Related papers: A_N-type Dunkl operators and new spin Calogero-Sut…
The family of Tremblay-Turbiner-Winternitz (TTW) Hamiltonians $H_k$ on a plane, corresponding to any positive real value of $k$, is shown to admit another ${\cal N} = 2$ supersymmetric extension than that previously introduced by the…
The family of quaternionic quasi-unitary (or quaternionic unitary Cayley--Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_{N+1}, as well as many…
We propose and analyze a family of approximately-symmetric neural networks for quantum spin liquid problems. These tailored architectures are parameter-efficient, scalable, and significantly outperform existing symmetry-unaware neural…
We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the…
We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various N=4 superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic…
We continue our systematic construction of Baxter Q-operators for spin chains, which is based on certain degenerate solutions of the Yang-Baxter equation. Here we generalize our approach from the fundamental representation of gl(n) to…
Heckman-Polychronakos operators form a prominent family of commuting differential-difference operators defined in terms of the Dunkl operators $\mathcal D_i$ as $\mathcal P_m= \sum_{i=1}^N (x_i \mathcal D_i)^m$. They have been known since…
We investigate a special class of the $\mathcal{PT}$-symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the…
Our work addresses the problem of generating maximally entangled two spin-1/2 (qubit) symmetric states using NMR, NQR, Lipkin-Meshkov-Glick Hamiltonians. Time evolution of such Hamiltonians provides various logic gates which can be used for…
Using a similarity transformation that maps the Calogero model into $N$ decoupled quantum harmonic oscillators, we construct a set of mutually commuting conserved operators of the model and their simultaneous eigenfunctions. The…
The relationship between the quasi-exactly solvable problems and W-algebras is revealed. This relationship enabled one to formulate a new general method for building multi-dimensional and multi-channel exactly and quasi-exactly solvable…
We develop a supersymmetric representation of spin operators which unifies the Schwinger and Abrikosov representations of SU(N) spin operators, allowing a second-quantized treatment of representations of the SU(N) group with both symmetric…
In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and…
We derive the necessary and sufficient condition for Type A N-fold supersymmetry by direct calculation of the intertwining relation and show the complete equivalence between this analytic construction and the sl(2) construction based on…
An efficient procedure for constructing quasi-exactly solvable matrix models is suggested. It is based on the fact that the representation spaces of representations of the algebra sl(2,R) within the class of first-order matrix differential…
We review a recently introduced set of N-dimensional quasi-maximally superintegrable Hamiltonian systems describing geodesic motions, that can be used to generate "dynamically" a large family of curved spaces. From an algebraic viewpoint,…
In this paper we study the su(m) spin Sutherland (trigonometric) model of D_N type and its related spin chain of Haldane-Shastry type obtained by means of Polychronakos's freezing trick. As in the rational case recently studied by the…
N-fold supersymmetry is an extension of the ordinary supersymmetry in one-dimensional quantum mechanics. One of its major property is quasi-solvability, which means that energy eigenvalues can be obtained for a portion of the spectra. We…
Quaternionic and octonionic spinors are introduced and their fundamental properties (such as the space-times supporting them) are reviewed. The conditions for the existence of their associated Dirac equations are analyzed. Quaternionic and…
A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the…