Volume 31, Issue 11, November 1990
Index of content:

Beltrami algebra and symmetry of the Beltrami equation on Riemann surfaces
View Description Hide DescriptionIt is shown that the Beltrami equation has an infinite‐dimensional symmetry, namely the Beltrami algebra, on its solution spaces. The Beltrami algebra with central extension and its supersymmetric version are explicitly found.

Three‐dimensional quantum groups from contractions of SU(2)_{ q }
View Description Hide DescriptionContractions of Lie algebras and of their representations are generalized to define new quantum groups. An explicit and complete exposition is made for the one‐dimensional Heisenberg H(1)_{ q } and the two‐dimensional Euclidean quantum group E(2)_{ q } obtained by contracting SU(2)_{ q }.

Classification of all star irreps of gl(m‖n)
View Description Hide DescriptionAll the finite‐dimensional star irreps of gl(m‖n) are classified in terms of their highest weights, thereby completing the classification of all finite‐dimensional star irreps of the basic classical Lie superalgebras. The lowest weights of such irreps are determined explicitly and it is shown that the contravariant and covariant tensor irreps of gl(m‖n) are star irreps of type (1) and (2), respectively. This explains the applicability of the Young diagram method to these two classes of representations. However, it is shown that one‐parameter families of typical star irreps occur that are intrinsically different from the tensorial irreps.

Automorphisms and general charge conjugations
View Description Hide DescriptionIt is shown how outer automorphisms of a semisimple Lie algebra and automorphisms of the corresponding root system describe conjugations of general charges. New results about representations of semidirect products of non‐Abelian Lie groups and about projective representations of finite groups are derived.

On a Fourier integral over SO(3)
View Description Hide DescriptionA two‐matrix function of general interest in the areas of configuration statistics of macromolecules,number theory, harmonic analysis, and multivariate statistics is studied. The function is defined as a Fourier integral over SO(3), the Lie group of orthogonal 3×3 matrices with unit determinant. This six‐variable function is first expressed as a product of a three‐variable function and an exponential function of an additional variable, thereby reducing the total number of independent variables by 2. The new function with three parameters is expressible either as a double integral or as a series in one of the variables with the coefficients being polynomials in the other two. A special, nontrivial case where one of three arguments of the function takes a particular value is explored thoroughly. The resulting two‐variable function is real valued and is an oscillating function of one of the variables when the other is fixed. When this function is expanded as a power series in one of the two variables, it generates polynomials in the other variable. Numerical analysis of this series shows it to be rapidly convergent and it is of practical use in the numerical evaluation of the function. Although the connection between these newly found polynomials and zonal polynomials has not been investigated, the parametrization for the four new variables of the two‐matrix function studied may well prove useful in the effective numerical evaluation of the function when expressed alternatively as a series in zonal polynomials with an exponential part factored out.

A parametrization for independent variables of a two‐matrix hypergeometric function
View Description Hide DescriptionA new parametrization for the independent variables of the two‐matrix hypergeometric function _{0} F ^{(k)} _{0}(a,b) is given. This parametrization results in a net reduction of two variables. The new arguments for the zonal polynomials in the series representation of _{0} F ^{(k)} _{0}(a,b), multiplied by an exponential factor, are all in the interval [0,1], which greatly improves the convergence of the series. The results are of practical use in a number of applications, such as the configuration statistics of macromolecules, shapes of random walks, and multivariate statistics.

A search for integrable bilinear equations: The Painlevé approach
View Description Hide DescriptionThe possibility of the existence of new integrable partial differential equations is investigated, the tools of singularity analysis. The equations treated are written in the Hirota bilinear formalism. It is shown here how to apply the Painlevé method directly under the bilinear form. Just by studying the dominant part of the equations, the number of cases to be considered can be limited drastically. Finally, the partial differential equations identified in a previous work [J. Hietarinta, J. Meth. Phys. 2 8, 1732, 2096, and 2586 (1987); 2 9, 628 (1988)] as possessing at least four soliton solutions, are shown to pass the Painlevé test as well, which is a strong indication of their integrability.

Nonperturbative square‐well approximation to a quantum theory
View Description Hide DescriptionThe possibility of expressing the solution to a φ^{2P }quantum field theory as a series in powers of 1/P is proposed. Such a series would be nonperturbative in its dependence on the fundamental parameters of the theory such as the mass and the coupling constant. The first term in such a series describes a field in an infinite‐dimensional square‐well potential. In this paper, the quantum‐mechanical Hamiltonian H=p ^{2}+q ^{2P } is studied as a model calculation and the expansion of the energy levels as series in powers of 1/P is examined. The method of matched asymptotic expansions to determine the first five terms in the series for all energy levels is used. The results are compared with extensive numerical calculations of the ground‐state energy and it is found that the series is extremely accurate: When P=2, the five‐term series has a relative error of 6%, when P=10 the relative error is 0.009%, and when P=200 the relative error is 3.4×10^{−9}%.

On the quantization of the modified Pöschl–Teller potential
View Description Hide DescriptionIt is shown that the connection one‐form on the prequantization line bundle that generates the correct energy spectrum for the modified Pöschl–Teller potential consists of the canonical one‐form on the corresponding cotangent bundle, together with an adjustment term that arises from the Berry’s connection.

Accuracy‐weighted variational principle for degenerate continuum states
View Description Hide DescriptionA variational principle for continuum states is given that permits numerical solution by the Ritz method. It allows one to maximize the accuracy of the solution in preselected regions of space, and also allows the selection of that solution from the perhaps infinitely degenerate solution set that is needed in the particular application.

Physical constraints on the coefficients of Fourier expansions in cylindrical coordinates
View Description Hide DescriptionIt is demonstrated that (i) the postulate of infinite differentiability in Cartesian coordinates and (ii) the physical assumption of regularity on the axis of a cylindrical coordinate system provide significant simplifying constraints on the coefficients of Fourier expansions in c y l i n d r i c a l c o o r d i n a t e s. These constraints are i n d e p e n d e n t of any governing equations. The simplification can provide considerable practical benefit for the analysis (especially numerical) of actual physical problems. Of equal importance, these constraints demonstrate that if A is a n y a r b i t r a r y physical vector, then the o n l y finite Fourier terms of A _{ r } and A _{θ} are those with m=1 symmetry. In the Appendix, it is further shown that postulate (i) may be inferred from a more primitive assumption, namely, the arbitrariness of the location of the cylindrical axis of the coordinate system.

Bäcklund transformations for generalized nonlinear Schrödinger equations
View Description Hide DescriptionA general class of Bäcklund transformations are considered for equations of the form i z _{ y }+z _{ x x }+f(z,z̄)=0, where f(z,z̄) is a function of z=x+i y and z̄=x−i y. The nonlinear forms of this equation that admit such transformations are completely classified and shown to exist only when f(z,z̄)=z ^{2} z̄ (the nonlinear Schrödinger equation), z ln z̄, z ln z, (z+z̄)^{2}, or suitable combinations of these functions. The form f(z,z̄)=(z+z̄)^{2} leads to auto‐Bäcklund transformations for the Boussinesq equation.

An alternate characterization of integrability
View Description Hide DescriptionThis paper will show that the existence of at least three independent symplectic forms (related in a simple way) on the phase space of a dynamical system is a sufficient condition for the integrability of the system.

Multi‐Hamiltonian structure of equations of hydrodynamic type
View Description Hide DescriptionThe discussion of the Hamiltonian structure of two‐component equations of hydrodynamic type is completed by presenting the Hamiltonian operators for Euler’s equation governing the motion of plane sound waves of finite amplitude and another quasilinear second‐order wave equation. There exists a doubly infinite family of conserved Hamiltonians for the equations of gas dynamics that degenerate into one, namely, the Benney sequence, for shallow‐water waves. Infinite sequences of conserved quantities for these equations are also presented. In the case of multicomponent equations of hydrodynamic type, it is shown, that Kodama’s generalization of the shallow‐water equations admits bi‐Hamiltonian structure.

Multidimensional solitons and their spectral transforms
View Description Hide DescriptionThe soliton solution to the hierarchy of two‐dimensional nonlinear evolution equations related to the Zakharov–Shabat spectral problem (including the Davey–Stewartson equation) are derived and studied. The solitons are localized two‐dimensional structures traveling on straight lines at constant velocities. Their spectral transform is not uniquely defined and this point is discussed by giving two explicit different spectral transforms of the one‐soliton solution and also by giving the general dependence of the spectral transform on the definition of the basic Jost‐like solutions.

Multiple scattering of elastic waves by a distribution of identical spheres
View Description Hide DescriptionSelf‐consistent integral equations for multiple scattering amplitudes and dispersion relations for coherent propagation of elastic waves, are obtained from general reciprocity, two space scatterer formalism and by averaging the functional equations relating the multiple and single scattering amplitudes of the spheres. Approximations for bulk wave propagation numbers are derived.

Reciprocity relations and energy theorems for multiple scattering of elastic waves
View Description Hide DescriptionReciprocity relations, energy theorems, and scattering cross sections are derived for a fixed configuration of N‐identical and similarly aligned lossless scatterers whose centers are uniformly distributed in a given volume. The general results are specialized to scattering in the forward direction.

Nonlinear corrections to the Schrödinger equation from geometric quantum mechanics
View Description Hide DescriptionThe nonlinear corrections to the Schrödinger equation based on the results of geometricquantum mechanics (GQM) are derived. Such theory derives quantum mechanics from the underlying Weyl space‐time geometry associated with the classical ensemble of particle paths given by the solutions of the Hamilton–Jacobi equation.

Quantum mechanics with higher time derivatives: Some mathematical facts and the inclusion of gravitational self‐interaction
View Description Hide DescriptionThe inclusion of self‐interaction, in particular, gravitational into the quantum mechanics of a particle (first quantization) implies two necessary features: dealing with quantum mechanics involving the second time derivative and the construction of a variational principle maintaining the rescaling invariance of the wave function. The relevant variational principle is constructed which is not the principle of extremality of any action. Some general mathematical facts lain in the basis of this construction are discussed.

An asymptotic analysis and its application to the nonrelativistic limit of the Pauli–Fierz and a spin‐boson model
View Description Hide DescriptionAn abstract asymptotic theory of a family of self‐adjoint operators {H _{κ}}_{κ>0} acting in the tensor product of two Hilbert spaces is presented and it is applied to the nonrelativistic limit of the Pauli–Fierz model in quantum electrodynamics and of a spin‐boson model. It is proven that the resolvent of H _{κ} converges strongly as κ→∞ and the limit is a pseudoresolvent, which defines an ‘‘effective operator’’ of H _{κ} at κ≊∞. As corollaries of this result, some limit theorems for H _{κ} are obtained, including a theorem on spectral concentration. An asymptotic estimate of the infimum of the spectrum (the ground state energy) of H _{κ} is also given. The application of the abstract theory to the above models yields some new rigorous results for them.