7720717
Solvable quantum field theories and polynomial conserved quantities for the quantum nonlinear Schrödinger equation.
119127302
Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape invariant Hamiltonians of the radial oscillator and the Darboux-Pöschl-Teller potential to the shape invariant potentials of Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of generalised Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.
444250
Hierarchies of spin models related to Calogero–Moser models
The universal formulation of spin exchange models related to Calogero–Moser models implies the existence of integrable hierarchies, which have not been explored. We show the general structures and features of the spin exchange model hierarchies by taking as examples the well-known Heisenberg spin chain with the nearest neighbour interactions. The energy spectra of the second member of the hierarchy belonging to the models based on the Ar root systems (r=3,4,5) are explicitly and exactly evaluated. They show many interesting features and in particular, much higher degree of degeneracy than the original Heisenberg model, as expected from the integrability.
122772294
Vertex Operators for a Bosonic String
Abstract Based on the operator formalism and the Virasoro algebra, we present a simple method of constructing vertex operators describing the emission and absorption of general particles in bosonic string theories.
14037238
Generalised Calogero-Moser Models and Universal Lax Pair Operators
Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones basedon crystallographic root systems, namely those associatedwith Lie algebras. Universal Lax pair operators for all of the generalisedCalogero-Moser mod els andfor any choices of the potentials are constructedas linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A2, B2, G2, and I2(m). The root type andthe minimal type Lax pairs, d erivedin our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models. Generalised Calogero-Moser models are integrable many-particle dynamical systems based on finite reflection groups. Finite reflection groups include the dihedral groups I2(m) and H3 and H4 together with the Weyl groups of the root systems associated with Lie algebras, called crystallographic root systems. Integrability of classical Calogero-Moser models based on the crystallographic root systems 1), 2) is shown in terms of Lax pairs. The root and the minimal type Lax pairs derived in our previous papers 3) provide a universal framework for these Calogero-Moser models, including those based on exceptional root systems and the twisted models. On the other hand, a theory of classical integrability for the models based on non-crystallographic root systems has been virtually non-existent. This is in sharp contrast with the quantum counterpart. Dunkl operators, which are useful for solving quantum Calogero-Moser models, were first explicitly constructed for the models based on the dihedral groups. 4)
119144805
Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice
We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the Korteweg de Vries (KdV) hierarchy. An N-soliton potential has the time t and 2N positive parameters, k1 < ⋯ < kN and {cj}, j = 1, …, N, corresponding to N discrete eigenvalues {−kj2}. The eigenfunctions are elementary functions expressed by the ratio of determinants. The Darboux-Crum-Krein-Adler transformations or the Abraham-Moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters {cj′}. We explore various identities satisfied by the eigenfunctions of the soliton potentials, which reflect the uniqueness theorem of Gel'fand-Levitan-Marchenko equations for separable (degenerate) kernels.
30945770
Euclidean vacuum mode functions for a scalar field on open de Sitter space.
Motivated by recent studies of the one-bubble inflationary universe scenario that predicts a low density, negative curvature universe, we investigate the Euclidean vacuum mode functions of a scalar field in a spatially open chart of de Sitter space which is foliated by hyperbolic time slices. When we consider the possibility of an open inflationary universe, we are faced with the problem of the initial condition for the quantum fluctuations of the inflaton field, because the inflationary era should not last too long to lose all information of the initial condition. In the one-bubble scenario, in which an open universe is created in an exponentially expanding false vacuum universe triggered by quantum decay of a false vacuum, it seems natural that the initial state is the de Sitter\char21{}invariant Euclidean vacuum. Here we present explicit expressions for the Euclidean vacuum mode functions in the open chart for a scalar field with arbitrary mass and curvature coupling.
19319131
Statistics of baryon isocurvature perturbations in the inflationary universe.
The statistical distribution of baryon-number fluctuations, which may provide a proper initial condition for the minimal isocurvature scenario, is carefully investigated both analytically and numerically. For fluctuations associated with power-law inflation, we find that the distribution is highly non-Gaussian on scales of pregalactic star formation while it is Gaussian on scales of large-scale structure. On the other hand, in the pure de Sitter universe, it is shown to be Gaussian on any astrophysical scale