Ryu Sasaki

Affine Toda field theory and exact S-matrices

Abstract The masses and three-point couplings for all affine Toda theories are calculated. The exact factorisable S-matrices are conjectured on the basis of the classical masses and couplings and found, in the case of theories based on simply-laced algebras, to give consistent solutions to the bootstrap. An investigation of the properties of the exact S-matrices in perturbation theory is begun but non-perturbative methods will be required to understand the conjectured duality between weak and strong coupling which appears to be a striking feature of these theories.

Infinitely many shape invariant potentials and new orthogonal polynomials

Three sets of exactly solvable one-dimensional quantum mechanical potentials are presented. These are shape invariant potentials obtained by deforming the radial oscillator and the trigonometric/hyperbolic Poschl-Teller potentials in terms of their degree l polynomial eigenfunctions. We present the entire eigenfunctions for these Hamiltonians (l = 1,2, . . .) in terms of new orthogonal polynomials. Two recently reported shape invariant potentials of Quesne and Gomez-Ullate et al.s are the first members of these infinitely many potentials.

Soliton equations and pseudospherical surfaces

Abstract All the soliton equations in 1 + 1 dimensions that can be solved by the AKNS 2 × 2 inverse scattering method (for example, the sine-Gordon, KdV or modified KdV equations) are shown to describe pseudospherical surfaces, i.e., surfaces of constant negative Gaussian curvature. This result provides a unified picture of all these soliton equations. Geometrical interpretations of characteristic properties like infinite numbers of conservation laws and Backlund transformations and of the soliton solutions themselves are presented. The important role of scale transformations as generating one parameter families of pseudospherical surfaces is pointed out.

Another set of infinitely many exceptional (Xℓ) Laguerre polynomials

Abstract We present a new set of infinitely many shape invariant potentials and the corresponding exceptional ( X l ) Laguerre polynomials. They are to supplement the recently derived two sets of infinitely many shape invariant thus exactly solvable potentials in one-dimensional quantum mechanics and the corresponding X l Laguerre and Jacobi polynomials [S. Odake, R. Sasaki, Phys. Lett. B 679 (2009) 414]. The new X l Laguerre polynomials and the potentials are obtained by a simple limiting procedure from the known X l Jacobi polynomials and the potentials, whereas the known X l Laguerre polynomials and the potentials are obtained in the same manner from the mirror image of the known X l Jacobi polynomials and the potentials.

Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux–Crum transformations

A simple derivation is presented of the four families of infinitely many shapeinvariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The Darboux‐Crum transformations are applied to connect the well-known shape-invariant Hamiltonians of the radial oscillator and the Darboux‐P¨ oschl‐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 the generalized Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.

Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials

Abstract Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of types I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the ‘ virtual state wavefunctions’ which are ‘deleted’ by the generalisation of Crum–Adler theorem. Each polynomial has another integer n which counts the nodes.

Infinitely many shape-invariant potentials and cubic identities of the Laguerre and Jacobi polynomials

We provide analytic proofs for the shape invariance of the recently discovered [Odake and Sasaki, Phys. Lett. B 679, 414 (2009)] two families of infinitely many exactly solvable one-dimensional quantum mechanical potentials. These potentials are obtained by deforming the well-known radial oscillator potential or the Darboux–Poschl–Teller potential by a degree l (l=1,2,…) eigenpolynomial. The shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the Laguerre or Jacobi polynomials. These identities are proved elementarily by combining simple identities.

Extended Toda Field Theory and Exact S Matrices

Abstract The existence of exact unitary crossing symmetric S -matrices associated with the a, d, e series of Toda field theories is conjectured and the main features illustrated within a discussion of the d 4 case.

Multiple poles and other features of affine Toda field theory

Some perturbative features of affine Toda field theory are explored, in particular the mechanisms responsible for he first-, second- and third-order poles in the conjectured exact factorisable S-matrices in the ADE series of models. It is found that generic collections of Feynman diagrams are responsible for the leading order poles in any of the theories. However, the complexity is such that it has not yet proved possible to analyse all the singularities that occur up to order twelve. Some comments are made on an associated tiling problem and on an interesting connection between the affine Toda couplings and the Clebsch-Gordan decomposition of tensor products.

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)