are given to ascertain that a given real 4×4 matrix transforms the four‐vector of Stokes parameters of a beam of light into the four‐vector of Stokes parameters of another beam of light, and to determine whether a given 4×4 matrix is a weighted sum of pure Mueller matrices. The latter result is derived for matrices satisfying a certain symmetry condition. To derive these results indefinite inner products are applied.
TL;DR: In this paper, the mode functions of the electromagnetic field in ideal cavity which boundary oscillates at a resonance frequency are obtained in the long-time limit, and the rate of photons creation from initial vacuum state is shown to be time independent and proportional to the amplitude of oscillations and the resonance frequency.
Abstract: New solutions for the mode functions of the electromagnetic field in ideal cavity which boundary oscillates at a resonance frequency are obtained in the long‐time limit. The rate of photons creation from initial vacuum state is shown to be time independent and proportional to the amplitude of oscillations and the resonance frequency. Temperature corrections are evaluated. The squeezing coefficients for the quantum states of the field generated are calculated, as well as the backward reaction of the field on the vibrating wall.
TL;DR: In this paper, the algebra of (D+1)dimensional Euclidean group is set up as fundamental commutation relation, which plays the role of the ordinary canonical algebra on the flat space.
Abstract: Quantum mechanics on SD is formulated. The algebra of (D+1)‐dimensional Euclidean group is set up as fundamental commutation relation, which plays the role of the ordinary canonical algebra on the flat space. Compared to the standard procedure of using Dirac’s formalism for quantization of the constrained system, our method leads to more general results. Monopolelike structure appears inside the SD, which becomes visible when the space is cut by 3‐dimensional hyperplane. Gauge potential for this structure is obtained. We also show, that no higher dimensional singularity can appear inside the SD.
TL;DR: In this paper, the authors studied spinor, scalar, and vector irreducible representations of generators of the Lorentz transformations in a four-dimensional Grassmann subspace.
Abstract: Spinor, scalar, and vector irreducible representations of generators of the Lorentz transformations in a four‐dimensional Grassmann subspace of a d‐dimensional Grassmann space are studied. For d=5 there are four spinor representations with the properties of Majorana spinors and two scalar, two three‐vector and two four‐vector representations with the properties of bosons, offering the possibility of the canonical quantization of not only bosonic but also fermionic fields.
TL;DR: In this article, a complete analysis of the projective unitary irreducible representations of the Poincare group in 1+2 dimensions applying the Mackey theorem and using an explicit formula for the universal covering group of the Lorentz group was given.
Abstract: A complete analysis is given of the projective unitary irreducible representations of the Poincare group in 1+2 dimensions applying the Mackey theorem and using an explicit formula for the universal covering group of the Lorentz group in 1+2 dimensions. Explicit formulas are given for all representations.
TL;DR: In this paper, the Yang-Baxter equation can be recast as a triple product equation and new solutions for the simplest triple systems which are called octonionic and quaternionic are found.
Abstract: The Yang–Baxter equation can be recast as a triple product equation. Assuming the triple product to satisfy some algebraic relations, new solutions of the Yang–Baxter equation can be found. This program has been completed here for the simplest triple systems which are called octonionic and quaternionic. The solutions are of rational type.
TL;DR: In this paper, the relativistic Toda lattice equation is decomposed into three Toda systems, the Toda Lattice itself, Backlund transformation of Toda lattice, and discrete time Toda-Lattice.
Abstract: The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, Backlund transformation of Toda lattice, and discrete time Toda lattice. It is shown that the solutions of the equation are given in terms of the Casorati determinant. By using the Casoratian technique, the bilinear equations of Toda systems are reduced to the Laplace expansion form for determinants. The N‐soliton solution is explicitly constructed in the form of the Casorati determinant.
TL;DR: The maximum entropy approach to the inverse problem of moments, in which one seeks to reconstruct a function p(x) from the values of a finite set N+1 of its moments, is studied in this paper.
Abstract: The maximum‐entropy approach to the solution of classical inverse problem of moments, in which one seeks to reconstruct a function p(x) [where x∈(0,+∞)] from the values of a finite set N+1 of its moments, is studied. It is shown that for N≥4 such a function always exists, while for N=2 and N=3 the acceptable values of the moments are singled out analytically. The paper extends to the general case where the results were previously bounded to the case N=2.
TL;DR: In this article, the analytical expressions for the energy gain and transition probabilities between energy levels of a nonrelativistic quantum particle confined in a box with uniformly moving walls, including the cases of adiabatic motion and a sudden change of the size of the box was obtained.
Abstract: The analytical expressions for the energy gain and transition probabilities between energy levels of a nonrelativistic quantum particle confined in a box with uniformly moving walls, including the cases of adiabatic motion and a sudden change of the size of the box was obtained.
TL;DR: In this article, new types of reductions of the Kadomtsev-Petviashvili (KP) hierarchy are considered on the basis of Sato's approach.
Abstract: New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy are considered on the basis of Sato’s approach. Within this approach the KP hierarchy is represented by infinite sets of equations for potentials u2,u3,..., of pseudodifferential operators and their eigenfunctions Ψ and adjoint eigenfunctions Ψ*. The KP hierarchy was studied under constraints of the following type (∑ni=1 ΨiΨ*i)x = Sκ,x where Sκ,x are symmetries for the KP equation and Ψi(λi), Ψ*i(λi) are eigenfunctions with eigenvalue λi. It is shown that for the first three cases κ=2,3,4 these constraints give rise to hierarchies of 1+1‐dimensional commuting flows for the variables u2, Ψ1,...,Ψn, Ψ*1,...,Ψ*n. Bi‐Hamiltonian structures for the new hierarchies are presented.
TL;DR: In this paper, the concept of positons was introduced for the sine-Gordon equation exploiting its covariance under Darboux transformations, where positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one.
Abstract: The concept of positons is introduced for the sine‐Gordon equation exploiting its covariance under Darboux transformations. Positon solutions represent potentials in the associated linear system corresponding to a vanishing reflection coefficient and a transmission coefficient identically one. The mutual interaction of positons and the interaction of positons with solitons is considered.