Oscillator Representation Research Articles

The Oscillator Representation and¶Groups of Heisenberg Type

We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms Φ of “Clifford type”-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G⋅ \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K⋅ \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a Φ: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra.

P-Matrix and J-Matrix Approaches: Coulomb Asymptotics in the Harmonic Oscillator Representation of Scattering Theory

The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use a well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. The P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.

Coupling a polarizable water model to the hydrated ion–water interaction potential: A test on the Cr3+ hydration

A strategy to build interaction potentials for describing ionic hydration of highly charged monoatomic cations by computer simulations, including the polarizable character of the solvent, is proposed. The method is based on the hydrated ion concept that has been previously tested for the case of Cr3+ aqueous solutions [J. Phys. Chem. 100, 11748 (1996)]. In the present work, the interaction potential of [Cr(H2O6)]3+ with water has been adapted to a water model that accounts for the polarizable character of the solvent by means of a mobile charge harmonic oscillator representation (MCHO model) [J. Chem. Phys. 93, 6448 (1990)]. Monte Carlo simulations of the Cr3+ hexahydrate plus 512 water molecules have been performed to study the energetics and structure of the ionic solution. The results show a significant improvement in the estimate of the hydration enthalpy [ΔHhydr(Cr3+)=−1109.6±70 kcal/mol] that now matches the experimental value within the uncertainty of this magnitude. The use of the polarizable water model lowers by ∼140 kcal/mol the statistical estimation of the [Cr(H2O6)]3+ hydration enthalpy compared to the nonpolarizable model. (−573 kcal/mol for the polarizable model vs −714 kcal/mol for the nonpolarizable one.) This improvement reflects a more accurate treatment of the many-body nonadditive effects.

Local Theory of Primitive Theta Functions

We study primitive theta functions, which were first introduced by Shintani, in a purely local setting. We investigate a metaplectic representation of U(1) acting on the space of local primitive theta functions and give its explicit irreducible decomposition. As a by-product, we give a new proof of epsilon dichotomy for (U(1),U}(1)).

Solvable gaudin models for higher rank symplectic algebras

We study generalisations of theXXX-model to higher rank underlying symplectic algebras in the semi-classical approximation. With the help of the metaplectic representation ofsp(2n,ℝ) we establish general results that are useful in the analysis of these algebras, and hence solve the algebraic Bethe ansatz explicitly in this representation. Thesu(n) multiplet structure of the Bethe eigenstates is revealed.

Refined algebraic quantization in the oscillator representation of SL(2, ℝ)

We investigate refined algebraic quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T*R4 and gauge group SL(2, ℝ). The reduced phase space ℳ is connected and contains four mutually disconnected “regular” sectors with topology R×S1, but these sectors are connected to each other through an exceptional set, where ℳ is not a manifold and where ℳ has non-Hausdorff topology. The RAQ physical Hilbert space Hphys decomposes as Hphys≃⊕Hi, where the four subspaces Hi naturally correspond to the four regular sectors of ℳ. The RAQ observable algebra Aobs, represented on Hphys, contains natural subalgebras represented on each Hi. The group averaging takes place in the oscillator representation of SL(2, ℝ) on L2(R2,2), and ensuring convergence requires a subtle choice for the test state space: the classical analog of this choice is to excise from ℳ the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space Hphys and a finitely generated observable subalgebra of Aobs is recovered through both Ashtekar’s algebraic quantization and Isham’s group theoretic quantization.

Quantum integrable systems and oscillator representations of c n

Solutions of the Yang-Baxter equation with spectral parameter for systems with in-variance under a Lie algebra \(\mathfrak{g}\) and for which the quantum space is a Hilbert space different from the auxiliary space are studied. In particular, for the case of \(\mathfrak{g}\)=cn= sp (2n, ℝ), solutions on infinite-dimensional state spaces are constructed.

Higher Heisenberg Lie algebras and metaplectic representations

Higher Heisenberg Lie algebras and metaplectic representations

INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS

For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.

Alpha and exotic cluster decay with a new single particle basis

A single particle basis consisting of two different harmonic oscillator representations is introduced with the aim of studying microscopically alpha and cluster decay processes. A correct description of the wave functions at large distances is obtained within a minimal single particle basis. Experimental data corresponding to a large number of alpha decay transitions from even-even nuclei are well reproduced. Applications to 14 C decay from Ra isotopes are presented.

Howe duality for the quantum groups Uqu(m,n), Uqu(M)

A dual pair is defined to be a pair of mutually centralizing subgroups of the real symplectic group. Let (G,G′) be a dual pair of reductive groups in which G is also compact and consider the decomposition of the metaplectic representation of the symplectic group into GG′-irreducibles. Each such irreducible is a tensor product of an irreducible of G with one of G′, and it turns out there is a bijective correspondence between them, a particular irreducible of G only occurring with a particular one of G′ and vice versa. This so-called Howe duality is here generalized to the quantum group case (Uqu(m,n),Uqu(M)) for q not a root of unity. A metaplectic representation for this dual pair is given in terms of the q-oscillator algebra and a Uqu(m,n)×Uqu(M)-covariant Heisenberg–Weyl algebra is also constructed and realized on Fock space. The heart of the proof lies in showing that Uqu(m,n) and Uqu(M) are essentially mutual commutants on Fock space. The duality follows using the compactness of Uqu(M). The proof is independent of the classical theory. As a consequence, given any two unitary highest weight representations of a quantum pseudo-unitary group (which arise from the restriction of a metaplectic representation), the decomposition of their tensor product is the same as in the classical case.

Hurwitz transformation and oscillator representation of a 5D “isospin” particle

The correspondence between 5D quantized Kepler problem and 8D harmonic oscillator in the case of topologically nontrivial constraints imposed on the wavefunction of the latter is considered. It is shown that both the “isospinor” and the “isovector” particles on the SU(2) instanton plus 1R background in 5D can be described in terms of an 8D singular oscillator.

Discrete multi-Gabor expansions

A discrete multi-Gabor expansion (DMGE) is developed to meet the requirements of localized and refined time-frequency (TF) representation of signals. The DMGE uses multiple windows and their translations and complex modulations as synthesis (or analysis) waveforms. It includes and generalizes the metaplectic (translation, modulation, and dilation) representations which are useful in signal analysis. Uniform, nonuniform, and proportional time sampling schemes are analyzed. The fundamental features and the importance of the DMGE are discussed. We focus on the construction of DMGE and deriving fast algorithms for the computation of related multi-analysis sequences. With matrix algebra, the algorithms derived apply to both multi-Gabor expansions and uni-(window) Gabor expansions. Another useful feature of the DMGE lies in the fact that the multi-Gabor transform can be realized in a parallel FFT-based implementation structure. Examples of DMGE and their applications to TF analysis are also discussed.

Optical functions of silicon at high temperatures

The optical spectra of silicon are studied ellipsometrically at temperatures between 300 and 1200 K in the spectral range from 2 to 4.3 eV. We present a database of optical constants for pure and heavily doped samples, with the highest concentration of free carriers of 4×1020 cm−3. We cover the photon energy range between 0.2 and 2 eV by using an oscillator representation of the visible and ultraviolet ellipsometric data. We also report the results for the temperature dependence of the E1 interband transition. The values of the energy shift and Lorentzian broadening of this spectral feature are obtained from analytical critical-point line shapes by fitting differentiated dielectric functions; alternatively, we apply the numerical convolution with Lorentzian contours to describe quantitatively the increased broadening with increasing temperature.

True many-particle scattering in the oscillator representation

Scattering theory in the oscillator representation is generalized to true many-particle scattering

Products and symmetrized powers of irreducible representations of

The calculation of branching rules, tensor products and plethysms of the infinite-dimensional harmonic series unitary irreducible representations of the non-compact group is considered and the duality between and Sp(2k) exploited. The branching rule for the restriction of an arbitrary harmonic series irreducible representation of to U(n) is derived, and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations, , of whose direct sum constitutes the metaplectic representation, H, of . A concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given. A general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of . Finally, relationships that arise from the embedding of the product groups and in the metaplectic group Mp(4nk) are discussed.

Sur la formule du caractère pour un groupe de Lie résoluble presque algébrique sur un corps p-adique

We give a global description for the characters of irreducible unitary representations of solvable almost algebraic groups over a p-adic field of characteristic zero. To obtain this, we prove a character formula on some neighborhoods of semi-simple elements. As a by product we derive a formula for the character of the metaplectic representation.

Quadratic Weyl Representations

We offer a new construction for the quadratic representations of the canonical commutation relations introduced by Proksch, Reents and Summers; our construction, which is related to the metaplectic representation, automatically produces the quadratic representations in Weyl form, circumventing the use of unbounded operators.

The level structure of exceptional metaplectic representations

The level structure of exceptional metaplectic representations

Degenerate principal series and local theta correspondence

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ωp,q and Ωp,q(1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].