Multiple Meixner polynomials and non-Hermitian oscillator Hamiltonians

Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind.Multiple Meixner polynomials are polynomials in one variable which satisfy orthogonality relations with respect to r > 1 different negative binomial distributions (Pascal distributions). There are two kinds of multiple Meixner polynomials, depending on the selection of the parameters in the negative binomial distribution. We recall their definition and some formulas and give generating functions and explicit expressions for the coefficients in the nearest neighbor recurrence relation. Following a recent construction of Miki, Tsujimoto, Vinet and Zhedanov (for multiple Meixner polynomials of the first kind), we construct r > 1 non-Hermitian oscillator Hamiltonians in r dimensions which are simultaneously diagonalizable and for which the common eigenstates are expressed in terms of multiple Meixner polynomials of the second kind.

arXiv:1310.0982Journal of Physics A Mathematical and TheoreticalNov 26, 20138 citations

Meixner Oscillators

Natig M. Atakishiyev, E. I. Jafarov, Шакир Мамед Нагиев +1 more

Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2, $\Re$ ). The reproducing kernel for the wavefunctions of these models is also found.Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,$\Re$). The reproducing kernel for the wavefunctions of these models is also found.

arXiv:math-ph/9807035Jul 31, 199825 citations

New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics

Satoru Odake

Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-)Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics.They are polynomials in sinusoidal coordinates η(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points.By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x.Except for the (q-)Racah case, they can be expressed in terms of η only, without explicit x-dependence...........Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-)Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics.They are polynomials in sinusoidal coordinates η(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points.By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x.Except for the (q-)Racah case, they can be expressed in terms of η only, without explicit x-dependence...........

arXiv:1702.03078Progress of Theoretical and Experimental PhysicsApr 4, 20174 citations

A superintegrable discrete oscillator and two-variable Meixner polynomials

Julien Gaboriaud, Vincent X. Genest, Jessica Lemieux +1 more

A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional N × N lattice. It is governed by a Hamiltonian expressed as a seven-point difference operator involving three parameters. The exact solutions of the model are given in terms of the two-variable Meixner polynomials orthogonal with respect to the negative trinomial distribution. The constants of motion of the system are constructed using the raising and lowering operators for these polynomials. They are shown to generate an su ( 2 ) invariance algebra. The two-variable Meixner polynomials are seen to support irreducible representations of this algebra. In the continuum limit, where the lattice constant tends to zero, the standard isotropic quantum oscillator in two-dimensions is recovered. The limit process from the two-variable Meixner polynomials to a product of two Hermite polynomials is carried out by involving the bivariate Charlier polynomials.A superintegrable, discrete model of the quantum isotropic oscillator in two-dimensions is introduced. The system is defined on the regular, infinite-dimensional N × N lattice. It is governed by a Hamiltonian expressed as a seven-point difference operator involving three parameters. The exact solutions of the model are given in terms of the two-variable Meixner polynomials orthogonal with respect to the negative trinomial distribution. The constants of motion of the system are constructed using the raising and lowering operators for these polynomials. They are shown to generate an su ( 2 ) invariance algebra. The two-variable Meixner polynomials are seen to support irreducible representations of this algebra. In the continuum limit, where the lattice constant tends to zero, the standard isotropic quantum oscillator in two-dimensions is recovered. The limit process from the two-variable Meixner polynomials to a product of two Hermite polynomials is carried out by involving the bivariate Charlier polynomials.

arXiv:1502.07293Journal of Physics A Mathematical and TheoreticalSep 16, 20154 citations

Exactly Solvable Multicomponent Spinless Fermions

Ryu Sasaki B

Abstract By generalizing the one-to-one correspondence between exactly solvable Hermitian matrices $\mathcal {H}=\mathcal {H}^\dagger$ and exactly solvable spinless fermion systems $\mathcal {H}_f=\sum _{x,y}c_x^\dagger \mathcal {H}(x,y)c_y$ , four types of exactly solvable multicomponent fermion systems are constructed explicitly. They are related to the multivariate Krawtchouk, Meixner, and two types of Rahman-like polynomials, constructed recently by myself. The Krawtchouk and Meixner polynomials are the eigenvectors of certain real symmetric matrices $\mathcal {H}$ which are related to the difference equations governing them. The corresponding fermions have nearest-neighbor interactions. The Rahman-like polynomials are eigenvectors of certain reversible Markov chain matrices $\mathcal {K}$ , from which real symmetric matrices $\mathcal {H}$ are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide-range interactions.Abstract By generalizing the one-to-one correspondence between exactly solvable Hermitian matrices $\mathcal {H}=\mathcal {H}^\dagger$ and exactly solvable spinless fermion systems $\mathcal {H}_f=\sum _{x,y}c_x^\dagger \mathcal {H}(x,y)c_y$, four types of exactly solvable multicomponent fermion systems are constructed explicitly. They are related to the multivariate Krawtchouk, Meixner, and two types of Rahman-like polynomials, constructed recently by myself. The Krawtchouk and Meixner polynomials are the eigenvectors of certain real symmetric matrices $\mathcal {H}$ which are related to the difference equations governing them. The corresponding fermions have nearest-neighbor interactions. The Rahman-like polynomials are eigenvectors of certain reversible Markov chain matrices $\mathcal {K}$, from which real symmetric matrices $\mathcal {H}$ are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide-range interactions.

arXiv:2502.05455Progress of Theoretical and Experimental PhysicsApr 8, 20250 citations

New determinant expressions of multi-indexed orthogonal polynomials in discrete quantum mechanics

Satoru Odake

Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-) Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics. They are polynomials in sinusoidal coordinates eta(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points. By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x. Except for the (q-) Racah case, they can be expressed in terms of eta only, without explicit x-dependence.Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-) Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics. They are polynomials in sinusoidal coordinates eta(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points. By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x. Except for the (q-) Racah case, they can be expressed in terms of eta only, without explicit x-dependence.

May 1, 20175 citations

\Re

). The reproducing kernel for the wavefunctions of these models is also found.

Abstract By generalizing the one-to-one correspondence between exactly solvable Hermitian matrices

\mathcal {H}=\mathcal {H}^\dagger

and exactly solvable spinless fermion systems

\mathcal {H}_f=\sum _{x,y}c_x^\dagger \mathcal {H}(x,y)c_y

, four types of exactly solvable multicomponent fermion systems are constructed explicitly. They are related to the multivariate Krawtchouk, Meixner, and two types of Rahman-like polynomials, constructed recently by myself. The Krawtchouk and Meixner polynomials are the eigenvectors of certain real symmetric matrices

\mathcal {H}

which are related to the difference equations governing them. The corresponding fermions have nearest-neighbor interactions. The Rahman-like polynomials are eigenvectors of certain reversible Markov chain matrices

\mathcal {K}

, from which real symmetric matrices

\mathcal {H}

are uniquely defined by the similarity transformation in terms of the square root of the stationary distribution. The fermions have wide-range interactions.