942 research outputs found

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

$GL_h(n) \times GL_{h'}(m)$-covariant (hh')-bosonic (or (hh')-fermionic)
algebras ${\cal A}_{hh'\pm}(n,m)$ are built in terms of the corresponding R_h
and $R_{h'}$-matrices by contracting the $GL_q(n) \times
GL_{q^{\pm1}}(m)$-covariant q-bosonic (or q-fermionic) algebras ${\cal
A}^{(\alpha)}_{q\pm}(n,m)$, $\alpha = 1, 2$. When using a basis of ${\cal
A}^{(\alpha)}_{q\pm}(n,m)$ 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 $U_q(gl(n)) \otimes
U_{q^{\pm1}}(gl(m))$, a contraction limit only exists for $n, m \in \{1, 2, 4,
6, ...\}$. 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
$U_h(sl(2))$ (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 ${\cal A}_{h\pm}(2,1)$.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

The construction of rationally-extended Morse potentials is analyzed in the
framework of first-order supersymmetric quantum mechanics. The known family of
extended potentials $V_{A,B,{\rm ext}}(x)$, obtained from a conventional Morse
potential $V_{A-1,B}(x)$ 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 $V_{A+1,B}(x)$, 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 $A$
and the degree $m$ 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

$GL_h(n) \times GL_h(m)$-covariant $h$-bosonic algebras are built by
contracting the $GL_q(n) \times GL_q(m)$-covariant $q$-bosonic algebras
considered by the present author some years ago. Their defining relations are
written in terms of the corresponding $R_h$-matrices. Whenever $n=2$, and $m=1$
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 $h$-bosonic algebra corresponding to $n=2$ and
$m=1$.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

We consider two families of extensions of the oscillator in a $d$-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
$W_+(r)$ (and its accompanying function $W_-(r)$).Comment: 30 pages, 4 figures, published versio

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

Fractional supersymmetric quantum mechanics of order $\lambda$ is realized in
terms of the generators of a generalized deformed oscillator algebra and a
Z$_{\lambda}$-grading structure is imposed on the Fock space of the latter.
This realization is shown to be fully reducible with the irreducible components
providing $\lambda$ sets of minimally bosonized operators corresponding to both
unbroken and broken cases. It also furnishes some examples of
Z$_{\lambda}$-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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