Search

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than n(max) so that their energies are lower than the groundstate energy., Article, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 46(23):235205 (2013)

For 11 examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations, deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials., Article, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 46(24):245201 (2013)

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of the P\"oschl-Teller potential, we obtain several families of explicit and global solutions of certain second order Fuchsian differential equations with an apparent singularity of characteristic exponent -2 and -1. They form orthogonal polynomials over $x\in(-1,1)$ with weight functions of the form $(1-x)^\alpha(1+x)^\beta/\{(ax+b)^4q(x)^2\}$, in which $q(x)$ is a polynomial in $x$., Comment: 27 pages, no figures. A Table added to summarize the main results, typos corrected, some statements added to improve presentation. Version in J. Phys. A

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of ?discrete quantum mechanics? with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey?Wilson polynomials. They are obtained from the original Wilson and Askey?Wilson polynomials by multiple applications of the discrete analogue of the Darboux transformations or the Crum?Krein?Adler deletion of ?virtual state solutions? of types I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.

Following the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the $su(2)$ coherent states, we propose ``interpretation'' of $su(1,1)$ and $su(r,1)$ coherent states ``in terms of probability theory''. They will be called the ``negative binomial'' (``multinomial'') ``states'' which correspond to the ``negative'' binomial (multinomial) distribution, the non-compact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here we show fruitful interplay of probability theory, group theory and quantum theory., 24 pages, latex, no figures

We show that the binomial states (BS) of Stoler {\it et al.} admit the ladder and displacement operator formalism. By generalizing the ladder operator formalism we propose an eigenvalue equation which possesses the number and the squeezed states as its limiting solutions. The explicit forms of the solutions, to be referred to as the {\it generalized binomial states} (GBS), are given. Corresponding to the wide range of the eigenvalue spectrum these GBS have as widely different properties. Their limits to number and {\it squeezed} states are investigated in detail. The time evolution of BS is obtained as a special case of the approach., LaTeX-2e, 11 pages, no figure. Accepted for publication in J.Phys.A:math.Gen

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., 18 pages, Six figures (in two pages), LaTeX, Accepted for publication in Phys.Rev.A

We introduce the negative binomial states with negative binomial distribution as their photon number distribution. They reduce to the ordinary coherent states and Susskind-Glogower phase states in different limits. The ladder and displacement operator formalisms are found and they are essentially the Perelomov's su(1,1) coherent states via its Holstein-Primakoff realisation. These states exhibit strong squeezing effect and they obey the super-Poissonian statistics. A method to generate these states is proposed., Comment: 10 pages, latex2e, 2 EPS figures

The generally deformed oscillator (GDO) and its multiphoton realization as well as the coherent and squeezed vacuum states are studied. We discuss, in particular, the GDO depending on a complex parameter q (therefore we call it q-GDO) together with the finite dimensional cyclic representations. As a realistic physical system of GDO the isospectral oscillator system is studied and it is found that its coherent and squeezed vacuum states are closely related to those of the oscillator. It is pointed out that starting from the q-GDO with q root of unity one can define the hermitian phase operators in quantum optics consistently and algebraically. The new creation and annihilation operators of the Pegg-Barnett type phase operator theory are defined by using the cyclic representations and these operators degenerate to those of the ordinary oscillator in the classical limit q->1., 21 pages, latex, no figures

The equation of motion of affine Toda field theory is a coupled equation for $r$ fields, $r$ is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than $r$ fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In this paper we present many new reductions. In other words the symmetry of affine Dynkin diagrams could be extended and it leads to non-canonical foldings. We investigate these reductions in detail and formulate general rules for possible reductions. We will show that eventually most of the theories end up in $a_{2n}^{(2)}$ that is the theory cannot have a further dimension $m$ reduction where $m, Comment: 26 pages, Latex2e, usepackage `graphics.sty', 15 figures

Abstarct: Boundary effects caused by the boundary interactions in various integrable field theories on a half line are discussed at the classical as well as the quantum level. Only the so-called ``integrable" boundary interactions are discussed. They are obtained by the requirement that certain combinations of the lower members of the infinite set of conserved quantities should be preserved. Contrary to the naive expectations, some ``integrable" boundary interactions can drastically change the character of the theory. In some cases, for example, the sinh-Gordon theory, the theory becomes ill-defined because of the instability introduced by ``integrable" boundary interactions for a certain range of the parameter., Comment: 12 pages

The question of the integrability of real-coupling affine toda field theory on a half-line is addressed. It is found, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, for the $a_n\ (n>1)$ series of models there can be no free parameters introduced by the boundary condition; indeed the only remaining freedom (apart from choosing the simple condition $\partial_1��=0$), resides in a choice of signs. For a special case of the boundary condition, it is argued that the classical boundary bound state spectrum is closely related to a consistent set of reflection factors in the quantum field theory., 16 pages, TEX (harvmac), DTP-94/7, YITP/U-94-11

A comprehensive review of the discrete quantum mechanics with the pure imaginary shifts and the real shifts is presented in parallel with the corresponding results in the ordinary quantum mechanics. The main subjects to be covered are the factorized Hamiltonians, the general structure of the solution spaces of the Schrodinger equation (Crum's theorem and its modification), the shape invariance, the exact solvability in the Schrodinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, the unified theory of exact and quasi-exact solvability based on the sinusoidal coordinates, and the infinite families of new orthogonal (the exceptional) polynomials. Two new infinite families of orthogonal polynomials, the X-l Meixner-Pollaczek and the X-l Meixner polynomials, are reported., Article, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL. 44(35):353001 (2011)

The S-matrices for non-simply-laced affine Toda field theories are considered in the context of a generalised bootstrap principle. The S-matrices, and in particular their poles, depend on a parameter whose range lies between the Coxeter numbers of dual pairs of the corresponding non-simply-laced algebras. It is proposed that only odd order poles in the physical strip with positive coefficients throughout this range should participate in the bootstrap. All other singularities have an explanation in principle in terms of a generalised Coleman-Thun mechanism. Besides the S-matrices introduced by Delius, Grisaru and Zanon, the missing case ($f_4^{(1)},e_6^{(2)}$), is also considered and provides many interesting examples of pole generation., Comment: 23 pages including two figures, harvmac

The reflection equations (RE) are a consistent extension of the Yang-Baxter equations (YBE) with an addition of one element, the so-called reflection matrix or $K$-matrix. For example, they describe the conditions for factorizable scattering on a half line just like the YBE give the conditions for factorizable scattering on an entire line. The YBE were generalized to define quadratic algebras, \lq Yang-Baxter algebras\rq\ (YBA), which were used intensively for the discussion of quantum groups. Similarly, the RE define quadratic algebras, \lq the reflection equation algebras\rq\ (REA), which enjoy various remarkable properties both new and inherited from the YBA. Here we focus on the various properties of the REA, in particular, the quantum group comodule properties, generation of a series of new solutions by composing known solutions, the extended REA and the central elements, etc., 31 pages, 8 figures (not included)

Recently, Zamolodchikov has suggested a way of exploring the properties of the Ising model in a background magnetic field, i.e. away from criticaliy(1).

Exact Heisenberg operator solutions for independent “sinusoidal coordinates” as many as the degree of freedom are derived for typical exactly solvable multiparticle quantum mechanical systems, the Calogero systems [J. Math. Phys.12, 419 (1971)] based on any root system. These Heisenberg operator solutions also present the explicit forms of the annihilation-creation operators for various quanta in the interacting multiparticle systems. At the same time they can be interpreted as multivariable generalization of the three term recurrence relations for multivariable orthogonal polynomials constituting the eigenfunctions., Article, JOURNAL OF MATHEMATICAL PHYSICS. 48(8):082106 (2007)

In this paper new multi-variable deformed Fokker-Planck (FP) equations are presented. These deformed FP equations are associated with the Ruijsenaars-Schneider-van Diejen (RSvD) type systems in the same way that the usual one variable FP equation is associated with the one particle Schr\"odinger equation. As the RSvD systems are the "discrete" counterparts of the celebrated exactly solvable many-body Calogero-Sutherland-Moser systems, the deformed FP equations presented here can be considered as "discrete" deformations of the ordinary multi-variable FP equations., Comment: 8 pages, no figures

The annihilation-creation operators a((+/-)) are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the "sinusoidal coordinate". Thus a((+/-)) are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the "discrete" quantum mechanics., Article, JOURNAL OF MATHEMATICAL PHYSICS. 47(10):102102 (2006)

Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems., Comment: 18 pages, LaTeX, no figure

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.

A simple derivation of the pseudopotentials of Wahlquist and Estabrook is given for the general AKNS system. A relationship between some pseudopotentials and the infinite numbers of conservation laws is pointed out.

A class of nonlinear equations that can be solved in terms of an n x n scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. An n x n problem is reduced to n sub ( n — 1) x ( n — 1) problems and finally to 2 x 2 problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite number of polynomial conservation laws for an n x n problem is presented. The 3 x 3 and 2 x 2 problems are discussed explicitly.

The canonical structure of the Kaup-Newell system, i.e. the family of the derivative nonlinear Schrodinger equation and its generalizations, is discussed.

In previous work one of the authors gave a geometric theory of those nonlinear evolution equations (n. e. es) that can be solved by the Zakharov & Shabat (1972) inverse scattering scheme as generalized by Ablowitz, Kaup, Newell & Segur (1973 b , 1974). In this paper we extend the geo­metric theory to include the Hamiltonian structure of those n. e. es solvable by the method, and we indicate the connection between the geometric theory and the theory of prolongation structures and pseudopotentials due to Wahlquist & Estabrook (1975, 1976). We exploit a ‘gauge’ in­variance of the geometric theory to derive both the well known polynomial conserved densities of the sine-Gordon equation and a non-local set of conserved densities. These act as Hamiltonian densities for a hierarchy of sine-Gordon equations which is analogous to that found by Lax (1968) for the Korteweg-de Vries equation and appears to be new. In an Appendix we derive an expression for the equation of motion for an arbitrary member of the sine-Gordon hierarchy by methods which can be applied in larger context. The results for the sine-Gordon equation lead to the conclusion that a complete set of conserved densities for an arbitrary n. e. e. solvable by the A. K. N. S. –Z. S. scheme must include non-local conserved densities.

On construit explicitement une representation unitaire des algebres de Virasoro avec la charge centrale c=1−6/(N+1) (N+2)

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.

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.

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.

Most of the integrable nonlinear evolution equations in 1 + 1 dimensions are known to be Hamiltonian systems. We show that the “Generalized Backlund Transformations” of Calogero and Degasperis form the group of canonical transformations that keeps these Hamiltonians invariant. In other words, they form a group of symmetry transformations for these Hamiltonian systems.

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.

We study the classical solutions of the complex Grassmannian nonlinear sigma models and of theCP N−1 model in two euclidean dimensions. Exact solutions of various types, which seem to be complete, are constructed explicitly in an elementary way, namely in terms of holomorphic functions and the Gramm-Schmidt orthonormalization procedure. A new type of discrete symmetry transformations which map one solution into another is presented.

On etudie des relations entre des algebres super-Virasoro et des systemes super-soliton aux niveaux classique et quantique. On deduit explicitement plusieurs membres de l'ensemble infini de quantites conservees

The canonical structure of the nonlinear evolution equations in 1 + 1 dimensions solvable in terms of an N × N inverse scattering problem is discussed. The simplest form of the scattering problems, that is those containing the spectral parameter linearly, is considered. It applies to most of the known soliton equations, like the Korteweg-de Vries eq., the sine-Gordon eq. and the Boussinesq eq. Discussion of various possible reductions of the number of dependent variables by imposing constraints consistent with the Hamiltonian flows is given together with the canonical structure of the reduced systems. A direct proof of the involutive character of the infinite number of conserved quantities is given for the general case as well as the reduced case. The relation between the conserved quantities and symmetry transformations (Lie-Backlund transformations) becomes very simple in this framework.