Heun Operator Research Articles

ON A FAMILY OF DISCRETE ND LADDER-TYPE OPERATORS CONSTRUCTED IN TERMS OF THE HERMITIAN TOEPLITZ COMMUTATOR OPERATOR Z_N

A development of an algebraic system with N-dimensional ladder-type operators associated with the discrete Fourier transform is described, following an analogy with the canonical commutation relations of the continuous case. It is found that a Hermitian Toeplitz matrix Z_N, which plays the role of the identity, is sufficient to satisfy the Jacobi identity and, by solving some compatibility relations, a family of ladder operators with corresponding Hamiltonians can be constructed. The behaviour of the matrix Z_N for large N is elaborated. It is shown that this system can be also realized in terms of the Heun operator W, associated with the discrete Fourier transform, thus providing deeper insight on the underlying algebraic structure.

Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes

This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and . The entanglement Hamiltonian and its relation to Heun operators are also investigated.

Entanglement of inhomogeneous free fermions on hyperplane lattices

We introduce an inhomogeneous model of free fermions on a (D−1)-dimensional lattice with D(D−1)/2 continuous parameters that control the hopping strength between adjacent sites. We solve this model exactly, and find that the eigenfunctions are given by multidimensional generalizations of Krawtchouk polynomials. We construct a Heun operator that commutes with the chopped correlation matrix, and compute the entanglement entropy numerically for D=2,3,4, for a wide range of parameters. For D=2, we observe oscillations in the sub-leading contribution to the entanglement entropy, for which we conjecture an exact expression. For D>2, we find logarithmic violations of the area law for the entanglement entropy with nontrivial dependence on the parameters.

Time and band limiting operator and Bethe ansatz

The time and band limiting operator is introduced to optimize the reconstruction of a signal from only a partial part of its spectrum. In the discrete case, this operator commutes with the so-called algebraic Heun operator which appears in the context of the quantum integrable systems. The construction of both operators and the proof of their commutativity is recalled. A direct connection between their spectra is obtained. Then, the Bethe ansatz, a well-known method to diagonalize integrable quantum Hamiltonians, is used to diagonalize the Heun operator and to obtain insights on the spectrum of the time and band limiting operator.

The rational Sklyanin algebra and the Wilson and para-Racah polynomials

The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second-order Heun operators on quadratic grids with no diagonal terms are determined. These special or S-Heun operators lead to the rational degeneration of the Sklyanin algebra; they also entail the contiguity and structure operators of the Wilson polynomials. The finite-dimensional restriction yields a representation that acts on the para-Racah polynomials.

Heun operator of Lie type and the modified algebraic Bethe ansatz

The generic Heun operator of Lie type is identified as a certain BC-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We also show that these Bethe roots are intimately associated with the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of O(3) and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.

Entanglement in fermionic chains and bispectrality

Entanglement in finite and semi-infinite free Fermionic chains is studied. A parallel is drawn with the analysis of time and band limiting in signal processing. It is shown that a tridiagonal matrix commuting with the entanglement Hamiltonian can be found using the algebraic Heun operator construct in instances when there is an underlying bispectral problem. Cases corresponding to the Lie algebras [Formula: see text] and [Formula: see text] as well as to the q-deformed algebra [Formula: see text] at [Formula: see text] a root of unity are presented.

The rational Heun operator and Wilson biorthogonal functions

We consider the rational Heun operator defined as the most general second-order q-difference operator which sends any rational function of type \([(n-1)/n]\) to a rational function of type \([n/(n+1)]\). We shall take the poles to be located on the Askey–Wilson grid. It is shown that this operator is related to the one-dimensional degeneration of the Ruijsenaars–van Diejen Hamiltonians. The Wilson biorthogonal functions of type \({_{10}}\Phi _9\) are found to be solutions of a generalized eigenvalue problem involving rational Heun operators of the special “classical” kind.

Sklyanin-like algebras for (q-)linear grids and (q-)para-Krawtchouk polynomials

S-Heun operators on linear and q-linear grids are introduced. These operators are special cases of Heun operators and are related to Sklyanin-like algebras. The continuous Hahn and big q-Jacobi polynomials are functions on which these S-Heun operators have natural actions. We show that the S-Heun operators encompass both the bispectral operators and Kalnins and Miller’s structure operators. These four structure operators realize special limit cases of the trigonometric degeneration of the original Sklyanin algebra. Finite-dimensional representations of these algebras are obtained from a truncation condition. The corresponding representation bases are finite families of polynomials: the para-Krawtchouk and q-para-Krawtchouk ones. A natural algebraic interpretation of these polynomials that had been missing is thus obtained. We also recover the Heun operators attached to the corresponding bispectral problems as quadratic combinations of the S-Heun operators.

Degenerate Sklyanin algebras, Askey–Wilson polynomials and Heun operators

The q-difference equation, the shift and the contiguity relations of the Askey–Wilson polynomials are cast in the framework of the three and four-dimensional degenerate Sklyanin algebras and . It is shown that the q-para Racah polynomials corresponding to a non-conventional truncation of the Askey–Wilson polynomials form a basis for a finite-dimensional representation of . The first order Heun operators defined by a degree raising condition on polynomials are shown to form a five-dimensional vector space that encompasses . The most general quadratic expression in the five basis operators and such that it raises degrees by no more than one is identified with the Heun–Askey–Wilson operator.

Entanglement of free Fermions on Hadamard graphs

Free Fermions on vertices of distance-regular graphs are considered. Bipartitions are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in detail within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.

Heun algebras of Lie type

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For s u ( 2 ) \mathfrak {su}(2) , this leads to the Heun–Krawtchouk algebra. The corresponding Heun–Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovsky–Voltera gyrostat. For s u ( 1 , 1 ) \mathfrak {su}(1,1) , one obtains the Heun algebras attached to the Meixner, Meixner–Pollaczek, and Laguerre polynomials. These Heun algebras are shown to be isomorphic to the the Hahn algebra. Focusing on the harmonic oscillator algebra h o \mathfrak {ho} leads to the Heun–Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the s u ( 1 , 1 ) \mathfrak {su}(1,1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn, and Confluent Heun operators, respectively.

Free-Fermion entanglement and orthogonal polynomials

We present a simple construction for a tridiagonal matrix T that commutes with the hopping matrix for the entanglement Hamiltonian of open finite free-Fermion chains associated with families of discrete orthogonal polynomials. It is based on the notion of algebraic Heun operator attached to bispectral problems, and the parallel between entanglement studies and the theory of time and band limiting. As examples, we consider Fermionic chains related to the Chebychev, Krawtchouk and dual Hahn polynomials. For the former case, which corresponds to a homogeneous chain, the outcome of our construction coincides with a recent result of Eisler and Peschel; the latter cases yield commuting operators for particular inhomogeneous chains. Since T is tridiagonal and non-degenerate, it can be readily diagonalized numerically, which in turn can be used to calculate the spectrum of , and therefore the entanglement entropy.

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

The Heun–Askey–Wilson algebra is introduced through generators $$\{{\textsf {X}},{\textsf {W}}\}$$ and relations. These relations can be understood as an extension of the usual Askey–Wilson ones. A central element is given, and a canonical form of the Heun–Askey–Wilson algebra is presented. A homomorphism from the Heun–Askey–Wilson algebra to the Askey–Wilson one is identified. On the vector space of the polynomials in the variable $$x=z+z^{-1}$$ , the Heun operator of Askey–Wilson type realizing $${\textsf {W}}$$ can be characterized as the most general second-order q-difference operator in the variable z that maps polynomials of degree n in $$x=z+z^{-1}$$ into polynomials of degree $$n+1$$ .

The Heun operator of Hahn-type

The Heun-Hahn operator on the uniform grid is defined. This operator is shown to map polynomials of degree $n$ to polynomials of degree $n+1$, to be tridiagonal in bases made out of either Pochhammer or Hahn polynomials and to be bilinear in the operators of the Hahn algebra. The extension of this algebra that includes the Heun-Hahn operator as generator is described. Biorthogonal rational functions on uniform grids are shown to be related to this framework.

Algebraic Heun Operator and Band-Time Limiting

We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to quite a few other situations. We also extend his algebraic construction to cover some purely finite cases where his proposal fails to give a useful commuting operator.

On q-Deformations of the Heun Equation

The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator.

Separation equations for 2D superintegrable systems on constant curvature spaces

Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

Tridiagonalization and the Heun equation

It is shown that the tridiagonalization of the hypergeometric operator L yields the generic Heun operator M. The algebra generated by the operators L, M and Z = [L, M] is quadratic and a one-parameter generalization of the Racah algebra. The Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators L and M. An interpretation in terms of the Racah problem for su(1,1) and separation of variables in a superintegrable system are discussed.

The Heun operator as a Hamiltonian

It is shown that the celebrated Heun operator + is the Hamiltonian of the -quantum Euler–Arnold top of spin ν in a constant magnetic field. For it is canonically equivalent to Calogero–Moser–Sutherland quantum models; if , ten known one-dimensional quasi-exactly-solvable problems are reproduced, and if, in addition, , then four well-known one-dimensional quantal exactly-solvable problems are reproduced. If spin ν of the top takes a (half)-integer value the Hamiltonian possesses a finite-dimensional invariant subspace and a number of polynomial eigenfunctions occur. Discrete systems on uniform and exponential lattices are introduced which are canonically equivalent to the one described by the Heun operator.