DOCSLIB.ORG

Lecture 2 : Quantum Integrable Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 2 : Quantum Integrable Systems Lecture 2 : Quantum Integrable Systems Maxime Fairon 2 December 2020 1 Motivation and definition Let us recall the classical picture which is at the basis of integrability. First, we are given a real phase space M of dimension 2n, which we assume for simplicity to admit global Darboux coordinates in terms of n positions q and n momenta p. In convenient situations, we may be able to split M with respect to these two sets of coordinates as M ' Mq × Mp. Then, we are also given a Hamiltonian H(p; q) 2 C1(M), which could take the suitable form n 1 X H = p2 + V;V = V (q) 2 C1(M ) : (1.1) 2 j q j=1 Finally, we have a canonically defined Poisson bracket which allows us to consider the following equations of motion : df = ff; Hg ; (1.2) dt for any function f 2 C1(M) whose evolution we are interested in studying. The quantum picture that we are going to introduce is motivated by quantum mechanics, and we need to reinterpret the classical construction. We are still considering the space Mq where we can measure positions, but quantum mechanics now tells us that we should be interested in the probability of finding a given system in a particular configuration. We are led to consider a complex 2 n Hilbert space S of states depending on Mq, for example the Hilbert space L (R ; dq) of complex- n valued square-integrable functions on Mq = R . What will be important from the point of view of 2 n physics is to consider the set S1 of states of unit norm, such as the unit-length vectors in L (R ; dq), which can be used to compute the probability of finding a particular system in a region of Mq or to make \measurements". To be more precise, let us introduce an algebra A of operators, called observables, acting on S. Given an element F 2 A, we can compute the expectation value of F in a state 2 S1 which is given by E [F] := h ; F i ; (1.3) where h−; −i denotes the inner product on S. A state will evolve over time according to Schr¨odinger'sequation : @ 2 i = − ~ ∆ + V ; V = V (q) 2 C1(M ) : (1.4) ~ @t 2m q Pn 2 2 1 Here, we use the Laplacian ∆f = j=1 @ f=@qj for f 2 C (Mq), while ~ is the Planck constant. From a more modest point of view, these considerations lead us to the problem of solving (1.4) with 1 m = 1 and ~ being seen as a formal parameter, from which we get the following simplified version n 2 @ 1 X @ i = H ; H = −i + V: (1.5) ~ @t 2 ~@q j=1 j The element H, called the quantum Hamiltonian, is an element of the algebra A of operators acting on S. If the operator H is time-independent, we can formally solve (1.5) by introducing the one- parameter group of operators i U(t) = exp − tH ; t 2 R : (1.6) ~ @U(t) It suffices to note that i~ @t = U(t)H = HU(t), in order to get the solution (t) = U(t) (0) for the initial condition (0). Furthermore, we can use U to obtain the evolution of an observable F 2 A. Mathematically, we note that1 −1 E (t)[F] = h (t); F (t)i = h (0); (U(t) FU(t)) (0)i = E (0)[F(t)] ; (1.7) for the one-parameter group of operators F(t) = U(t)−1FU(t). This can be reformulated as saying that the expectation value of the observable F in the state (t) coincides with the expectation value of the evolution operator F(t) in the initial state (0). We obtain in that way the Heisenberg picture of dynamics for observables n 2 dF(t) −i 1 X @ = [F(t); H] ; H = −i + V; (1.8) dt 2 ~@q ~ j=1 j where [−; −] denotes the commutator [A; B] = AB − BA. The classical (1.2) and quantum (1.8) equations of motion are quite similar. The role of the Pois- son bracket {−; −} is now played by the (rescaled) commutator, and both operations are bideriva- tions. Comparing the quantum Hamiltonian H in (1.5) with its classical counterpart (1.1) suggests that each momentum pj has its role now being played by the derivation −i~@=@qj. Hence we are \replacing" the Poisson algebra C1(M) by the algebra of operators A acting on S. Put together, we can give a rough correspondence between the classical and the quantum settings as in Table1. In order to understand what is the analogue of an integrable system in the quantum case, let us try to solve (1.5) by decomposing this equation with respect to an eigenbasis for H of normalised 2 eigenfunctions (φ`) corresponding to a discrete set of complex eigenvalues (E`), i.e. φ` 2 S1 is such that Hφ` = E`φ` : (1.9) P Then the evolution of some state (q; 0) comes from its decomposition as (q; 0) = ` c`φ`(q) for some constants (c`). Indeed, it is simply given by X −iE`t=~ (q; t) = c` e φ`(q) : (1.10) ` 1Here we require that H is a self-adjoint operator, which means that hHφ, i = hφ, H i holds for all φ, 2 S. This implies that U(t) is unitary, that is hU(t)φ, i = hφ, U(t)−1 i. 2The eigenvalues are real if H is self-adjoint. 2 Classical Quantum Poisson algebra of functions C1(M) A Algebra of operators Classical Hamiltonian H H Quantum Hamiltonian Poisson bracket {−; −} −i [−; −] Commutator ~ Classical equation of motion df = ff; Hg dF = −i [F; H] Heisenberg equation dt dt ~ Table 1: correspondence between classical and quantum notions. So now the question reduces to find an eigenbasis for H. By definition, an operator F 2 A commutes with H if [F; H] = FH − HF = 0 : (1.11) The key remark is to realise that if F commutes with H, then HFφ` = FHφ` = E`Fφ` ; (1.12) so that Fφ` is itself an eigenvector of H of eigenvalue E`. Therefore, we can consider the more precise problem of finding a common normalised eigenbasis of H and F, that is we want 0 0 φ` 2 S1 such that Hφ` = E`φ`; Fφ` = E`φ`;E`;E` 2 C : (1.13) This strategy can be repeated with an operator commuting with both H and F, then a fourth operator commuting with the first three ones, and so on and so forth. But when should we stop? 1 In the classical case, we limited our choice of Poisson commuting elements to n = 2 dim(M) = dim(Mq) functions which are functionally independent, so we shall introduce a similar property of independence. Definition 1.1. A set of operators H1;:::; Hn 2 A is a quantum (completely) integrable system if these operators pairwise commute (i.e. [Hi; Hj] = 0 for all 1 ≤ i; j ≤ n) and there exists no relation between them. A word of warning here : the existence of a relation usually means that we can write the identity P (H1;:::; Hn) = 0 ; (1.14) for some nonzero polynomial P 2 C[z1; : : : ; zn] in n variables. (Since the operators (Hi) pair- wise commute, replacing the variables (zi) by them yields a well-defined operator.) Depending on the context, P may have a slightly different form, e.g. we could seek independence over real- algebraic/analytic functions in n variables. As an example, note that for i 6= j there exists a relation 2 2 2 2 between the three commuting operators F1 = @ =@qi , F2 = @ =@qj , and F3 = @=@qi + @=@qj, since 2 2 for P (z1; z2; z3) = (z3 − z1 − z2) − 4z1z2 ; we have P (F1; F2; F3) = 0 : (1.15) There is a widespread use of a more convenient definition of quantum integrability which is motivated by the classical case, and it has the advantage of getting the independence of the operators \for free". This is based on the method of deformation/quantization. The quantization process consists in making the formal replacement @ pj −i~ ; (1.16) @qj 3 to go from a polynomial function on M to an operator belonging to A. There is an obvious issue of ordering here, since the function qjpj can be quantized in the many forms @ @ @ αqjpj + (1 − α)pjqj −αi~qj − (1 − α)i~ qj = −i~qj + (α − 1)i~ ; α 2 C : @qj @qj @qj Nevertheless, it is well-defined for the Hamiltonian operators of interest since n n 2 1 X 1 X @ H = p2 + V (q) H = −i + V (q) : (1.17) 2 j 2 ~@q j=1 j=1 j The \opposite" of the map (1.16) is given by the replacement @ − i~ pj ; then ~ ! 0 ; (1.18) @qj and it is called the reduction. For example, H is the reduction of H in (1.17). Note that performing quantization (1.16) then reduction (1.18) returns the original function, but the opposite may not be true. With this set up, we have the following alternative definition of quantum integrability. Definition 1.2 (Alternative to Definition 1.1). A set of operators H1;:::; Hn 2 A is a quan- tum (completely) integrable system if these operators pairwise commute ([Hi; Hj] = 0) and their reductions H1;:::;Hn are functionally independent. 2 Examples of quantum integrable systems Remark 2.1. To ease notations when dealing with examples, it is usual to fix ~ to some value in C n f0g. In our examples, we will work with ~ = −i so that the Hamiltonian operators are of the 1 Pn @2 @ form H = 2 j=1 2 + V , and quantization is just pj @q .
Load more

Recommended publications
  • Contact Lax Pairs and Associated (3+ 1)-Dimensional Integrable Dispersionless Systems
    Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems Maciej B laszak a and Artur Sergyeyev b a Faculty of Physics, Division of Mathematical Physics, A. Mickiewicz University Umultowska 85, 61-614 Pozna´n, Poland E-mail [email protected] b Mathematical Institute, Silesian University in Opava, Na Rybn´ıˇcku 1, 74601 Opava, Czech Republic E-mail [email protected] January 17, 2019 Abstract We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related R-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems. 1 Introduction Integrable systems play an important role in modern mathematics and theoretical and mathematical arXiv:1901.05181v1 [nlin.SI] 16 Jan 2019 physics, cf. e.g. [15, 34], and, since according to general relativity our spacetime is four-dimensional, integrable systems in four independent variables ((3+1)D for short; likewise (n+1)D is shorthand for n + 1 independent variables) are particularly interesting. For a long time it appeared that such systems were very difficult to find but in a recent paper by one of us [39] a novel systematic and effective construction for a large new class of integrable (3+1)D systems was introduced.
    [Show full text]
  • Arxiv:2012.03456V1 [Nlin.SI] 7 Dec 2020 Emphasis on the Kdv Equation
    Integrable systems: From the inverse spectral transform to zero curvature condition Basir Ahamed Khan,1, ∗ Supriya Chatterjee,2 Sekh Golam Ali,3 and Benoy Talukdar4 1Department of Physics, Krishnath College, Berhampore, Murshidabad 742101, India 2Department of Physics, Bidhannagar College, EB-2, Sector-1, Salt Lake, Kolkata 700064, India 3Department of Physics, Kazi Nazrul University, Asansol 713303, India 4Department of Physics, Visva-Bharati University, Santiniketan 731235, India This `research-survey' is meant for beginners in the studies of integrable systems. Here we outline some analytical methods for dealing with a class of nonlinear partial differential equations. We pay special attention to `inverse spectral transform', `Lax pair representation', and `zero-curvature condition' as applied to these equations. We provide a number of interesting exmples to gain some physico-mathematical feeling for the methods presented. PACS numbers: Keywords: Nonlinear Partial Differential Equations, Integrable Systems, Inverse Spectral Method, Lax Pairs, Zero Curvature Condition 1. Introduction Integrable systems are represented by nonlinear partial differential equations (NLPDEs) which, in principle, can be solved by analytic methods. This necessarily implies that the solution of such equations can be constructed using a finite number of algebraic operations and integrations. The inverse scattering method as discovered by Gardner, Greene, Kruskal and Miura [1] represents a very useful tool to analytically solve a class of nonlinear differential equa- tions which support soliton solutions. Solitons are localized waves that propagate without change in their properties (shape, velocity etc.). These waves are stable against mutual collision and retain their identities except for some trivial phase change. Mechanistically, the linear and nonlinear terms in NLPDEs have opposite effects on the wave propagation.
    [Show full text]
  • Aspects of Integrable Models
    Aspects of Integrable Models Undergraduate Thesis Submitted in partial fulfillment of the requirements of BITS F421T Thesis By Pranav Diwakar ID No. 2014B5TS0711P Under the supervision of: Dr. Sujay Ashok & Dr. Rishikesh Vaidya BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI, PILANI CAMPUS January 2018 Declaration of Authorship I, Pranav Diwakar, declare that this Undergraduate Thesis titled, `Aspects of Integrable Models' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Signed: Date: i Certificate This is to certify that the thesis entitled \Aspects of Integrable Models" and submitted by Pranav Diwakar, ID No. 2014B5TS0711P, in partial fulfillment of the requirements of BITS F421T Thesis embodies the work done by him under my supervision. Supervisor Co-Supervisor Dr. Sujay Ashok Dr. Rishikesh Vaidya Associate Professor, Assistant Professor, The Institute of Mathematical Sciences BITS Pilani, Pilani Campus Date: Date: ii BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE PILANI, PILANI CAMPUS Abstract Master of Science (Hons.) Aspects of Integrable Models by Pranav Diwakar The objective of this thesis is to study the isotropic XXX-1/2 spin chain model using the Algebraic Bethe Ansatz.
    [Show full text]
  • The Idea of a Lax Pair–Part II∗ Continuum Wave Equations
    GENERAL ARTICLE The Idea of a Lax Pair–Part II∗ Continuum Wave Equations Govind S. Krishnaswami and T R Vishnu In Part I [1], we introduced the idea of a Lax pair and ex- plained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to con- tinuum mechanical systems of fields such as the linear wave equation for vibrations of a stretched string and the Korteweg- de Vries (KdV) equation for water waves. Unlike the Lax ma- trices for systems of particles, here Lax pairs are differential operators. A key idea is to view the Lax equation as a com- Govind Krishnaswami is on the faculty of the Chennai patibility condition between a pair of linear equations. This is Mathematical Institute. He used to obtain a geometric reformulation of the Lax equation works on various problems in as the condition for a certain curvature to vanish. This ‘zero theoretical and mathematical curvature representation’ then leads to a recipe for finding physics. (typically an infinite sequence of) conserved quantities. 1. Introduction In the first part of this article [1], we introduced the idea of a dy- namical system: one whose variables evolve in time, typically via T R Vishnu is a PhD student at the Chennai Mathematical differential equations. It was pointed out that conserved quanti- Institute. He has been ties, which are dynamical variables that are constant along trajec- working on integrable tories help in simplifying the dynamics and solving the equations systems. of motion (EOM) both in the classical and quantum settings.
    [Show full text]
  • A New Two-Component Integrable System with Peakon Solutions
    A new two-component integrable system with peakon solutions 1 2 Baoqiang Xia ∗, Zhijun Qiao † 1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, P. R. China 2Department of Mathematics, University of Texas-Pan American, Edinburg, Texas 78541, USA Abstract A new two-component system with cubic nonlinearity and linear dispersion: 1 1 mt = bux + 2 [m(uv uxvx)]x 2 m(uvx uxv), 1 − −1 − nt = bvx + [n(uv uxvx)]x + n(uvx uxv), 2 − 2 − m = u uxx, n = v vxx, − − where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically, this system describes a nontrivial one-parameter family of pseudo-spherical surfaces. In the case b = 0, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion 1 2 2 1 ∗ ∗ arXiv:1211.5727v4 [nlin.SI] 10 May 2015 mt = bux + [m( u ux )]x m(uux uxu ), m = u uxx, 2 | | − | | − 2 − − is obtained by imposing the complex conjugate reduction v = u∗ to the two-component system. The complex valued N-peakon solution and kink wave solution to this complex equation are also derived. Keywords: Integrable system, Lax pair, Peakon. Mathematics Subject Classifications (2010): 37K10, 35Q51. ∗E-mail address: [email protected] †Corresponding author, E-mail address: [email protected] 1 2 1 Introduction In recent years, the Camassa-Holm (CH) equation [1] mt bux + 2mux + mxu = 0, m = u uxx, (1.1) − − where b is an arbitrary constant, derived by Camassa and Holm [1] as a shallow water wave model, has attracted much attention in the theory of soliton and integrable system.
    [Show full text]
  • Remarks on the Complete Integrability of Quantum and Classical Dynamical
    Remarks on the complete integrability of quantum and classical dynamical systems Igor V. Volovich Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia [email protected] Abstract It is noted that the Schr¨odinger equation with any self-adjoint Hamiltonian is unitary equivalent to a set of non-interacting classical harmonic oscillators and in this sense any quantum dynamics is completely integrable. Higher order integrals of motion are presented. That does not mean that we can explicitly compute the time dependence for expectation value of any quantum observable. A similar re- sult is indicated for classical dynamical systems in terms of Koopman’s approach. Explicit transformations of quantum and classical dynamics to the free evolution by using direct methods of scattering theory and wave operators are considered. Examples from classical and quantum mechanics, and also from nonlinear partial differential equations and quantum field theory are discussed. Higher order inte- grals of motion for the multi-dimensional nonlinear Klein-Gordon and Schr¨odinger equations are mentioned. arXiv:1911.01335v1 [math-ph] 4 Nov 2019 1 1 Introduction The problem of integration of Hamiltonian systems and complete integrability has already been discussed in works of Euler, Bernoulli, Lagrange, and Kovalevskaya on the motion of a rigid body in mechanics. Liouville’s theorem states that if a Hamiltonian system with n degrees of freedom has n independent integrals in involution, then it can be integrated in quadratures, see [1]. Such a system is called completely Liouville integrable; for it there is a canonical change of variables that reduces the Hamilton equations to the action-angle form.
    [Show full text]
  • "Integrable Systems, Random Matrices and Random Processes"
    Contents Integrable Systems, Random Matrices and Random Processes Mark Adler ..................................................... 1 1 Matrix Integrals and Solitons . 4 1.1 Random matrix ensembles . 4 1.2 Large n–limits . 7 1.3 KP hierarchy . 9 1.4 Vertex operators, soliton formulas and Fredholm determinants . 11 1.5 Virasoro relations satisfied by the Fredholm determinant . 14 1.6 Differential equations for the probability in scaling limits . 16 2 Recursion Relations for Unitary Integrals . 21 2.1 Results concerning unitary integrals . 21 2.2 Examples from combinatorics . 25 2.3 Bi-orthogonal polynomials on the circle and the Toeplitz lattice 28 2.4 Virasoro constraints and difference relations . 30 2.5 Singularity confinement of recursion relations . 33 3 Coupled Random Matrices and the 2–Toda Lattice . 37 3.1 Main results for coupled random matrices . 37 3.2 Link with the 2–Toda hierarchy . 39 3.3 L U decomposition of the moment matrix, bi-orthogonal polynomials and 2–Toda wave operators . 41 3.4 Bilinear identities and τ-function PDE’s . 44 3.5 Virasoro constraints for the τ-functions . 47 3.6 Consequences of the Virasoro relations . 49 3.7 Final equations . 51 4 Dyson Brownian Motion and the Airy Process . 53 4.1 Processes . 53 4.2 PDE’s and asymptotics for the processes . 59 4.3 Proof of the results . 62 5 The Pearcey Distribution . 70 5.1 GUE with an external source and Brownian motion . 70 5.2 MOPS and a Riemann–Hilbert problem . 73 VI Contents 5.3 Results concerning universal behavior . 75 5.4 3-KP deformation of the random matrix problem .
    [Show full text]
  • On the Non-Integrability of the Popowicz Peakon System
    DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2009 pp. 359–366 ON THE NON-INTEGRABILITY OF THE POPOWICZ PEAKON SYSTEM Andrew N.W. Hone Institute of Mathematics, Statistics & Actuarial Science University of Kent, Canterbury CT2 7NF, UK Michael V. Irle Institute of Mathematics, Statistics & Actuarial Science University of Kent, Canterbury CT2 7NF, UK Abstract. We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev´eanalysis provides strong evidence that the Popowicz system is non-integrable. Nevertheless, we are able to con- struct exact travelling wave solutions in terms of an elliptic integral, together with a degenerate travelling wave corresponding to a single peakon. We also describe the dynamics of N-peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N. 1. Introduction. The members of a one-parameter family of partial differential equations, namely mt + umx + buxm =0, m = u − uxx (1) with parameter b, have been studied recently. The case b = 2 is the Camassa-Holm equation [1], while b = 3 is the Degasperis-Procesi equation [3], and it is known that (with the possible exception of b = 0) these are the only integrable cases [14], while all of these equations (apart from b = −1) arise as a shallow water approximation to the Euler equations [6].
    [Show full text]
  • What Are Completely Integrable Hamilton Systems
    THE TEACHING OF MATHEMATICS 2011, Vol. XIII, 1, pp. 1–14 WHAT ARE COMPLETELY INTEGRABLE HAMILTON SYSTEMS BoˇzidarJovanovi´c Abstract. This paper is aimed to undergraduate students of mathematics and mechanics to draw their attention to a modern and exciting field of mathematics with applications to mechanics and astronomy. We cared to keep our exposition not to go beyond the supposed knowledge of these students. ZDM Subject Classification: M55; AMS Subject Classification: 97M50. Key words and phrases: Hamilton system; completely integrable system. 1. Introduction Solving concrete mechanical and astronomical problems was one of the main mathematical tasks until the beginning of XX century (among others, see the work of Euler, Lagrange, Hamilton, Abel, Jacobi, Kovalevskaya, Chaplygin, Poincare). The majority of problems are unsolvable. Therefore, finding of solvable systems and their analysis is of a great importance. In the second half of XX century, there was a breakthrough in the research that gave a basis of a modern theory of integrable systems. Many great mathemati- cians such as P. Laks, S. Novikov, V. Arnold, J. Mozer, B. Dubrovin, V. Kozlov contributed to the development of the theory, which connects the beauty of classi- cal mechanics and differential equations with algebraic, symplectic and differential geometry, theory of Lie groups and algebras (e.g., see [1–4] and references therein). The aim of this article is to present the basic concepts of the theory of integrable systems to readers with a minimal prior knowledge in the graduate mathematics, so we shall not use a notion of a manifold, symplectic structure, etc. The most of mechanical and physical systems are modelled by Hamiltonian equations.
    [Show full text]
  • INVERSE SCATTERING TRANSFORM, Kdv, and SOLITONS
    INVERSE SCATTERING TRANSFORM, KdV, AND SOLITONS Tuncay Aktosun Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762, USA Abstract: In this review paper, the Korteweg-de Vries equation (KdV) is considered, and it is derived by using the Lax method and the AKNS method. An outline of the inverse scattering problem and of its solution is presented for the associated Schr¨odinger equation on the line. The inverse scattering transform is described to solve the initial-value problem for the KdV, and the time evolution of the corresponding scattering data is obtained. Soliton solutions to the KdV are derived in several ways. Mathematics Subject Classification (2000): 35Q53, 35Q51, 81U40 Keywords: Korteweg-de Vries equation, inverse scattering transform, soliton Short title: KdV and inverse scattering 1 1. INTRODUCTION The Korteweg-de Vries equation (KdV, for short) is used to model propagation of water waves in long, narrow, and shallow canals. It was first formulated [1] in 1895 by the Dutch mathematicians Diederik Johannes Korteweg and Gustav de Vries. Korteweg was a well known mathematician of his time, and de Vries wrote a doctoral thesis on the subject under Korteweg. After some scaling, it is customary to write the KdV in the form @u @u @3u 6u + = 0; x R; t > 0; (1.1) @t − @x @x3 2 where u(x; t) corresponds to the vertical displacement of the water from the equilibrium − at the location x at time t: Replacing u by u amounts to replacing 6 by +6 in (1.1). − − Also, by scaling x; t; and u; i.e.
    [Show full text]
  • Orthogonal Functions Related to Lax Pairs in Lie Algebras
    The Ramanujan Journal https://doi.org/10.1007/s11139-021-00424-9 Orthogonal functions related to Lax pairs in Lie algebras Wolter Groenevelt1 · Erik Koelink2 Dedicated to the memory of Richard Askey Received: 5 October 2020 / Accepted: 2 March 2021 © The Author(s) 2021 Abstract We study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for sl(d + 1, C) is studied in certain representations. Keywords Special functions · Orthogonal polynomials · Toda lattice Mathematics Subject Classification 33C80 · 37K10 · 17B80 1 Introduction The link of the Toda lattice to three-term recurrence relations via the Lax pair after the Flaschka coordinate transform is well understood, see e.g. [2,27]. We consider a Lax pair in a specific Lie algebra, such that in irreducible ∗-representations the Lax operator is a Jacobi operator. A Lax pair is a pair of time-dependent matrices or operators L(t) and M(t) satisfying the Lax equation L˙ (t) =[M(t), L(t)], where [ , ] is the commutator and the dot represents differentiation with respect to time. The Lax operator L is isospectral, i.e. the spectrum of L is independent of time. A famous example is the Lax pair for the Toda chain in which L is a self-adjoint Jacobi operator, B Erik Koelink [email protected] Wolter Groenevelt [email protected] 1 Delft Institute of Applied Mathematics, Delft University of Technology, P.O.
    [Show full text]
  • Introduction to Integrable Systems 33Rd SERB Theoretical High Energy Physics Main School, Khalsa College, Delhi Univ
    Introduction to Integrable Systems 33rd SERB Theoretical High Energy Physics Main School, Khalsa College, Delhi Univ. December 17-26, 2019 (updated Feb 26, 2020) Govind S. Krishnaswami, Chennai Mathematical Institute Comments and corrections may be sent to [email protected] https://www.cmi.ac.in/˜govind/teaching/integ-serb-thep-del-19/ Contents 1 Some reference books/review articles 2 2 Introduction 3 2.1 Integrability in Hamiltonian mechanics.............................................3 2.2 Liouville-Arnold Theorem...................................................6 2.3 Remarks on commuting flows.................................................9 2.4 Integrability in field theory...................................................9 2.5 Some features of integrable classical field theories.......................................9 2.6 Integrability of quantum systems................................................ 10 2.7 Examples of integrable systems................................................ 11 2.8 Integrable systems in the wider physical context........................................ 12 3 Lax pairs and conserved quantities 12 3.1 Lax pair for the harmonic oscillator.............................................. 12 3.2 Isospectral evolution of the Lax matrix............................................. 13 4 KdV equation and its integrability 15 4.1 Physical background and discovery of inverse scattereing................................... 15 4.2 Similarity and Traveling wave solutions............................................ 17 4.3
    [Show full text]