Quasiexactly solvable lie algebras of differential operators. Invariant solutions of the twodimensional shallow water equations with a particular class of bottoms. Quantum superintegrable systems as exactly solvable models 3 2 a metric and its symmetries f or an n dimensional pseudoriemannian space, with local co ordinates x 1. The derived exactlysolvable generalized gaudin models include the hamiltonians of bardeencooperschrieffer, suhlmatthiaswalker, lipkinmeshkovglick, the generalized dicke and atommolecule, the nuclear interacting boson model, a new exactlysolvable kondolike impurity model, and many more that have not been exploited in the. Correlation functions of quantum integrable systems and.
The summary at the end of the work recapitulates the main results and the relations. Solvable quantum versions of the classical dynamical system characterized by the hamiltonian h. Dusselc a bartol research institute, university of delaware, newark, delaware 19716, usa. The first group of papers published here is concerned with integrable structures of quantum lattice models related to quantum group symmetries. This transformation is called canonical if it preserves the poisson bracket xn k1. However, from our knowledge of partial differential equations we know that there are two potential namely and which this can be exactly solved. Exactlysolvable generalization of the jaynescummings model and its application to atommolecule systems s. Offdiagonal bethe ansatz for exactly solvable models.
Abstract it is shown that the f 4 rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Infinite family of solvable and integrable planar quantum. Advanced series in mathematical physics exactly solvable models of strongly correlated electrons, pp. We also recall a matrix version of the binary darboux transformation and, inspired by the socalled cauchy matrix approach. Eigenfunctions are constructed through the action of the commuting operators. This obscures the distinction between integrability in the hamiltonian sense, and the more general dynamical systems sense. To the best of my knowledge, the complete understanding of what is an integrable system for the case of three 3d or more independent variables is still missing. However, among integrable systems there are exactly solvable systems.
It turns out that the model is still exactly solvable. Exactly solvable models in the cold atomic systems. Superintegrability and exactly solvable problems in classical. Quantum superintegrable systems as exactly solvable models. The pairing model pm hamiltonian was solved exactly for both boson and fermion systems by richardson in the 1960s. It is shown that the threebody trigonometric g 2 integrable system is exactly solvable. Exactly solvable and integrable systems authorstitles. Feb 14, 2007 quantum super integrable systems as exactly solvable models 3 2 a metric and its symmetries f or an n dimensional pseudoriemannian space, with local co ordinates x 1.
Pdf it is shown that the f 4 rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Samples of exactly and completely integrable wave and evolution equations are considered in detail, including the generalized periodic and finite nonperiodic toda lattice, nonlinear. There are relatively few models in physics which have nonlinear terms or couplings, and can still be solved exactly that is. This survey is devoted to a consistent exposition of the groupalgebraic methods for the integration of systems of nonlinear partial differential equations possessing a nontrivial internal symmetry algebra. Exactlysolvable generalization of the jaynescummings. Pdf it is shown that the f4 rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Exactly solvable and integrable systems authorstitles dec. Solvability of the f4 integrable system international. Claeys pieter correlation functions and inner products in integrable richardsongaudin models. As examples, we present two new 3body models and determine explicit n expressions for their eigenvalues and eigenfunctions. The eigenvalues of the quantum hamiltonians are exhibited, as well as the corresponding eigenfunctions. Intuitively, an integrable system is a di erential equation we can \integrate up directly.
It is the beauty of integrable systems that completely classical and nonlinear field theories share common fundamental structures with their quantum counterparts. On darboux integrable semidiscrete systems of exponential type. Spectral intertwining relations in exactly solvable quantum. Version accepted for publication in the proceedings of the x international symposium quantum theory and symmetries and xii international worshop lie theory and its applications in physics. Exactlysolvable models derived from a generalized gaudin. Quantum superintegrable systems are exactly solvable quantum eigenvalue problems. The intimate relationship between superintegrability and exact solvability is illustrated. Exactlysolvable models derived from the gaudin algebra. Their solvability is due to symmetry, but the symmetry is often hidden. Their spectra are found explicitly while eigenfunctions are by pure algebraic means. Some personal perspectives will be offered on the current status of the field and its future outlook, with an eye on new areas of applicability such as quench dynamics and driven quantum systems.
Some time ago, it was conjectured that all maximally superintegrable systems for n 2 are exactly solvable 1. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. Formal recursion operators of integrable nonevolutionary equations and lagrangian systems. While the shape invariance is very powerful to construct exactly solvable quantummechanical models, it is important to obtain other concepts to characterize such models. Topological 1soliton solutions to some conformable fractional partial differential equations. Integrable structures of exactly solvable twodimensional. In physics, completely integrable systems, especially in the infinitedimensional setting, are often referred to as exactly solvable models. For both systems new variables are introduced in which the hamiltonian has an algebraic form being also blocktriangular. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The riemannhilbert problem and integrable systems alexander r. Superintegrability and quasiexactly solvable eigenvalue. Superintegrability and quasiexactly solvable eigenvalue problems in quantum mechanics willard miller joint with e. So doing we shall encounter various remarkable combinatorial models, such as tilings, pipedreams and puzzles.
Therefore, in this terminology, a quasiexactly solvable differential operator is a lie algebraic differential operator corre sponding to a quasiexactly solvable lie algebra of differential operators, the hidden symmetry algebra. The marsdenweinstein reduction structure of integrable. Although the wavefunction of equation 18 is represented in a general form with d j. However, the project can be tailored to the background of most students, regardless of whether they have studied any of these. This book serves as an introduction of the offdiagonal bethe ansatz method, an analytic theory for the eigenvalue problem of quantum integrable models. Their spectra are found explicitly while eigenfunctions are obtained by pure algebraic means.
Exactly solvable and integrable systems authorstitles dec 2017. Exactly solvable and integrable systems authorstitles new. Pdf quantum superintegrable systems as exactly solvable. This mechanics can be used to deform toric systems into semitoric systems, and also to deform semitoric systems into hyperbolic semitoric systems. I quantum integrable systems exactly solvable models of statistical physics. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for. Quantum superintegrable systems as exactly solvable models 3 2 a metric and its symmetries f or an n dimensional pseudoriemannian space, with. To connect to other answers, this should only work out when there are enough symmetries for us to write down and integrate. We establish a fundamental isomorphism between discretetime balanced urn processes and certain ordinary differen. This category deals with exactly solvable models or integrable systems in physics, and specifically in statistical mechanics, quantum field theory and fluid mechanics. Such notions have direct applications to discrete integrable systems and raise the question of the general geometric picture in the context of poissonlie groups. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the hamiltonian can be expressed as a quadratic polynomial in the generators of some lie algebra of differential operators in a finitedimensional. Solvability of the g2 integrable system international. Integrability in qft and adscft, lecture notes eth.
I will discuss some examples of such deformations joint work with joachim worthington 2 and the behaviour of the joint spectrum of the corresponding integrable quantum systems. An infinite family of exactly solvable and integrable potentials on a plane is introduced. These systems are shape invariant and their eigenfunctions are described by the multiindexed continuous hahn and meixnerpollaczek orthogonal polynomials. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the hamiltonian can be expressed as a quadratic polynomial in the generators of some lie algebra of differential operators in a finitedimensional representation. Condensed matter statistical mechanics, high energy physics theory, nonlinear sciences exactly solvable and integrable systems, quantum physics. One of the results of this chapter, which to my knowledge has not been reported before, is the existence of exactly solvable potentials as limiting cases of the quasi exactly solvable ones. Hence, solubility is more strict condition than integrablity. An infinite family of solvable and integrable quantum systems.
Nonlinear sciences exactly solvable and integrable systems. Quantum super integrable systems as exactly solvable models fordy, allan p. Exactly solvable discrete quantum mechanical systems and. Exactlysolvable models derived from a generalized gaudin algebra.
It is shown that the f 4 rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Exactly solvable models for trapped boson systems sciencedirect. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Pdf on jan 1, 1999, ken umeno and others published exactly solvable chaos and addition theorems of elliptic functions applied mathematics of discrete integrable systems find, read and cite. So, literally, an integrable system in this view is one that can be solved by a sequence of integrals which may not be explicitly solvable in elementary functions, of course. Exactly solvable multidimensional nonlinear equations and inverse scattering m j ablowitz aspects of integrability in one and several dimensions a s fokas a geometrical approach to integrable systems admitting time dependent invariants w oevel direct linearization method and integrable lattice systems h w capel. Exactly solvable generalizations of the uniform pairing hamiltonian were proposed and subsequently applied to various finite bose systems. Ashton notes taken by dexter chua michaelmas 2016 these notes are not endorsed by the lecturers, and i have modi ed them often signi cantly after lectures. Solvable quantum version of an integrable hamiltonian system. D 2n, in this work we study only the case with d j.
The intimate relationship between superintegrability and ex. Quantum integrable systems via quantum ktheory anton m. There are relatively few models in physics which have nonlinear terms or couplings, and can still be solved exactly that is, without resorting to perturbation theory. Exactly solvable and integrable systems authorstitles sep 2019. Its i n its original setting, the riemann hilbertproblem is the question of surjectivity of the monodromy map in the theory of fuchsian systems. In the past, many exactly solvable quantummechanical systems have been studied by means of shape invariance see, e.
Integrability does not help to solve the schroedinger equation. A twodimensional system is integrable if it allows two integrals of motion and maximally superintegrable if it allows three. Exactly solvable and integrable systems authorstitles feb. Quantum superintegrable systems as exactly solvable. While in theory, integrable systems should be very rare, it happens that in nature, a lot of systems happen to be integrable. Exactly solvable and integrable systems authorstitles apr 2018. Exactly solvable and integrable systems authorstitles may. We shall discuss some applications of exactly solvable models of twodimensional statistical mechanics to the theory of symmetric polynomials and to schubert calculus.
We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of coordinates is a whole real line. Pdf exactly solvable chaos and addition theorems of. Nijhoff frank lagrangian multiform theory and integrability recent developments in the theory of discrete integrable systems, in particular the notion of multidimensional consistency and its consequences, has led to the formulation of a novel variational principle based on the notion of lagrangian multiforms. This website uses cookies to ensure you get the best experience on our website. Solvability of f 4 quantum integrable systems springerlink. Topics in soliton theory and exactly solvable nonlinear equations. The riemann hilbert approach to quantum exactly solvable models was initiated in the beginning of the 1980s by jimbo, miwa, mori, and sato, and it was further developed in the late 1980s and in the 1990s in the series of works of izergin, korepin, slavnov, deift, zhou, and this author. Exactly solvable and integrable systems authorstitles recent. Here we will show that several exactly solvable systems are in fact maximally superintegrable. Although important in their own right, these systems form an archipelago of solvable models in a sea of unknown, and can be used as stepping stones to investigate properties of nearby nonintegrable systems. In particular, for the case of three independent variables a. For each point in m there exists precisely one torus tn passing. Pdf quantum superintegrable systems as exactly solvable models.
Abstract an approach to describing nonlinear lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the marsdenweinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Integrable systems have a rich mathematical structure, which means that many interesting exact solutions to the pdes can be found. Exactly solvable and integrable systems authorstitles apr. Download fulltext pdf exactly solvable chaos and addition theorems of elliptic functions applied mathematics of discrete integrable systems article pdf available january 1999 with 19 reads. We propose a systematic procedure for the construction of exactly solvable knbody systems which are natural generalisations of calogero models. Equivalence groupoid and enhanced group classification of a class of generalized kawahara equations. Exactly and completely integrable nonlinear dynamical systems.
Hence this systems in the sense of liouville is integrable. It also presents some fundamental knowledge about quantum integrability and the algebraic bethe ansatz method. Integrable systems, etc, one of which is available as a 4mm module each year. We consider some examples of quantum superintegrable systems and the as sociated nonlinear extensions of lie algebras. High energy physics theory, condensed matter statistical mechanics, mathematical physics, nonlinear sciences exactly solvable and integrable systems, quantum physics. Perturbations of integrable systems and dysonmehta. After an introduction to some aspects of bidifferential calculus on associative algebras, we focus on the notion of a symmetry of a generalized zero curvature equation and derive backlund and forward, backward and binary darboux transformations from it. The second group deals with the description of integrable structures in twodimensional quantum field theories, especially boundary problems, thermodynamic bethe ansatz and form factor problems. We consider some examples of quantum super integrable systems and the associated nonlinear extensions of lie algebras.