Superintegrable systems in classical and quantum mechanics. Background, ideas and new developments Pavel Winternitz E-mail address: wintern@crm.umontreal.ca Centre de recherches mathématiques et Département de Mathématiques et de Statistique, Université de Montréal, CP 6128, Succ. Centre-Ville, Montréal, Quebec H3C 3J7, Canada Analytic and algebraic methods in physics Prague, Czech Republic, 6 - 9 June 2016 organised in honour of Miloslav Znojil’s 70th birthday in the year 2016 Pavel Winternitz (CRM) Superintegrability and exact solvability 1 / 48
Outline Introduction 1 Second Order Superintegrability 2 Third order superintegrability 3 General setting One first order integral Second order "Cartesian" integral Example of Schrödinger equation with Painlevé potential Second order "polar" integral Infinite families of superintegrable potentials with integrals of arbitrary order 4 5 Superintegrability with spin Conclusion and open problems 6 Pavel Winternitz (CRM) Superintegrability and exact solvability 2 / 48
Introduction Introduction Let us first consider a classical system in an n-dimensional Riemannian space with Hamiltonian n � x ∈ R n H = g ik p i p k + V ( � x ) ,� (1.1) i , k = 1 The system is called integrable (or Liouville integrable) if it allows n − 1 integrals of motion (in addition to H) X a = f a ( � x ,� a = 1 , . . . , n − 1 p ) , dX a = { H , X a } = 0 , { X a , X b } = 0 (1.2) dt This system is superintegrable if it allows further integrals Y b = f b ( � x ,� 1 ≤ k ≤ n − 1 p ) , b = 1 , . . . , k dY b = { H , Y b } = 0 . (1.3) dt Pavel Winternitz (CRM) Superintegrability and exact solvability 3 / 48
Introduction The integrals must satisfy 1. The integrals H , X a , Y b are well defined functions on phase space, i.e. polynomials or convergent power series on phase space (or an open submanifold of phase space). 2. The integrals H , X a are in involution, i.e. Poisson commute as indicated in (1.3). The integrals Y b Poisson commute with H but not necessarily with each other, nor with X a . 3. The entire set of integrals is functionally independent, i.e., the Jacobian matrix satisfies rank ∂ ( H , X 1 , . . . , X n − 1 , Y 1 , . . . , Y k ) = n + k (1.4) ∂ ( x 1 , . . . , x n , p 1 , . . . , p n ) Pavel Winternitz (CRM) Superintegrability and exact solvability 4 / 48
Introduction In quantum mechanics we define integrability and superintegrability in the same way, however in this case, H , X a and Y b are operators. The condition on the integrals of motion must also be modified e.g. as follows : 1. H , X a and Y b are well defined Hermitian operators in the enveloping algebra of the Heisenberg algebra H n ∼ { � x ,� p , � } or some generalization thereof. 2. The integrals satisfy the Lie bracket relations [ H , X a ] = [ H , Y b ] = 0 , [ X i , X k ] = 0 (1.5) 3. No polynomial in the operators H , X a , Y b formed entirely using Lie anticommutators should vanish identically. Pavel Winternitz (CRM) Superintegrability and exact solvability 5 / 48
Introduction The two best known superintegrable systems are the Kepler-Coulomb system with potential V ( r ) = α r and the isotropic harmonic oscillator V ( r ) = α r 2 . In both cases the integrals X a correspond to angular momentum, the additional integrals Y a to the Laplace-Runge-Lenz vector for V ( r ) = α r and to the quadrapole tensor T ik = p i p k + α x i x k , respectively. No further ones were discovered until a 1940 paper by Jauch and Hill on the rational anisotropic harmonic oscillator x ) = α � n i = 1 n i x 2 V ( � i , n i ∈ Z . A systematic search for superintegrable systems was started in 1965 and a real proliferation of them was observed during the last couple of years. Pavel Winternitz (CRM) Superintegrability and exact solvability 6 / 48
Introduction Let us just list some of the reasons why superintegrable systems are interesting both in classical and quantum physics. In classical mechanics, superintegrability restricts trajectories to an n − k dimensional subspace of phase space. For k = n − 1 (maximal superintegrability), this implies that all finite trajectories are closed and motion is periodic. Moreover, at least in principle, the trajectories can be calculated without any calculus. Bertrand’s theorem states that the only spherically symmetric potentials V ( r ) for which all bounded trajectories are closed are α r and α r 2 , hence no other superintegrable systems are spherically symmetric. The algebra of integrals of motion { H , X a , Y b } is a non-Abelian and interesting one. Usually it is a finitely generated polynomial algebra, only exceptionally a finite dimensional Lie algebra. In the special case of quadratic superintegrability (all integrals of motion are at most quadratic polynomials in the moments), integrability is related to separation of variables in the Hamilton-Jacobi equation, or Schrödinger equation, respectively. Pavel Winternitz (CRM) Superintegrability and exact solvability 7 / 48
Introduction In quantum mechanics, superintegrability leads to an additional degeneracy of energy levels, sometimes called "accidental degeneracy". The term was coined by Fok and used by Moshinsky and collaborators, though the point of their studies was to show that this degeneracy is certainly no accident. A conjecture, born out by all known examples, is that all maximally superintegrable systems are exactly solvable. If the conjecture is true, then the energy levels can be calculated algebraically. The wave functions are polynomials (in appropriately chosen variables) multiplied by some gauge factor. The non-Abelian polynomial algebra of integrals of motion provides energy spectra and information on wave functions. Interesting relations exist between superintegrability and supersymmetry in quantum mechanics. Pavel Winternitz (CRM) Superintegrability and exact solvability 8 / 48
Introduction As a comment, let us mention that superintegrability has also been called non-Abelian integrability. From this point of view, infinite dimensional integrable systems (soliton systems) described e.g. by the Korteweg-de-Vries equation, the nonlinear Schrödinger equation, the Kadomtsev-Petviashvili equation, etc. are actually superintegrable. Indeed, the generalized symmetries of these equations form infinite dimensional non-Abelian algebras (the Orlov-Shulman symmetries) with infinite dimensional Abelian subalgebras of commuting flows. Pavel Winternitz (CRM) Superintegrability and exact solvability 9 / 48
Second Order Superintegrability Second Order Superintegrability Let us consider the Hamiltonian (1.1) in the Euclidian space E 2 and search for second order integrals of motion. We have 2 � � � f jk ( x 1 , x 2 ) , p j 1 p k H = 1 p 2 1 + p 2 � � + V ( x 1 , x 2 ) , X = (2.1) 2 2 2 j + k = 0 In the quantum case we have p j = − i � ∂ L 3 = x 1 p 2 − x 2 p 1 , (2.2) ∂ x j The commutativity condition [ H , X ] = 0 implies that the even terms j + k = 0 , 2 and odd terms j + k = 1 in X commute with H separately. Hence we can, with no loss of generality, set f 10 = f 01 = 0. Further we find that the leading (second order) term in X lies in the enveloping algebra of the Euclidian algebra e ( 2 ) . Thus we obtain X = aL 2 3 + b 1 ( L 3 p 1 + p 1 L 3 ) + b 2 ( L 3 p 2 + p 2 L 3 ) + c 1 ( P 2 1 − P 2 2 ) (2.3) + 2 c 2 P 1 P 2 + φ ( x 1 , x 2 ) where a , b i , c i are constants. Pavel Winternitz (CRM) Superintegrability and exact solvability 10 / 48
Second Order Superintegrability The function φ ( x 1 , x 2 ) must satisfy the determining equations − 2 ( ax 2 2 + 2 b 1 x 2 + c 1 ) V x 1 + 2 ( ax 1 x 2 + b 1 x 1 − b 2 x 2 − c 2 ) V x 2 φ x 1 = − 2 ( ax 1 x 2 + b 1 x 1 − b 2 x 2 − c 2 ) V x 1 + 2 ( − ax 2 φ x 2 = 1 + 2 b 2 x 1 + c 1 ) V x 2 (2.4) The compatibility condition φ x 1 x 2 = φ x 2 x 1 implies ( − ax 1 x 2 − b 1 x 1 + b 2 x 2 + c 2 )( V x 1 x 1 − V x 2 x 2 ) − ( a ( x 2 1 + x 2 2 ) + 2 b 1 x 1 + 2 b 2 x 2 + 2 c 1 ) V x 1 x 1 − ( ax 2 + b 1 ) V x 1 + 3 ( ax 1 − b 2 ) V x 2 = 0 (2.5) Eq. (2.5) is exactly the same equation that we would have obtained if we had required that the potential should allow the separation of variables in the Schrödinger equation in one of the coordinate system in which the Helmholtz equation allows separation. Another important observation is that (2.4) and (2.5) do not involve the Planck constant. Indeed, if we consider the classical functions H and X in (2.1) and require that they Poisson commute, we arrive at exactly the same conclusions and to equations (2.4) and (2.5). Thus for quadratic integrability (and superintegrability) the potentials and integrals of motion coincide in classical and quantum mechanics (up to a possible symmetrization). Pavel Winternitz (CRM) Superintegrability and exact solvability 11 / 48
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