8073859
Exponential and Laguerre squeezed states for su(1,1) algebra and the Calogero-Sutherland model.
A class of squeezed states for the su(1,1) algebra is found and expressed by the exponential and Laguerre-polynomial operators acting on the vacuum states. As a special case it is proved that the Perelomov's coherent state is a ladder-operator squeezed state and therefore a minimum uncertainty state. The theory is applied to the two-particle Calogero-Sutherland model. We find some new squeezed states and compared them with the classical trajectories. The connection with some su(1,1) quantum optical systems (amplitude-squared realization, Holstein-Primakoff realization, the two mode realization and a four mode realization) is also discussed.
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)
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.
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.
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.
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.
12175080
Quantum state during and after O(4)-symmetric bubble nucleation with gravitational effects.
We extend our previous analysis of the quantum state during and after $O(4)$-symmetric bubble nucleation to the case including gravitational effects. We find that there exists a simple relationship between the case with and without gravitational effects. In a special case of a conformally coupled scalar field which is massless except on the bubble wall, the state is found to be conformally equivalent to the case without gravity.
45655606
Particle spectrum created through bubble nucleation and quantum field theory in the Milne universe.
Using the multi-dimensional wave function formalism, we investigate the quantum state of a scalar field inside a true vacuum bubble nucleated through false vacuum decay in flat spacetime. We developed a formalism which allows us a mode-by-mode analysis. To demonstrate its advantage, we describe in detail the evolution of the quantum state during the tunneling process in terms of individual mode functions and interpret the result in the language of particle creation. The spectrum of the created particles is examined based on quantum field theory in the Milne universe.