Oscillator Representation Research Articles

Theoretical study of the deuteron + deuteron radiative capture

In the present work, the d + d radiative capture process is studied. This process is of significant interest for astrophysical applications. The theoretical framework of the study is based on the microscopic cluster approach in the oscillator representation. The total and partial cross sections for the reaction in terms of the astrophysical S factor are calculated. A good agreement with experimental data is achieved. The tensor force of the nuclear interaction is shown to play a key role in describing the low-energy dependence of the total cross section (astrophysical S factor).

Teleportation of a qubit using quasi-Bell states

In this paper, we study the exotic Landau problem at the classical level where two conserved quantities are derived. At the quantum level, the corresponding quantum operators of the conserved quantities provide two oscillator representations from which we derive two Boson Fock spaces. Using the normalized coherent states which are the minimum uncertainty states on noncommutative configuration space isomorphic to each of the boson Fock space, we form entangled coherent states which are Bell- like states labeled quasi-Bell states. The effect of non-maximality of a quasi-Bell state based quantum channel is investigated in the context of a teleportation of a qubit.

Equal rank local theta correspondence as a strong Morita equivalence

Let (G, H) be one of the equal rank reductive dual pairs Mp2n,O2n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( Mp_{2n},O_{2n+1} \\right) $$\\end{document} or Un,Un\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( U_n,U_n \\right) $$\\end{document} over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say G^θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{G}_\ heta $$\\end{document} and H^θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{H}_\ heta $$\\end{document}, of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras whose spectra are G^θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{G}_\ heta $$\\end{document} and H^θ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widehat{H}_\ heta $$\\end{document}. This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees.

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group

The classical Stone–von Neumann theorem relates the irreducible unitary representations of the Heisenberg group Hn to non-trivial unitary characters of its center Z, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of Z and those of Hn. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara’s lemma from the theory of D-modules.

Differential Operator Approach to ıquantum Groups and Their Oscillator Representations

Differential Operator Approach to ıquantum Groups and Their Oscillator Representations

Oscillator Representations of Quantum Affine Orthosymplectic Superalgebras

Oscillator Representations of Quantum Affine Orthosymplectic Superalgebras

Complex Weyl symbols of the extended metaplectic representation operators

Complex Weyl symbols of the extended metaplectic representation operators

Bosonic Representation of Matrices and Angular Momentum Probabilistic Representation of Cyclic States.

The Jordan-Schwinger map allows us to go from a matrix representation of any arbitrary Lie algebra to an oscillator (bosonic) representation. We show that any Lie algebra can be considered for this map by expressing the algebra generators in terms of the oscillator creation and annihilation operators acting in the Hilbert space of quantum oscillator states. Then, to describe quantum states in the probability representation of quantum oscillator states, we express their density operators in terms of conditional probability distributions (symplectic tomograms) or Husimi-like probability distributions. We illustrate this general scheme by examples of qubit states (spin-1/2 su(2)-group states) and even and odd Schrödinger cat states related to the other representation of su(2)-algebra (spin-j representation). The two-mode coherent-state superpositions associated with cyclic groups are studied, using the Jordan-Schwinger map. This map allows us to visualize and compare different properties of the mentioned states. For this, the su(2) coherent states for different angular momenta j are used to define a Husimi-like Q representation. Some properties of these states are explicitly presented for the cyclic groups C2 and C3. Also, their use in quantum information and computing is mentioned.

Molecular electronic stopping cross section for H and He beams colliding with water: projectile charge state contribution

Abstract When an ion beam impinges on a target material, electron stripping and capture processes involve different charge fraction states in the beam, such that each projectile charge state produces a different energy-loss. In this work, the energy deposition of swift hydrogen and helium ion beams colliding with a water target in the gas phase is studied. The electronic structure of the molecular target is represented in terms of core, bond, and lone pair orbital decomposition within a Harmonic Oscillator representation. In this way, the stopping cross section becomes only a function of the orbital mean excitation energy, I 0i . The ion beam charge fraction compositions colliding on water is determined from the work of Wedlund et al (2019) Astronomy & Astrophysics, 630, A36) by accounting for the electron transfer cross sections. We find that the larger the projectile charge state, the larger the electronic stopping cross section and that the beam charge fraction determines the position of the maximum of the electronic stopping curve. Also, in agreement with the experiment, evidence is given on the dominant role of the largest projectile charge state in defining the stopping cross section for high energy collisions, while for low collision energies it is the lowest charge state together with all possible charge states contributing at the maximum of the electronic stopping cross section curve. Our results are reported and compared to available experimental data showing an excellent agreement to the available literature.

The Plancherel formula of Fourier integral operators: The case of Sp(1,1)

In this paper, we consider the Plancherel formula for Fourier integral operator on the space [Formula: see text], where [Formula: see text] is the quaternion skew field. By using the metaplectic representations of a noncompact classical dual pair [Formula: see text] and combining with the Plancherel formula of Poisson transform, we obtain the direct integral decomposition of all irreducible unitary representations of Fourier integral operator.

The Binding Energy and Mass of Λ-Hypernuclei

Abstract: The binding energies, mass spectrum, and constituent masses of single -exotic hypernuclei have been investigated in the framework of quarks’ structure and the non-relativistic Schrödinger equation, with the linear interaction potential describing the electrostatic interaction between the -hyperon and the nucleus core. The ground states of the hypernuclei have been studied in the framework of the oscillator representation method. The obtained results are compared with theoretical and experimental data, showing a good agreement with other values wherever available. This indicates that the interaction between the -hyperon and the nucleus core predicted by the semi-Cornell potential based on the quark-antiquark structure of clusters acts reasonably well. Keywords: Bound states, Constituent mass, Hypernuclei, Schrödinger equation.

Affinization of q-oscillator representations of $$U_q(\mathfrak {gl}_n)$$

We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible representations, which naturally corresponds to finite-dimensional irreducible representations of quantum affine algebra of untwisted affine type $A$. It is done by constructing a category of $q$-oscillator representations of the quantum affine superalgebra of type $A$, which interpolates these two family of irreducible representations. The category $\widehat{\mathcal{O}}_{\rm osc}$ can be viewed as a quantum affine analogue of the semisimple tensor category generated by unitarizable highest weight representations of $\mathfrak{gl}_{u+v}$ ($n=u+v$) appearing in the $(\mathfrak{gl}_{u+v},\mathfrak{gl}_\ell)$-duality on a bosonic Fock space.

Heisenberg parabolically induced representations of Hermitian Lie groups, Part I: Intertwining operators and Weyl transform

For a Hermitian Lie group G, we study the family of representations induced from a character of the maximal parabolic subgroup P=MAN whose unipotent radical N is a Heisenberg group. Realizing these representations in the non-compact picture on a space I(ν) of functions on the opposite unipotent radical N¯, we apply the Heisenberg group Fourier transform mapping functions on N¯ to operators on Fock spaces. The main result is an explicit expression for the Knapp–Stein intertwining operators I(ν)→I(−ν) on the Fourier transformed side. This gives a new construction of the complementary series and of certain unitarizable subrepresentations at points of reducibility. Further auxiliary results are a Bernstein–Sato identity for the Knapp–Stein kernel on N‾ and the decomposition of the metaplectic representation under the non-compact group M.

From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators

We probe the contraction from $2d$ relativistic CFTs to theories with Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, using diagnostics of quantum chaos. Starting from an Ultrarelativistic limit on a relativistic scalar field theory and following through at the quantum level using an oscillator representation of states, one can show the CFT$_2$ vacuum evolves smoothly into a BMS$_3$ vacuum in the form of a squeezed state. Computing circuit complexity of this transmutation using the covariance matrix approach shows clear divergences when the BMS point is hit or equivalently when the target state becomes a boundary state. We also find similar behaviour of the circuit complexity calculated from methods of information geometry. Furthermore, we discuss the hamiltonian evolution of the system and investigate Out-of-time-ordered correlators (OTOCs) and operator growth complexity, both of which turn out to scale polynomially with time at the BMS point.

Singular Conformal Oscillator Representations of Orthosymplectic Lie Superalgebras

Singular Conformal Oscillator Representations of Orthosymplectic Lie Superalgebras

Determinantal Ideals and the Canonical Commutation Relations: Classically or Quantized

We construct homomorphic images of $$su(n,n)^{{\mathbb {C}}}$$ in Weyl Algebras $${{\mathcal {H}}}_{2nr}$$ . More precisely, and using the Bernstein filtration of $${{\mathcal {H}}}_{2nr}$$ , $$su(n,n)^{{\mathbb {C}}}$$ is mapped into degree 2 elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of $${{\mathcal {H}}}_{2nr}$$ , these homomorphisms give all unitary highest weight representations of $$su(n,n)^{{\mathbb {C}}}$$ thus reconstructing the Kashiwara–Vergne List for the Segal–Shale–Weil representation. Using an idea from the derivation of the their list, we construct a homomorphism of $$u(r)^{{\mathbb {C}}}$$ into $${{\mathcal {H}}}_{2nr}$$ whose image commutes with the image of $$su(n,n)^{{\mathbb {C}}}$$ , and vice versa. This gives the multiplicities. The construction also gives an easy proof that the ideal of $$(r+1)\times (r+1)$$ minors is prime. Here, of course, $$r\le n-1$$ and for a fixed such r, the space of any irreducible representation of $$su(n,n)^{{\mathbb {C}}}$$ is annihilated by this ideal. As a consequence, these representations can be realized in spaces of solutions to Maxwell type equations. We actually go one step further and determine exactly for which representations from our list there is a non-trivial homomorphism between generalized Verma modules, thereby revealing, by duality, exactly which covariant differential operators have unitary null spaces. We prove the analogous results for $${{\mathcal {U}}}_q(su(n,n)^{{\mathbb {C}}})$$ . The Weyl Algebras are replaced by the Hayashi–Weyl Algebras $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ and the Fock space by a q-Fock space. Further, determinants are replaced by q-determinants, and a homomorphism of $${{\mathcal {U}}}_q(u(r)^{{\mathbb {C}}})$$ into $${{\mathcal {H}}}{{\mathcal {W}}}_{2nr}$$ is constructed with analogous properties. For this purpose a Drinfeld Double is used. We mention one difference: The quantized negative non-compact root spaces, while still of degree 2, are no longer given entirely by second order creation operators.

Random Walk on Quantum Blobs

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables — which should lead to fractal patterns on the manifold of squeezed states.

Frequency-independent dynamic stiffness and damping of laterally loaded single piles

A practical analytical model for the analysis of a pile subjected to lateral dynamic loads is presented. The model is based on the damped oscillator representation of a pile-soil system. The stiffness and damping were obtained by fitting the equation that governs the vibration of a damped oscillator to the dynamic displacements of the pile-soil system obtained by a three-dimensional finite element analysis. Unlike existing models, the model is frequency independent. Thus, the stiffness and damping can be used in the analysis of a soil-pile system in the time domain. The numerically obtained stiffness and damping are presented in dimensionless charts and influence factor Tables and can be used for analyses of massless piles and piles with added mass subjected to different loading conditions such as harmonic, impact, or earthquake loading. Time-domain analyses of piles that have been subjected to impact loading and earthquake motion are presented. Using the analytical model with the numerically obtained frequency-independent stiffness and damping compared well with existing models and with three dimensional finite element analyses results.

Estimates for matrix coefficients of representations

Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, $L^p$ estimates, which are a dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of ${\rm SL}(2,\Bbb{R})$ and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new $L^2$ estimate of dispersive type for the free Schr\"odinger equation in $\Bbb{R}^n$.