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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.

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 e 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., Article, PHYSICS LETTERS B. 679(4):414-417 (2009)

By factorization of the Hamiltonian describing the quantum mechanics of the continuous q-Hermite polynomial, the creation and annihilation operators of the q-oscillator are obtained. They satisfy a q-oscillator algebra as a consequence of the shape-invariance of the Hamiltonian. A second set of q-oscillator is derived from the exact Heisenberg operator solution. Now the q-oscillator stands on the equal footing to the ordinary harmonic oscillator., Article, PHYSICS LETTERS B. 663(1-2):141-145 (2008)

The annihilation–creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called ‘discrete’ quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation–creation operators (a(±)a(±)) as the positive/negative frequency parts of the exact Heisenberg operator solution., Article, PHYSICS LETTERS B. 641(1):112-117 (2006)

Affine Toda field theories are solvable 1 + 1 dimensional quantum field theories closely related to integrable deformations of conformal field theory. The S -matrix elements for an affine Toda field theory are believed to depend on the coupling constant β through one universal function B ( β ) which cannot be determined by unitarity, crossing and the bootstrap. From the requirement of nonexistence of extra poles in the physical region its form is conjectured to be B(β) = (2π) −1 β 2 (1 + β 2 4π ) . We show that the above conjecture is correct up to one-loop order (i.e., β 4 ) of perturbation for simply laced, i.e., a, d and e affine Toda field theories using a general argument which exhibits much of the richness of these theories.

General classical solutions are constructed for the complex grassmannian non-linear sigma models in two euclidean dimensions in terms of holomorphic functions. The grassmannian sigma models are a simple generalization of the well-known CPN−1 model in two dimensions and they share various interesting properties; existence of (anti-) instantons, an infinite number of conserved quantities and complete integrability.

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.

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.

For a wide class of solition equations in 1 + 1 dimensions an infinite set of symmetry transformations obeying the Kac-Moody Lie algebra is obtained. These symmetry transformations, however, can be applied only within a limited domain of the x (space) variable. This restriction applies, in particular, to the Heisenberg spin chain and the nonlinear Schrodinger equation for which a Kac-Moody structure was recently derived by Eichenherr.

Dispersion calculations are performed for electromagnetic form factors of leptons and scattering amplitude using a renormalizable model of weak interaction. An unambiguous result is obtained for the anomalous magnetic moment and an unsubtracted dispersion relation holds for elastic νν-scattering.