Search Results (68)

The para-positronium system $\left({}^{1}S_{0}Ps\right)$ is described by means of specially constructed coherent states (CSs) in the Klauder–Perelomov sense. It is analyzed from the physical point of view and from the geometry underlying the relevant symmetry group establishing the dynamics of the processes. In this new theoretical context, the possibility of a gamma-ray laser emission is investigated within a QFT context, showing explicitly that, in addition to the oscillator solution based only on a Bogoliubov approximation for the condensate, there is a second phase or “squeezed” stage by which physical features beyond the classical ones appear. Explicitly, while the generated photons are in the active medium (e.g., Ps-BEC), the evolution is described by a Heisenberg–Weyl coherent state with displacement operators dependent on the interaction time, which is related to the condensate shape. After the interaction time has elapsed, we explicitly demonstrate that the displacement operator of the ${}^{1}S_{0}Ps$ is transformed into a squeezed operator of the photonic fields modulated by the matrix element of the Positronium decay ${\mathcal{M}}_{{}^{1}S_{0}Ps}\to 2\gamma$. We also show that this squeezed operator (belonging to the Metaplectic group) generates a non-classical radiation state spanning only even (s = 1/4) levels in the number of photons. The implications in astrophysical systems of interest, considering gamma-ray coherent emission and the possibility of an ${}^{1}S_{0}Ps-BEC$ in the context of pulsars, blazars, and quasars, are briefly discussed. Full article

The q-Heisenberg algebra ${\mathfrak{h}}_n\left(q\right)$ is a significant class of solvable polynomial algebras, and it unifies the canonical commutation relations of Heisenberg algebras and the deformation theory of quantum groups. In this paper, we employ Gröbner-Shirshov basis theory and PBW (Poincar$\acute{e}$-Birkhoff-Witt) basis techniques to systematically investigate ${\mathfrak{h}}_n\left(q\right)$. Our main results establish that: ${\mathfrak{h}}_n\left(q\right)$ possesses an iterated skew-polynomial algebra structure, and it satisfies the important homological regularity properties of being Auslander regular, Artin-Schelter regular, and Cohen-Macaulay. These findings provide deep insights into the algebraic structure of ${\mathfrak{h}}_n\left(q\right)$, while simultaneously bridging the gap between noncommutative algebra and quantum representation theory. Furthermore, our constructive approach yields computable methods for studying modules over ${\mathfrak{h}}_n\left(q\right)$, opening new avenues for further research in deformation quantization and quantum algebra. Full article

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. Full article

A real-space renormalization group (RG) framework is formulated for classical SO(n) spin models defined on d-dimensional crystal lattices composed of corner-sharing hyper-tetrahedra, a class of geometrically frustrated crystal structures. This includes, as specific instances, the classical Heisenberg model on the kagome and pyrochlore crystals. The approach involves computing the partition function and corresponding order parameters for spin clusters embedded in the crystal, to leading order in symmetry-breaking fields generated by surrounding spins. The crystal geometry plays a central role in determining the scaling relations and the associated critical behavior. To illustrate the efficacy of the method, a reduced manifold of symmetry-allowed ordered states for isotropic nearest-neighbor interactions is analyzed. The RG flow systematically excludes the emergence of a $\overrightarrow{q}=\mathbf{0}$ ordered phase within the antiferromagnetic sector, independently of both the spatial dimensionality of the crystal and the number of spin components. Extensions to incorporate more elaborate crystal-symmetry-induced ordering patterns and fluctuation-driven phenomena—such as order-by-disorder—are also discussed. Full article

In this paper, we are interested in a class of critical fractional Choquard–Kirchhoff equations with p-Laplacian on the Heisenberg group. By employing several critical point theorems, we obtain the existence and multiplicity of nontrivial solutions under different perturbation terms. Due to the critical convolution term, the compactness condition may fail. To overcome this, we apply the concentration-compactness principle. The results in this paper can be viewed as complementary to the previous results under the conditions of $s=1$, $p=2$, and in the subcritical case. Full article

This study systematically investigates the magnetic ordering and magnetocaloric properties of a series of polycrystalline compounds, Gd_1-xDy_xScO₃ (x = 0, 0.1, 0.2 and 1). X-ray powder diffraction (XRD) analysis confirms that all samples exhibit an orthorhombic perovskite structure with a space group of Pbnm. The zero-field cooling and field cooling magnetization curves demonstrate a transition from antiferromagnetic to paramagnetic phases, with Néel temperatures of about 3 K for GdScO₃ and 4 K for DyScO₃. The doping of Dy³⁺ weakened long-range antiferromagnetic order and enhanced short-range magnetic disorder in GdScO₃, leading to vanished antiferromagnetic transition between 2 and 100 K for the sample of x = 0.2. Using the Arrott–Noakes equation, we constructed Arrott plots to analyze the system’s critical behavior. Both the compounds with x = 0.1 and x = 0.2 conform to the 3D-Heisenberg model. These results indicate the weakened long-range antiferromagnetic order induced by Dy³⁺ doping. Significant maximal magnetic entropy change (−Δ$S_{\mathrm{M}}^{\mathrm{M}\mathrm{a}\mathrm{x}}$) of 36.03 J/kg K at 3 K for the sample Gd_0.9Dy_0.1ScO₃ is achieved as the magnetic field changes from 0 to 50 kOe, which is higher than that of GdScO₃ (−Δ$S_{\mathrm{M}}^{\mathrm{M}\mathrm{a}\mathrm{x}}$ = 34.32 J/kg K) and DyScO₃ (−Δ$S_{\mathrm{M}}^{\mathrm{M}\mathrm{a}\mathrm{x}}$ = 15.63 J/kg K). The considerable magnetocaloric effects (MCEs) suggest that these compounds can be used in the development of low-temperature magnetic refrigeration materials. Full article

In the Heisenberg group ${\mathbb{H}}^n$, we establish the local regularity theory for weak solutions to non-homogeneous degenerate nonlinear parabolic equations of the form ${\partial }_{t}u-{\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i\left(Xu\right)=K(x,t,u,Xu)$, where the nonlinear structure is modeled on non-homogeneous parabolic p-Laplacian-type operators. Specifically, we prove two main local regularities: (i) For $2\le p\le 4$, we establish the local Lipschitz regularity ($u\in C_{\mathrm{loc}}^{0,1}$), with the horizontal gradient satisfying $Xu\in L_{\mathrm{loc}}^{\infty }$; (ii) For $2\le p<3$, we establish the local second-order horizontal Sobolev regularity ($u\in H{W}_{\mathrm{loc}}^{2,2}$), with the second-order horizontal derivative satisfying $XXu\in L_{\mathrm{loc}}^2$. These results solve an open problem proposed by Capogna et al. Full article

While either a spin or point-group adaptation is straightforward when considered independently, the standard technique for factoring isotropic spin Hamiltonians by the total spin S and the irreducible representation $\mathsf{\Gamma }$ of the point group is limited by the complexity of the transformations between different coupling schemes that are related in terms of their site permutations. To overcome these challenges, we apply projection operators directly to uncoupled basis states, enabling the simultaneous treatment of spin and point-group symmetry without the need for recoupling transformations. This provides a simple and efficient approach for the exact diagonalization of isotropic spin models, which we illustrate, with applications in Heisenberg spin rings and polyhedra, including systems that are computationally inaccessible with conventional coupling techniques. Full article

In this paper, we study some homogeneity properties of a semi-direct extension of the Heisenberg group, known in literature as the hyperbolic oscillator (or Boidol) group, equipped with the left-invariant metrics corresponding to the ones of the oscillator group. We identify the naturally reductive case by the existence of the corresponding special homogeneous structures. For the cases where these special homogeneous structures do not exist, we exhibit a complete description of the homogeneous geodesics. Full article

In the Heisenberg group ${\mathbb{H}}^n$, we obtain the local second-order $H{W}_{\mathrm{loc}}^{2,2}$-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ${\partial }_{t}u={\displaystyle \sum _{i=1}^{2n}}{X}_i{A}_i(Xu)$ modeled on the parabolic p-Laplacian equation. Specifically, when $2\le p\le 4$, we demonstrate the integrability of ${({\partial }_{t}u)}^2$, namely, ${\partial }_{t}u\in L_{\mathrm{loc}}^2$; when $2\le p<3$, we demonstrate the $H{W}_{\mathrm{loc}}^{2,2}$-regularity of u, namely, $XXu\in L_{\mathrm{loc}}^2$. For the $H{W}_{\mathrm{loc}}^{2,2}$-regularity, when $p\ge 2$, the range of p is optimal compared to the Euclidean case. Full article

The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type $H^p$ multiplier theorem on the Heisenberg group. If an operator-valued function $M\left(\lambda \right)$ satisfies certain conditions, the right-multiplier operator $T_M$ is bounded on the Hardy space $H^p\left({\mathbb{H}}^n\right),$ which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems. Full article

Spin Hamiltonians, like the Heisenberg model, are used to describe the magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities, such as heat capacities, magnetic susceptibilities or neutron-scattering spectra, can be calculated based on energies and eigenstates obtained by exact diagonalization (ED). Utilizing spin-rotational symmetry SU(2) to factor the Hamiltonian with respect to total spin S facilitates ED, but the conventional approach to spin-adapting the basis is more intricate than selecting states with a given magnetic quantum number M (the spin z-component), as it relies on irreducible tensor-operator techniques and spin-coupling coefficients. Here, we present a simpler technique based on applying a spin projector to uncoupled basis states. As an alternative to Löwdin’s projection operator, we consider a group-theoretical formulation of the projector, which can be evaluated either exactly or approximately using an integration grid. An important aspect is the choice of uncoupled basis states. We present an extension of Löwdin’s theorem for $s=\frac{1}{2}$ to arbitrary local spin quantum numbers s, which allows for the direct selection of configurations that span a complete, linearly independent basis in an S sector upon the spin projection. We illustrate the procedure with a few examples. Full article

Spin models like the Heisenberg Hamiltonian effectively describe the interactions of open-shell transition-metal ions on a lattice and can account for various properties of magnetic solids and molecules. Numerical methods are usually required to find exact or approximate eigenstates, but for small clusters with spatial symmetry, analytical solutions exist, and a few Heisenberg systems have been solved in closed form. This paper presents a simple, generally applicable approach to analytically solve isotropic spin clusters, based on adapting the basis to both total spin and point group symmetry to factor the Hamiltonian matrix into sufficiently small blocks. We demonstrate applications to small rings and polyhedra, some of which are straightforward to solve by successive spin-coupling for Heisenberg terms only; additional interactions, such as biquadratic exchange or multi-center terms necessitate symmetry adaptation. Full article

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel’d twists is presented, Lie bialgebra structures of which have been investigated by the authors recently. Full article