Volume 27, Issue 11, November 1986
Index of content:

On irreducible endospectral graphs
View Description Hide DescriptionThe paper considers e n d o s p e c t r a l t r e e s, a special class of graphs associated with the production of numerous isospectral graphs. Endospectral graphs have been considered in the literature sporadically (the name was suggested very recently [M. Randić, SIAM J. Algebraic Discrete Meth. 6, 145 (1985)]). They are characterized by the presence of a pair of special vertices that, if replaced by any fragment, produce an isospectral pair of graphs. Recently Jiang [Y. Jiang, Sci. Sin. 2 7, 236 (1984)] and Randić and Kleiner (M. Randić and A. F. Kleiner, ‘‘On the construction of endospectral trees,’’ submitted to Ann. NY Acad. Sci.) considered alternative c o n s t r u c t i v e approaches to endospectral trees and listed numerous such graphs. The listing of a l l such trees having n=16 or fewer vertices has been undertaken here. It has been found that relatively few endospectral trees have novel structural features and cannot be reduced to some already known endospectral tree. These few have been named ‘‘irreducible endospectral trees.’’ They are responsible for the occurrence of a large number of isospectral trees, leading to, when one considers trees of increasing size, the situation that led Schwenk [A. J. Schwenk, in N e w D i r e c t i o n s i n t h e T h e o r y o f G r a p h s, edited by F. Harary (Academic, New York, 1973), pp. 275–307] to conclude that ‘‘almost all trees are isospectral.’’

A complete determination of the zeros of weight‐1 6j coefficients
View Description Hide DescriptionIt is shown how all zeros of weight‐1 6j coefficients arise as particular cases of a four‐parameter family of such zeros. The parametrization is given explicitly.

Algebraic expressions for some multiplicity‐free 6j symbols and isoscalar factors for G_{2}
View Description Hide DescriptionGeneralized 6j symbols for G_{2} in which all four triangular conditions involve the seven‐dimensional irreducible representation (irrep) (10) are multiplicity‐free. Algebraic expressions depending on the highest weights of the accompanying irreps are found by using generalizations of the Racah‐Wigner algebra. A rule is given for generalizing the SO(3) phase factors. The results are applied to finding algebraic expressions for a class of isoscalar factors for SO(7)⊇G_{2}.

Group analysis of the three‐wave resonant system in (2+1) dimensions
View Description Hide DescriptionThe three‐wave resonant interaction equations (2D‐3WR) in two spatial and one temporal dimension within a group framework are analyzed. The symmetry algebra of this system, which turns out to be an infinite‐dimensional Lie algebra whose subalgebra is of the Kac–Moody type, is found. The one‐ and two‐dimensional symmetry subalgebras are classified and the corresponding reduction equations are obtained. From these the new invariant and the partially invariant solutions of the original 2D‐3WR equations are obtained.

Generalized Green’s functions and spectral densities in the complex energy plane
View Description Hide DescriptionThe Titchmarsh–Weyl theory is applied to the Schrödinger equation in the case when the asymptotic form of the solution is not known. It is assumed that the potential belongs to the Weyl’s limit‐point classification. A rigorous analytical continuation of the Green’s function, obtained from the solution regular at the origin and the square integrable Weyl’s solution (regular at infinity), to the ‘‘unphysical’’ Riemann energy sheet is carried out. It is demonstrated how the Green’s function can be uniquely constructed from the Titchmarsh–Weyl m‐function and its Nevanlinna representation. The behavior of the m‐function in the neighborhood of poles is investigated. The m‐function is decomposed in a, so called, generalized real part (Reg) and a generalized imaginary part (Img). Reg(m) is found to have a significant argument change upon pole passages. Img(m) is found to be a generalized spectral density. From the generalized spectral density, a spectral resolution of the differential operator and its resolvent is derived. In the expansion contributions are obtained from bound states, resonance states (Gamow states), and the ‘‘deformed continuum’’ given by the generalized spectral density.
The present expansion theorem is applicable to the general partial differential operator via a decomposition into partial waves. The numerical procedure involves all quantum numbers l and m, but for convenience, and with the inverse problem in mind, this study is focused on the case when the rotational quantum number equals zero. The theory is tested numerically and analyzed for an analytic model potential exhibiting a barrier and decreasing exponentially at infinity. The potential is Weyl’s limit point at infinity and allows for an analytical continuation into a sector in the complex plane. An attractive feature of the generalized spectral density of the present potential is that the poles close to the real axis seem to exhaust or deflate the above‐mentioned density inside the pole string. Outside this string the density rapidly approaches that of a free particle. This information is used to derive an approximate representation of the m‐function in terms of poles and residues as well as free‐particle background. In order to display the features mentioned above, the present study is accompanied with several plots of analytically continued quantities related to the Green’s function.

Auto‐Bäcklund transformation, Lax pairs, and Painlevé property of a variable coefficient Korteweg–de Vries equation. I
View Description Hide DescriptionUsing the Painlevé property of partial differential equations, the auto‐Bäcklund transformation and Lax pairs for a Korteweg–de Vries (KdV) equation with time‐dependent coefficients are obtained. The Lax pair criterion also makes it possible for some new models of the variable coefficient KdV equation to be found that can represent nonsoliton dynamical systems. This can explain the wave breaking phenomenon in variable depth shallow water.

A variable coefficient Korteweg–de Vries equation: Similarity analysis and exact solution. II
View Description Hide DescriptionA Korteweg–de Vries (KdV) equation with time‐dependent coefficients is studied in this paper. The similarity transformation for this system is investigated and an exact solution in a particular case is obtained. The Ablowitz–Ramani–Segur (ARS) conjecture is used to identify the integrability of the system. It is found that in some special cases the system may be integrable.

Periodic fixed points of Bäcklund transformations and the Korteweg–de Vries equation
View Description Hide DescriptionA new method for studying integrable systems based on the ‘‘periodic fixed points’’ of Bäcklund transformations (BT’s) is presented. Normally the BT maps an ‘‘old’’ solution into a ‘‘new’’ solution and requires a known ‘‘seed’’ solution to get started. Besides this limitation, it can also be difficult to qualitatively classify the result of applying the BT several times to a known solution. By studying the periodic fixed points of the BT (regarded as a nonlinear map in a function space), integrable systems of equations of finite degree (equal to the order of the fixed point) and a method for the systematic classification of the solutions of the original system are obtained.

On the stable analytic continuation with a condition of uniform boundedness
View Description Hide DescriptionIt is shown that, if h(x) is any continuous function defined on some interval [−a,b]⊆(−1,1) of the real axis, then, in general, its best L ^{2} approximant, in the class of functions holomorphic and bounded by unity in the unit disk of the complex plane, is a finite Blaschke product. An upper bound is placed on the number of factors of the latter and a method for its construction is given. The paper contains a discussion of the use of these results in performing a stable analytic continuation of a set of data points under a condition of uniform boundedness, as well as some numerical examples.

Inverse scattering for geophysical problems when the background is variable
View Description Hide DescriptionAn exact method for finding an inhomogeneity for a variable background from the knowledge of the scattered field on some manifolds is given.

A note on topology of supermanifolds
View Description Hide DescriptionA topological structure on the space of supernumbers is introduced, with which this space becomes a Fréchet space. The definition of supermanifold and ‘‘superprojective space’’ are given. The superprojective space is one of paracompact Hausdorff supermanifolds.

Killing tensors in spaces of constant curvature
View Description Hide DescriptionA Killing tensor is one possible way of generalizing the notion of a Killing vector on a Riemannian or pseudo‐Riemannian manifold. It is explained how Killing tensors may be identified with functions that are homogeneous polynomials in the fibers on the associated cotangent bundle. As such, Killing tensors may be identified with first integrals of the Hamiltonian geodesic flow, which are homogeneous polynomials in the momenta. Again using this identification, it is shown that in flat spaces the dimension of the vector space of Killing tensors is maximal and that the Killing tensors are generated by the Killing vectors. Finally, using Riemann’s model for the metric in spaces of constant curvature, a comparison argument is used to show that similar results are valid in that more general context.

Some problems of spinor and algebraic spinor structures
View Description Hide DescriptionA concrete realization of the Milnor–Lichnerowicz spinor bundle by algebraic spinors is considered in the case when the holonomy group of the Levi‐Civita connection is equal to the Crumeyrolle group. Some relationships between the existence of parallel spinor fields on a space‐time manifoldM and its topological invariants are given.

Hydrodynamics of a topologically nontrivial metric
View Description Hide DescriptionThe curvature and Einstein tensor are computed for a metric having one or more kinks (solitons) present. It is pointed out that the components of the fluid velocity four‐vector can be identified in a natural way with certain parameters present in the metric. Making this identification, a number of hydrodynamical quantities are computed.

Quantum causal structures
View Description Hide DescriptionThe axioms of a causal structure are reformulated. A natural generalization is suggested for the case when the subset lattice of space‐time events is replaced by a lattice coming from the quantum theory.

Special solutions of singular nonlinear Schrödinger equations with polynomial nonlinearities
View Description Hide DescriptionThe singular nonlinear Schrödinger equation i u _{ t } =−u _{ x x } +f(‖u‖^{2})u+χh(‖u‖^{2})h’(‖u‖^{2})u, where f has the form f(s)=a s ^{ n }, n≥1, a∈R, is investigated. A classification is given of those nonlinearities f and h that allow the existence of solitary waves and kink solutions. Further, in several cases the solutions are given in explicit form.

On renormalization in nonrelativistic interaction models with infinities
View Description Hide DescriptionThe Green’s function of two models with nonrelativistic separable interaction giving rise to infinities in the perturbation expansion is studied. These infinities do not arise from the E+i0 limit, but come from the slow falloff behavior of the vertices, modeled after the infinities in Feynman graphs of field theory. Both models are analytically solvable. It is found that the Green’s function obtained from summing the renormalized perturbation series is identical to the direct solution of the Green’s function, which requires only an intermediate regularization. In the first model the interaction is split in a singular part giving raise to infinities and a regular part. It is shown that the Green’s function is the same as the Green’s function derived from only the regular part. This effect is similar to the effect occurring in φ^{4} field theory in 3+1 dimensions, where the φ^{4} interaction vanishes after renormalization and the S matrix is trivial. The second model is constructed such that parts of the singular interaction survive in the Green’s function.

Low‐energy scattering for medium‐range potentials
View Description Hide DescriptionThe low‐energy behavior of the transmission coefficient in one dimension and of the phase shifts in two and three dimensions is studied for the Schrödinger equation with central potentials that have finite absolute moments of order between 1 and 2. Resulting modifications of Levinson’s theorem are also derived.

New classical properties of quantum coherent states
View Description Hide DescriptionA noncommutative version of the Cramer theorem is used to show that if two quantum systems are prepared independently, and if their center of mass is found to be in a coherent state, then each of the component systems is also in a coherent state, centered around the position in phase space predicted by the classical theory. Thermal coherent states are also shown to possess properties similar to classical ones.

Phase‐integral formulas for Bessel functions and their relation to already existing asymptotic formulas
View Description Hide DescriptionThe phase‐integral method devised by Fröman and Fröman [N. Fröman and P. O. Fröman, J W K B A p p r o x i m a t i o n, C o n t r i b u t i o n s t o t h e T h e o r y (North‐Holland, Amsterdam, 1965); Ann. Phys. (NY) 8 3, 103 (1974); Nuovo Cimento B 2 0, 121 (1974); N. Fröman, Ark. Fys. 3 2, 541 (1966); Ann. Phys. (NY) 6 1, 451 (1970)], involving a general phase‐integral approximation of arbitrary order, which is generated from an unspecified base function, is used for deriving first‐ and higher‐order phase‐integral formulas for Bessel functions. For different choices of the base function one thus obtains in a systematic way different kinds of asymptotic formulas. By series expansion of these formulas one obtains already existing asymptotic formulas presented is standard handbooks. The phase‐integral formulas are seen to have certain advantages that those latter formulas do not possess.