937 research outputs found

    Covariant (hh')-Deformed Bosonic and Fermionic Algebras as Contraction Limits of q-Deformed Ones

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    GLh(n)×GLh′(m)GL_h(n) \times GL_{h'}(m)-covariant (hh')-bosonic (or (hh')-fermionic) algebras Ahh′±(n,m){\cal A}_{hh'\pm}(n,m) are built in terms of the corresponding R_h and Rh′R_{h'}-matrices by contracting the GLq(n)×GLq±1(m)GL_q(n) \times GL_{q^{\pm1}}(m)-covariant q-bosonic (or q-fermionic) algebras Aq±(α)(n,m){\cal A}^{(\alpha)}_{q\pm}(n,m), α=1,2\alpha = 1, 2. When using a basis of Aq±(α)(n,m){\cal A}^{(\alpha)}_{q\pm}(n,m) wherein the annihilation operators are contragredient to the creation ones, this contraction procedure can be carried out for any n, m values. When employing instead a basis wherein the annihilation operators, as the creation ones, are irreducible tensor operators with respect to the dual quantum algebra Uq(gl(n))⊗Uq±1(gl(m))U_q(gl(n)) \otimes U_{q^{\pm1}}(gl(m)), a contraction limit only exists for n,m∈{1,2,4,6,...}n, m \in \{1, 2, 4, 6, ...\}. For n=2, m=1, and n=m=2, the resulting relations can be expressed in terms of coupled (anti)commutators (as in the classical case), by using Uh(sl(2))U_h(sl(2)) (instead of sl(2)) Clebsch-Gordan coefficients. Some U_h(sl(2)) rank-1/2 irreducible tensor operators, recently constructed by Aizawa, are shown to provide a realization of Ah±(2,1){\cal A}_{h\pm}(2,1).Comment: LaTeX, uses amssym.sty, 24 pages, no figure, to be published in Int. J. Theor. Phy

    Revisiting (quasi-)exactly solvable rational extensions of the Morse potential

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    The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials VA,B,ext(x)V_{A,B,{\rm ext}}(x), obtained from a conventional Morse potential VA−1,B(x)V_{A-1,B}(x) by the addition of a bound state below the spectrum of the latter, is re-obtained. More importantly, the existence of another family of extended potentials, strictly isospectral to VA+1,B(x)V_{A+1,B}(x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of `enlarged' shape invariance property, in the sense that their partner, obtained by translating both the parameter AA and the degree mm of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones.Comment: 24 pages, no figure, published versio

    Nonstandard GL_h(n) quantum groups and contraction of covariant q-bosonic algebras

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    GLh(n)×GLh(m)GL_h(n) \times GL_h(m)-covariant hh-bosonic algebras are built by contracting the GLq(n)×GLq(m)GL_q(n) \times GL_q(m)-covariant qq-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding RhR_h-matrices. Whenever n=2n=2, and m=1m=1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-1/2 irreducible tensor operators, recently contructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the hh-bosonic algebra corresponding to n=2n=2 and m=1m=1.Comment: 7 pages, LaTeX, no figure, presented at the 7th Colloquium ``Quantum Groups and Integrable Systems'', Prague, 18--20 June 1998, submitted to Czech. J. Phy

    Deformed shape invariance symmetry and potentials in curved space with two known eigenstates

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    We consider two families of extensions of the oscillator in a dd-dimensional constant-curvature space and analyze them in a deformed supersymmetric framework, wherein the starting oscillator is known to exhibit a deformed shape invariance property. We show that the first two members of each extension family are also endowed with such a property provided some constraint conditions relating the potential parameters are satisfied, in other words they are conditionally deformed shape invariant. Since, in the second step of the construction of a partner potential hierarchy, the constraint conditions change, we impose compatibility conditions between the two sets to build potentials with known ground and first excited states. To extend such results to any members of the two families, we devise a general method wherein the first two superpotentials, the first two partner potentials, and the first two eigenstates of the starting potential are built from some generating function W+(r)W_+(r) (and its accompanying function W−(r)W_-(r)).Comment: 30 pages, 4 figures, published versio

    Fractional supersymmetric quantum mechanics, topological invariants and generalized deformed oscillator algebras

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    Fractional supersymmetric quantum mechanics of order λ\lambda is realized in terms of the generators of a generalized deformed oscillator algebra and a Zλ_{\lambda}-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing λ\lambda sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of Zλ_{\lambda}-graded uniform topological symmetry of type (1, 1, ..., 1) with topological invariants generalizing the Witten index.Comment: LaTeX 2e, amssym, 16 pages, no figur
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