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arXiv:2010.15276 · Symmetry Integrability and Geometry Methods and Applications

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

Ian Marquette, C. Quesne, Université Libre de Bruxelles, Belgium

Published Jan 14, 2022• 2 citations
arXivHTMLPDF

Abstract

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze.However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied.We present a set of six operators {A ± , B ± , C ± } that can be combined to build a gl(3) hidden algebra.The latter can be embedded in an sp(6) algebra, as well as in an osp(1/6) superalgebra.The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state.We present the action of these operators and study the construction of an extended biorthogonal basis.These rely on establishing various nontrivial polynomial and commutator identities.We also make a connection between the hidden symmetry and the underlying superintegrability property of the model.Interestingly, the integrals generate a cubic algebra.This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

DOI
10.3842/sigma.2022.005

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A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze.However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied.We present a set of six operators {A ± , B ± , C ± } that can be combined to build a gl(3) hidden algebra.The latter can be embedded in an sp(6) algebra, as well as in an osp(1/6) superalgebra.The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state.We present the action of these operators and study the construction of an extended biorthogonal basis.These rely on establishing various nontrivial polynomial and commutator identities.We also make a connection between the hidden symmetry and the underlying superintegrability property of the model.Interestingly, the integrals generate a cubic algebra.This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.
A shape invariant nonseparable and nondiagonalizable two-dimensional model with quadratic complex interaction, first studied by Cannata, Ioffe, and Nishnianidze, is re-examined with the purpose of exhibiting its hidden algebraic structure. The two operators A+ and A−, coming from the shape invariant supersymmetrical approach, where A+ acts as a raising operator while A− annihilates all wavefunctions, are completed by introducing a novel pair of operators B+ and B−, where B− acts as the missing lowering operator. These four operators then serve as building blocks for constructing gl(2) generators, acting within the set of associated functions belonging to the Jordan block corresponding to a given energy eigenvalue. This analysis is extended to the set of Jordan blocks by constructing two pairs of bosonic operators, finally yielding an sp(4) algebra, as well as an osp(1/4) superalgebra. Hence, the hidden algebraic structure of the model is very similar to that known for the two-dimensional real harmonic oscillator.
The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by an explicit construction of a new inner product employing a ‘quasi-fermion’ algebra in momentum space, where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long-range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c = 1, while the latter has been connected to a logarithmic CFT with central charge c = −2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.
We study two three-dimensional quantum superintegrable systems in nonvanishing axially symmetric magnetic fields. For each model the integrals generate a quadratic algebra with a Casimir, and finite-dimensional deformed-oscillator representations yield algebraic spectra. The same spectra are recovered by separation of variables. In circular parabolic coordinates the separated equations are sextic biconfluent-Heun equations with finite polynomial sectors. We formulate them as Sturm eigenvalue problems with the physical energy as a parameter and the separation constant diagonalized. After gauge transformation, each equation admits an sl(2)sl(2)sl(2) algebraization and reduces to a finite tridiagonal matrix. We also construct a pre-separation Sturm representation of each complete Hamiltonian, taking the inverse-square coupling as the Sturm eigenvalue. The transformed integrals remain commuting and, after fixing the axial integral, changing variables and gauge rotating, the two-variable Sturm Hamiltonian and its non-central integrals belong to the universal enveloping algebra U(sl(3))U(sl(3))U(sl(3)). We thus obtain three complementary algebraic descriptions: the quadratic-algebra/deformed-oscillator representation of the original Hamiltonian, the sl(2)sl(2)sl(2) algebraization after parabolic separation, and the sl(3)sl(3)sl(3) algebraization of the complete Sturm problem before separation. Although the two magnetic Hamiltonians are distinct in the original classification, fixing the axial integral reduces them to the same singular 2:1 oscillator normal form with different effective parameter embeddings. This explains the common sl(3)sl(3)sl(3) structure without implying equivalence of the original magnetic systems. These results show how distinct hidden algebraic structures and exact or quasi-exact solvability can emerge at different stages of a multiseparable superintegrable system.
summary:In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the ddd-dimensional quantum harmonic oscillator are considered. In this case the usual set of the creation and annihilation operators is used as basis of the Weyl algebra under consideration. From this model are deduced the underlying symplectic structure and some of its consequences. In particular, the transitions from the initial $\text{Sp}(d,\bbfC)$-invariant model to the $\text{GL}(d,\bbfC)$- and, at last, U(d)\text{U}(d)U(d)-invariant models are performed.
We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-12\tfrac1221​ degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series su(1,1)su(1,1)su(1,1) representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional j=0j=0j=0 sector, complete energy-shell multiplicities, and the effect of particle interchange.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, G2G_2G2​ rational, or I6I_6I6​) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index kkk. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure g(k)g^{(k)}g(k) with various kkk, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra g(k)g^{(k)}g(k) for a certain kkk. In all presented cases the algebra of integrals is a 4-generated (H,I1,I2,I12≡[I1,I2])(H, I_1, I_2, I_{12}\equiv[I_1, I_2])(H,I1​,I2​,I12​≡[I1​,I2​]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.