We investigate the dynamical symmetry superalgebras of the one-dimensional matrix superconformal quantum mechanics with inverse-square potential. They act as spectrumgenerating superalgebras for the systems with the addition of the de Alfaro-Fubini-Furlan oscillator term. The undeformed quantum oscillators are expressed by 2 n × 2 n supermatrices; their corresponding spectrum-generating superalgebras are given by the osp(2n|2) series. For n = 1 the addition of a inverse-square potential does not break the osp(2|2) spectrum-generating superalgebra. For n = 2 two cases of inverse-square potential deformations arise. The first one produces Klein deformed quantum oscillators; the corresponding spectrum-generating superalgebras are given by the D(2, 1; α) class, with α determining the inverse-square potential coupling constants. The second n = 2 case corresponds to deformed quantum oscillators of non-Klein type. In this case the osp(4|2) spectrum-generating superalgebra of the undeformed theory is broken to osp(2|2). The choice of the Hilbert spaces corresponding to the admissible range of the inverse-square potential coupling constants and the possible direct sum of lowest weight representations of the spectrum-generating superalgebras is presented. The undeformed one-dimensional quantum oscillators and their osp(2n|2) spectrum-generating superalgebrasAs recalled in [24], the (96) superalgebra (see Appendix A) of the supersymmetric quantum mechanics can be constructed by Hermitian matrix differential operators Q I , H acting on a supermultiplet of real-valued fields. On the other hand the introduction of a dynamical symmetry realized by Hermitian operators closing a superconformal algebra requires a complex structure. The reason is the presence of non-vanishing commutators (such as [Q I , K] = i Q I ); they imply that the imaginary unit has to be introduced in order to have Hermitian operators on the right hand side. Therefore, without loss of generality, we can investigate superconformal dynamical symmetries (and spectrum-generating superalgebras) acting on supermultiplets of complex fields.

In this paper we quantize superconformal σ-models defined by worldline supermultiplets. Two types of superconformal mechanics, with and without a DFF term, are considered. Without a DFF term (Calogero potential only) the supersymmetry is unbroken. The models with a DFF term correspond to deformed (if the Calogero potential is present) or undeformed oscillators. For these (un)deformed oscillators the classical invariant superconformal algebra acts as a spectrum-generating algebra of the quantum theory.Besides the osp(1|2) examples, we explicitly quantize the superconformally-invariant worldine σ-models defined by the N = 4 (1, 4, 3) supermultiplet (with D(2, 1; α) invariance, for α = 0, −1) and by the N = 2 (2, 2, 0) supermultiplet (with two-dimensional target and sl(2|1) invariance). The parameter α is the scaling dimension of the (1, 4, 3) supermultiplet and, in the DFF case, has a direct interpretation as a vacuum energy. In the DFF case, for the sl(2|1) models, the scaling dimension λ is quantized (either λ = 1 2 + Z or λ = Z). The ordinary two-dimensional oscillator is recovered, after imposing a superselection restriction, from the λ = − 1 2 model. In particular a single bosonic vacuum is selected. The spectrum of the unrestricted two-dimensional theory is decomposed into an infinite set of lowest weight representations of sl(2|1). Extra fermionic raising operators, not belonging to the original sl(2|1) superalgebra, allow (for λ = 1 2 + Z) to construct the whole spectrum from the two degenerate (one bosonic and one fermionic) vacua.

We construct a three-dimensional superconformal quantum mechanics (and its associated de Alfaro-Fubini-Furlan deformed oscillator) possessing an sl(2|1) dynamical symmetry. At a coupling parameter β = 0 the Hamiltonian contains a 1 r 2 potential and a spin-orbit (hence, a first-order differential operator) interacting term. At β = 0 four copies of undeformed threedimensional oscillators are recovered. The Hamiltonian gets diagonalized in each sector of total j and orbital l angular momentum (the spin of the system is 1 2 ). The Hilbert space of the deformed oscillator is given by a direct sum of sl(2|1) lowest weight representations. The selection of the admissible Hilbert spaces at given values of the coupling constant β is discussed. The spectrum of the model is computed. The vacuum energy (as a function of β) consists of a recursive zigzag pattern. The degeneracy of the energy eigenvalues grows linearly up to E ∼ β (in proper units) and quadratically for E > β. The orthonormal energy eigenstates are expressed in terms of the associated Laguerre polynomials and the spin spherical harmonics. The dimensional reduction of the model to d = 2 produces two copies (for β and −β, respectively) of the two-dimensional sl(2|1) deformed oscillator. The dimensional reduction to d = 1 produces the one-dimensional D(2, 1; α) deformed oscillator, with α determined by β.

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