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We present numerical solutions for two problems in one-dimensional supersymmetric quantum mechanics. The first case deals with the superpotential W ( x ) = x 4 , which is about the simplest case with no known analytical solution. We compute the eigenvalues for the states above the supersymmetric E = O state of the corresponding ordinary potential ~~i x ) = . u~-a x~ for a=3; those states are also the bound states of the partner convex potential V + ( x ) = x 6 + 3 x 2 . We discuss in which sense the double-well potential V-(x) is a critical potential; increasing a 2 3 we obtain a welldefined grouping of positive-and negative-parity levels, corresponding to an increasing barrier between the two wells. For the second case we start with a ground-state wave function of Lorentzian shape, namely, u o =( 1 + x 2 i -' , which gives rise to the superpotential W ( x ) = Z x /( 1 + x 2 ) and the ordinary potentials V + i x ) = 2 / ( 1 + x 2 ) and V ( x ) = ( 6 x 2 -2 ) / 1 1 +x2)'. There is only a bound state at zero energy for V-ix), the partner potential V + ( x ) being a repulsive barrier. The potentials V + ( x ) and V_ ( x i decay like l i l + 1 ) / x 2 for 1 = 1 and 2, respectlvely, at large distances ("intermediate range" potentials) and this produces in particular anomalous phase shifts 6,,,, and Sodd at very low energies. We calculate these phase shifts for V,, those of the partner potential V-(x) being fixed by supersymmetry. We also show the peculiar character of these potentials by changing the parameters. In particular for the "craterlike" potential V-(x) the replacement of 6 x 2 -2 by 6 x 2 -1 gives rise to a distinctive resonance in the even phase shift. Some considerations regarding x -* potentials, factorization, and scale invariance are relegated to an appendix.

Using the new value for a ; '", fits of the hyperon-decay data are presented. They show that there is excellent agreement between the predictions of the Cabibbo model and the experimental data, except for perhaps two indications of minor corrections.The experimental value of the electron asymmetry in I--nev was of considerable concern because the world average (consistent with four old experiments) of a$-"= 0.26 f 0.19 was in contradiction with the prediction of the standard theory. The only explanation of this value-which was consistent with all other experimental data-was that the pseudotensor form factor for this process, gf'", was huge and caused by a large value of the SU(3)-invariant g3 form factor.' The new value obtained for this asymmetry2 by the Fermilab experiment 715 of a:' "--0.58 k0.16 shows that this concern was unjustified.To demonstrate how well the standard theory agrees now with a large number of experimental data, but also to expose the few minor deviations, we present here a fit of the Cabibbo model to the hyperon-semileptonic-decay data. In our fits we have used exact expressions for the rate R and for R a in. terms of the form factors f l , f2, f3, gl, g2, and g3 in which the phase-space factors are obtained by numerical integrati~n.~ We have also included radiative corrections4 and q2 dependence of the leading form factors in a linear approximation with the slope determined from the slope of the electromagnetic form factors and from neutrino ~cattering.~ We report here two different kinds of fits. In fit I of the ordinary Cabibbo model, the form factors fBB and g~" ( i = 1,2,3) for each individual process B-B'lv are expressed in terms of multiplet form factors fiF, FP, Gf, and GP by formulas like where the CF and CD are the F-and D-type ClebschGordan coefficients. With FpD and FgD determined from CVC (conserved vector current) and with FfD, GFD, and GfD equal to zero,6 it has three free parameters [sinQc,( 1 ' 4 ) G f = F, -m G f = D 1 to be determined from 25 experimental data. In fit I1 the form factors fB1' and g ,~" are expressed in terms of multiplet form factors by formulas like7 FI and F2 are again determined by CVC and the Gz are zero (second class). Thus one has five free parameters (sinec, Gf, Gf, Gf, Gf ) to be determined from the 25 data. For this spectrum-generating (SG) model the mass differences between the hyperons have been taken into account. In the limit of zero mass differences the two models are identical. But, as one can see from the form (2) and the corresponding expressions for the other form factors,' these two models become virtually indistinguishable if GfD is for some reason close to zero.In Table I we present the comparison between 25 experimental values (given in the second column of the table) and the prediction of the Cabibbo model (given in the third column) without any symmetry breaking. The contribution that each of the fitted predictions makes to x2 is listed in the fourth column, and we see that the 15% discrepancy8 for R (I-Aev) is the only significant...

The analogy between the representations of SU(1,1/1) and SU(2,2/1) is explained and used to suggest the latter as a spectrum supergroup of superconformal relativistic quantum mechanics for systems with rotational degrees of freedom.

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