Oscillator Representation Research Articles

Two-Parameter Harmonic Oscillator Expansion Method for Calculating Non-Relativistic Ground State Energy of Coulomb Three-Particle Systems with Two Identical Particles

The variational method in oscillator representation with individual parameters for each Jacobi coordinate is applied to the non-relativistic calculation of the ground state energy of a number of three-particle Coulomb systems, consisting of two identical particles and a different one. The accuracy and convergence rate of the calculations in the constructed oscillator basis are studied up to a total of 28 oscillator quanta. The results are compared with those of the traditional approach using only one such nonlinear variational parameter. The method with individual parameters for Jacobi coordinates is found to possess a number of advantages as compared to the traditional approach.

SS-HORSE Method for Analysis of Resonances: Charged-Particle Scattering

The method for analyzing resonance states based on the harmoni©oscillator representation of scattering equations (HORSE) and on analytic properties of partial-wave scattering amplitudes which has been proposed earlier, is generalized to the case of charged-particle scattering. The method in question is tested by applying it to the model problem of pα scattering and can be used to study resonance states on the basis of microscopic calculations within various versions of the shell model.

The moment map on symplectic vector space and oscillator representation

Let G denote Sp(n,R), U(p,q), or O∗(2n). The main aim of this article is to show that the canonical quantization of the moment map on a symplectic G-vector space (W,ω) naturally gives rise to the oscillator (or Segal–Shale–Weil) representation of g:=Lie(G)⊗C. More precisely, after taking a complex Lagrangian subspace V of the complexification of W, we assign an element of the Weyl algebra for V to 〈μ,X〉 for each X∈g, which we denote by 〈μˆ,X〉. Then we show that the map X↦i〈μˆ,X〉 gives a representation of g. With a suitable choice of V in each case, the representation coincides with the oscillator representation of g.

Oscillator versus prefundamental representations. II. Arbitrary higher ranks

We find the ℓ-weights and the ℓ-weight vectors for the highest ℓ-weight q-oscillator representations of the positive Borel subalgebra of the quantum loop algebra Uq(L(sll+1)) for arbitrary values of l. Having this, we establish the explicit relationship between the q-oscillator and prefundamental representations. Our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring q-oscillator representations.

Differential invariants on symplectic spinors in contact projective geometry

We present a complete classification and the construction of Mp(2n+2,R)-equivariant differential operators acting on the principal series representations, associated with the contact projective geometry on RP2n+1 and induced from the irreducible Mp(2n,R)-submodules of the Segal–Shale–Weil representation twisted by a one-parameter family of characters. The proof is based on the classification of homomorphisms of generalized Verma modules for the Segal–Shale–Weil representation twisted by a one-parameter family of characters, together with a generalization of the well-known duality between homomorphisms of generalized Verma modules and equivariant differential operators in the category of inducing smooth admissible modules.

Highest l-Weight Representations and Functional Relations

We discuss highest $\ell$-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and $q$-oscillator representations of the positive Borel subalgebras of the quantum group $\mathrm{U}_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the $L$-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.

Infinite rank spinor and oscillator representations

We develop a functorial theory of spinor and oscillator representations parallel to the theory of Schur functors for general linear groups. This continues our work on developing orthogonal and symplectic analogues of Schur functors. As such, there are a few main points in common. We define a category of representations of what might be thought of as the infinite rank pin and metaplectic groups, and give three models of this category in terms of: multilinear algebra, diagram categories, and twisted Lie algebras. We also define specialization functors to the finite rank groups and calculate the derived functors.

Square-integrability of multivariate metaplectic wave-packet representations

This paper presents a systematic study for harmonic analysis of metaplectic wave-packet representations on the Hilbert function space . The abstract notions of symplectic wave-packet groups and metaplectic wave-packet representations will be introduced. We then present an admissibility condition on closed subgroups of the real symplectic group , which guarantees the square-integrability of the associated metaplectic wave-packet representation on .

Square-integrability of metaplectic wave-packet representations on [formula omitted

This paper presents a systematic study for harmonic analysis of metaplectic wave-packets in L2(R) via group representation theory. The abstract notions of metaplectic wave-packet groups and metaplectic wave-packet representations will be presented and as the main result, we prove an admissibility condition on closed subgroups of the special linear group SL(2,R), which guarantees square integrability of the associated metaplectic wave-packet representation on L2(R). Finally, we present an analytic study of metaplectic wave-packet representations over admissible subgroups of SL(2,R).

Symbol calculus of square-integrable operator-valued maps

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method, we show that it behaves well with respect to infinite tensor products. This construction subsumes examples from the study of magnetic Weyl calculus, magnetic pseudo-differential Weyl calculus, metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.

Orthogonal and symplectic Yangians and Yang–Baxter R-operators

Yang–Baxter R operators symmetric with respect to the orthogonal and symplectic algebras are considered in an uniform way. Explicit forms for the spinorial and metaplectic R operators are obtained. L operators, obeying the RLL relation with the orthogonal or symplectic fundamental R matrix, are considered in the interesting cases, where their expansion in inverse powers of the spectral parameter is truncated. Unlike the case of special linear algebra symmetry the truncation results in additional conditions on the Lie algebra generators of which the L operators is built and which can be fulfilled in distinguished representations only. Further, generalized L operators, obeying the modified RLL relation with the fundamental R matrix replaced by the spinorial or metaplectic one, are considered in the particular case of linear dependence on the spectral parameter. It is shown how by fusion with respect to the spinorial or metaplectic representation these first order spinorial L operators reproduce the ordinary L operators with second order truncation.

The Segal–Shale–Weil representation, the indices of Kashiwara and Maslov, and quantum mechanics

We produce a connection between the Weil 2-cocycles defining the local and adèlic metaplectic groups defined over a global field, i.e. the double covers of the attendant local and adèlic symplectic groups, and local and adèlic Maslov indices of the type considered by Souriau and Leray. With the latter tied to phase integrals occurring in quantum mechanics, we provide a formulation of quadratic reciprocity for the underlying field, first in terms of an adèlic phase integral, and then in terms of generalized time evolution unitary operators.

Dynamics of a Generalized Two-mode Husimi Function in a Kerr Medium

We describe the dynamics of a generalized Husimi function for a two-mode harmonic oscillator representation. The time evolution equation is derived through a normally ordered technique which is shown to be convenient for describing dynamical systems with SU(2) symmetry. We obtain an exact solution in phase space for a quantum nonlinear system described by a Kerr medium initially in a two-mode coherent state. We study in detail the evolution of this system and explain some particular effects exhibited during the process.

Conformal oscillator representations of orthogonal Lie algebras

The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n+2, C). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\vec a \in \mathbb{C}^n $ , we prove that the space forms an irreducible o(n+2, C)-module for any c ∈ C; if $\vec a$ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2, C) on the polynomial algebra C in n variables. Moreover, we prove that C forms an infinite-dimensional irreducible weight o(n+2, C)-module with finite-dimensional weight subspaces if c ∉ Z/2.

Fine structure inαdecay of even-even nuclei using a finite-range nucleon-nucleon interaction

A systematic study on $\ensuremath{\alpha}$-decay fine structure is presented for even-even nuclei in the range $78\ensuremath{\le}Z\ensuremath{\le}102$. The penetration probability is obtained from the WKB approximation in combination with the Bohr-Sommerfeld quantization condition. The potential barrier is numerically constructed in the well-established double-folding model for both Coulomb and nuclear potentials. A realistic M3Y interaction, based on the $G$-matrix elements of the Paris $NN$ potential, has been used in the folding calculation. The local approximation for the nondiagonal one-body density matrix in the calculation of the exchange potential was included by using the harmonic oscillator representation of the nondiagonal density matrix of the $\ensuremath{\alpha}$ particle. The computed partial half-lives and branching ratios are compared with the recent experimental data and they are in good agreement.

Paths of canonical transformations and their quantization

In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics; the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.

Linear symplectomorphisms asR-Lagrangian subspaces

The graph of a real symplectic linear transformation is an R-Lagrangian subspace of a complex symplectic vector space. The restriction of the complex symplectic form is thus purely imaginary and may be expressed in terms of the generating function of the transformation. We provide explicit formulas; moreover, as an application, we give an explicit general formula for the metaplectic representation of the real symplectic group.

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.

T-Matrix in Discrete Oscillator Representation

We investigate T-matrix for bound and continuous-spectrum states in the discrete oscillator representation. The investigation is carried out for a model problem - the particle in the field of a central potential. A system of linear equations is derived to determine the coefficients of the T-matrix expansion in the oscillator functions. We selected four potentials (Gaussian, exponential, Yukawa, and square-well ones) to demonstrate peculiarities of the T-matrix and its dependence on the potential shape. We also study how the T-matrix expansion coefficients depend on the parameters of the oscillator basis such as the oscillator length and the number of basis functions involved in calculations.

Graßmannian integrals as matrix models for non-compact Yangian invariants

In the past years, there have been tremendous advances in the field of planar N=4 super Yang–Mills scattering amplitudes. At tree-level they were formulated as Graßmannian integrals and were shown to be invariant under the Yangian of the superconformal algebra psu(2,2|4). Recently, Yangian invariant deformations of these integrals were introduced as a step towards regulated loop-amplitudes. However, in most cases it is still unclear how to evaluate these deformed integrals. In this work, we propose that changing variables to oscillator representations of psu(2,2|4) turns the deformed Graßmannian integrals into certain matrix models. We exemplify our proposal by formulating Yangian invariants with oscillator representations of the non-compact algebra u(p,q) as Graßmannian integrals. These generalize the Brezin–Gross–Witten and Leutwyler–Smilga matrix models. This approach might make elaborate matrix model technology available for the evaluation of Graßmannian integrals. Our invariants also include a matrix model formulation of the u(p,q) R-matrix, which generates non-compact integrable spin chains.