Related papers: Quasi-Exactly Solvable Systems and Orthogonal Poly…

We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a $\mathcal{D}$-sphere ($\mathcal{D} \ge 1$), is a polynomial in the interelectronic distance $u$…

This work identifies a solvable (in the sense that spectral correlation functions can be expressed in terms of orthogonal polynomials), rotationally invariant random matrix ensemble with a logarithmic weakly confining potential. The…

A function g, with domain the natural numbers, is a quasi-polynomial if there exists a period m and polynomials p_0,p_1,...,p_{m-1} such that g(t)=p_i(t) for t=i mod m. Quasi-polynomials classically -- and "reasonably" -- appear in Ehrhart…

We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are…

We extend the notion of quasi-exactly solvable (QES) models from potential ones and differential equations to Bose systems. We obtain conditions under which algebraization of the part of the spectrum occurs. In some particular cases simple…

A new two-parameter family of quasi-exactly solvable quartic polynomial potentials $V(x)=-x^4+2iax^3+(a^2-2b)x^2+2i(ab-J)x$ is introduced. Until now, it was believed that the lowest-degree one-dimensional quasi-exactly solvable polynomial…

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to…

We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed.

We present a general procedure for determining possible (nonuniform) magnetic fields such that the Pauli equation becomes quasi-exactly solvable (QES) with an underlying $sl(2)$ symmetry. This procedure makes full use of the close…

We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…

Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…

We provide a rigorous definition of quasi-normal modes for a rotating black hole. They are given by the poles of a certain meromorphic family of operators and agree with the heuristic definition in the physics literature. If the black hole…

Various examples of exactly solvable `discrete' quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilation-creation operators, the dynamical symmetry algebras and coherent…

The q-Laguerre polynomials correspond to an indetermined moment problem. For explicit discrete non-N-extremal measures corresponding to Ramanujan's ${}_1\psi_1$-summation we complement the orthogonal q-Laguerre polynomials into an explicit…

We introduce the quasi-topological electromagnetism which is defined to be the squared norm of the topological 4-form $F\wedge F$. A salient property is that its energy-momentum tensor is of the isotropic perfect fluid with the pressure…

Quantum signal processing is a powerful framework in quantum algorithms, playing a central role in Hamiltonian simulation and related applications. The sequence of polynomials implemented at each step of this protocol provides a polynomial…

Analysis of edge-state energies in the integer quantum Hall effect is carried out within the semiclassical approximation. When the system is wide so that each edge can be considered separatly, this problem is equivalent to that of a one…

We first present a quasinormal mode (QNM) theory for coupled loss-gain resonators working near an exceptional point. Assuming linear media, which can be fully quantified using the complex pole properties of the QNMs, we show how the QNMs…

A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new…

Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians…