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Volume 23, Issue 11, November 1982

Path integral for coherent states of the dynamical group SU(1,1)
View Description Hide DescriptionPath integrals over coherent states of the dynamical group (noninvariance group) SU(1,1) are constructed. From the continuous limit the relevant classical dynamics is extracted and is shown to take place in a curved phase space of the form of a Lobachevskii plane. Applications are made to the harmonic oscillator, a model of superfluidhelium, the Morse oscillator, and the hydrogen atom. It is shown that when SU(1,1) is the relevant dynamical group the motion will appear oscillator‐like on the Lobachevskii plane.

Dynamical group of microscopic collective states. II. Boson representations in d dimensions
View Description Hide DescriptionThe present series of papers deals with various realizations of the dynamical group Sp_{c}(2d,R) of microscopic collective states for an Anucleon system in d dimensions, defined as those A particle states invariant under the orthogonal group O(n) associated with the n=A−1 Jacobi vectors. In the present paper, we derive two boson representations of Sp_{c}(2d,R), namely the Dyson representation and the Holstein–Primakoff (HP) one. Our starting point is a representation of microscopic collective states, as introduced in the first paper of the present series, in a Barut Hilbert spaceF_{c} of analytic functions in ν =(1/2)d(d+1) complex variables. Basis functions in F_{c}, classified according to the chain Sp_{c}(2d,R)⊇U_{c}(d), can be put into one‐to‐one correspondence with basis functions, classified according to the chain U(ν)⊇U(d), in a Bargmann Hilbert spaceB of analytic functions in ν complex variables representing ν‐dimensional boson states. By equating the complex variables of F_{c} and their conjugate momenta with those of B, we get the non‐Hermitian Dyson representation of Sp_{c}(d,R). We then go from the latter to the Hermitian HP representation by means of a canonical transformation that restores the Hermiticity properties of the variables and conjugate momenta. The inverse of the HP representation gives the unitary representation in quantum mechanics of the classical canonical transformation relating the oscillator Hamiltonians of the microscopic collective model and the boson macroscopic one. From the ν boson creation and annihilation operators, it is possible to build the generators of a U(ν) group, which in the physical three‐dimensional case reduces to U(6). The latter is finally compared with the U(6) group appearing in the interacting boson model.

G _{2} van der Waerden invariant
View Description Hide DescriptionThe G _{2} van der Waerden invariant is given. It solves the external labeling problem connected with direct products of irreducible representations of G _{2}.

Cartan–Gram determinants for the simple Lie groups
View Description Hide DescriptionThe Cartan–Gram determinants for the simple root systems are evaluated for the simple Lie groupsA _{ n }, B _{ n }, C _{ n }, D _{ n }, and E _{ k } (k=6,7,8). The determinants satisfy a linear recursion relation which turns out to be the same for all these groups. For the E _{ n } family, the Cartan–Gram determinant contains an explicit factor of (9−n) which vanishes for n=9 and is negative for n>9. This gives a simple explanation why the E _{ n } family terminates at E _{8}. The Cartan–Gram determinant affords a systematic explanation for the nonexistence of the forbidden Dynkin diagrams.

A closed formula for the product of irreducible representations of SU(3)
View Description Hide DescriptionWe determine a closed formula in terms of p, q, r, and s for the decomposition of the product [p, q] [r, s] of finite‐dimensional irreducible representations of SU(3). We also determine in terms of p, q, r, s, m, and n necessary and sufficient conditions that a term [m, n] appears in this decomposition and its multiplicity.

Addition formula for the Z _{ N }×Z _{ N } symmetric solutions of the factorization equations
View Description Hide DescriptionAn addition formula is derived which contains the addition relations for theta functions with characteristic proportional to 1/N, N integer.

On a unified approach to transformations and elementary solutions of Painlevé equations
View Description Hide DescriptionAn algorithmic method is developed for investigating the transformation properties of second‐order equations of Painlevé type. This method, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painlevé equations II–VI, with different parameters. These transformations easily generate rational and other elementary solutions of the equations. The relationship between Painlevé equations and certain new equations quadratic in the second derivative of Painlevé type is also discussed.

The equivalence problem for the heat equation
View Description Hide DescriptionIn this note we ask for the classes of equations of the second order which can be transformed into the heat equationu _{ t }=u _{ x x }. To give a partial answer to the question we express the heat equation by differential forms and prolong it by the Estabrook–Wahlquist method. This is motivated by the fact that our analysis is based upon conservation laws for which ideals of differential forms are a very suitable framework. Necessary conditions are derived for deciding whether a given equation can be transformed by some invertible point transformation into the heat equation or into its prolongation. In particular, the prolongation method enables us to understand the connection of various equations to the heat equation.

Space–time memory functions and solution of nonlinear evolution equations
View Description Hide DescriptionA new approach is presented for solving a certain class of nonlinear partial differential equations. A space–time memory function Λ(r,t) is introduced to exactly convert a given nonlinear evolution equation into the following linear form: (∂/∂t) f(r,t)=Ω(r) f(r,t) +∫^{ t } _{0} d t′∫d r′Λ(r−r′,t−t′)f(r′,t′). A Markovian integro‐differential operator Ω(r) and the memory function Λ(r,t) reflect the nonlinearity, and are determined depending on a given initial condition. The approach is useful if higher‐order memory functions associated with Λ are insensitive to approximation. The Korteweg–de Vries equation is treated as an example. For certain initial profiles the memory function is shown to be identically zero, and we find exact l i n e a rpartial differential equations leading to the single‐ and the two‐soliton solution. In the case of the three‐soliton solution, the second‐order memory function vanishes exactly, and Λ(r,t) is found to be a single exponential function of t.

Discontinuous solutions for first‐order systems through a limiting process
View Description Hide DescriptionThe possibility for physically general quasilinear differential systems to have discontinuous solutions or solutions with discontinuous derivatives is investigated using the method of asymptotic regularizations.

Laplace asymptotic expansions of conditional Wiener integrals and generalized Mehler kernel formulas
View Description Hide DescriptionImitating Schilder’s results for Wiener integrals rigorous Laplace asymptotic expansions are proven for conditional Wiener integrals. Applications are given for deriving generalized Mehler kernel formulas, up to arbitrarily high orders in powers of ℏ, for exp{−T H(ℏ)/ℏ}(x, y), T>0 where H(ℏ)=[(−ℏ^{2}/2)Δ_{1}+V], Δ_{1} being the one‐dimensional Laplacian, V being a real‐valued potential V∈C ^{∞}(R), bounded below, together with its second derivative.

Moving frames and prolongation algebras
View Description Hide DescriptionWe consider differential ideals generated by sets of 2‐forms which can be written with constant coefficients in a canonical basis of 1‐forms. By setting up a Cartan–Ehresmann connection, in a fiber bundle over a base space in which the 2‐forms live, one finds an incomplete Lie algebra of vector fields in the fibers. Conversely, g i v e n this algebra (a p r o l o n g a t i o nalgebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg–de Vries and Harrison–Ernst systems.

The differential geometric structure of general mechanical systems from the Lagrangian point of view
View Description Hide DescriptionSeveral differential‐geometric points of view on analytical mechanics of systems with a finite number of degrees of freedom are developed in generality, emphasizing Cartan’s calculus of differential forms and Ehresmann’s theory of jet spaces. The classical theory of Lagrange’s equationsw i t h e x t e r n a l f o r c e s a n d c o n s t r a i n t s (‘‘holonomic’’ or ‘‘nonholonomic’’) is put into an invariant and coordinate‐free form. The relation between this ‘‘Lagrangian’’ and the ‘‘Hamiltonian‐symplectic’’ approach, which is that most extensively used in the contemporary mathematical physics literature, is also developed.

The equivalence of two approaches to the Feynman integral
View Description Hide DescriptionTwo apparently quite different Banach algebras of functions have been introduced and studied recently in connection with the theory of the ‘‘Feynman integral.’’ The functions in both spaces have been shown to be ‘‘Feynman integrable,’’ but two different definitions of the ‘‘Feynman integral’’ were used. We show here that the two spaces are in fact isometrically isomorphic as Banach algebras where the correspondence is given by what is essentially an extension (or restriction) map. Further, the ‘‘Feynman integrals,’’ in the two different senses, of corresponding functions are equal. The equivalence between these two theories is surprisingly easy to prove but has a number of consequences for both theories. In the last section of the paper we give a few simple but useful consequences and make some remarks about our experience so far in using the equivalence.

Approximate methods for the solution of the Chandrasekhar H‐equation
View Description Hide DescriptionWe consider two methods of approximate solution to matrix valued analogs of the Chandrasekhar H‐equation. We give conditions under which they converge. The first method is a generalization of approximation of the integral by a quadrature. The second is Newton’s method.

Lagrangians for spherically symmetric potentials
View Description Hide DescriptionTwo Lagrangians are s‐equivalent (s for ‘‘solution’’) if they yield equations of motion having the same set of solutions. We consider Lagrangians s‐equivalent to T−V, where T is flat space kinetic energy and V is a spherically symmetric potential. We show that for n=dimension of space ≥3, there are many s‐equivalent Lagrangians which cannot be formed from T−V by multiplication by a constant or addition of a total time derivative. In general these s‐equivalent Lagrangians lead to inequivalent quantum theories in the sense that the energy spectra are different.

Structure of three‐twistor particles
View Description Hide DescriptionThe simplest physical system to have a nontrivial intrinsic structure in Minkowski space‐time is a three‐twistor particle. We investigate this structure and the two pictures of the particle as an extended object in space‐time and as a point in unitary space. We consider the effect of twistor translations on the mass triangle defined by the partial center of mass points in space‐time. Finally we consider the connections between twistor rotations and spin and we establish the spin deficiency formula.

Trace identities in the inverse scattering transform method associated with matrix Schrödinger operators
View Description Hide DescriptionTrace identities arising in the scattering theory of one‐dimensional matrix Schrödinger operators are deduced. They derive from the properties of an asymptotic expansion of the trace of the resolvent kernel in inverse powers of the spectral parameter. Applications of these trace identities for characterizing infinite families of conservation laws for nonlinear evolution equations are given.

Langer’s method for weakly bound states of the Helmholtz equation with symmetric profiles
View Description Hide DescriptionUse of the harmonic oscillatorequation as the comparison equation in the application of Langer’s method to bound states of the Helmholtz equation,w″+k ^{2} _{0} g(z)w(z)=0, with symmetric profiles k ^{2} _{0} g(z), produces the WKB eigenvalue condition, which asserts the equality of the phase integral of the original equation between the turning points to (n+1/2)π. In the case of weakly bound states, this condition gives eigenvalue estimates of low accuracy. Use of the Helmholtz equation with the symmetric Epstein profile, G(x)=[Ẽ+U _{0}(cosh αx)^{−} ^{2}], as the comparison equation provides the basis for a convenient method to obtain eigenvalue estimates of substantially increased accuracy in the case of weakly bound states. In addition to the usual condition of equality of the phase integrals of the original and comparison equations between the turning points, the conditions k ^{2} _{0} g(0) =G(0) and k ^{2} _{0} g(∞) =G(∞) are imposed. An eigenvalue condition which is a simple generalization of the usual WKB eigenvalue condition is obtained. Its application to selected diverse examples of the Helmholtz equation indicates that it has a broad range of utility.

Formal solutions of inverse scattering problems. IV. Error estimates
View Description Hide DescriptionThe formal solutions of certain three‐dimensional inverse scattering problems presented in papers I–III in this series [J. Math. Phys. 1 0, 1819 (1969); 1 7, 1175 (1976); 2 1, 2648 (1980)] are employed here to obtain quantitative estimates on the error resulting from the use of the Born approximations in both direct and inverse potential scattering problems. These estimates are uniformly valid at all energies, and for all sufficiently weak potentials.