In a recent paper [1] Cariñena et al analyzed a non-polynomial onedimensional quantum potential representing an oscillator which they argued was intermediate between the harmonic and isotonic oscillators. In particular they proved that it is Schrödinger soluble, and explicitly obtained the wavefunctions and energies of the bound states. In this paper we show that these results can be obtained much more simply by noting that this potential is a supersymmetric partner potential of the harmonic oscillator. We then use this observation to generate an infinite set of potentials which can be exactly solved in a similar manner.

We examine the destruction of superconducting pairing in metallic grains as their size is decreased for both even and odd numbers of electrons. This occurs when the average level spacing d is of the same order as the BCS order parameter ∆. The energy levels of these grains are randomly distributed according to random matrix theory, and we must work statistically. We find that the average value of the critical level spacing is larger than that for the model of equally spaced levels for both parities, and derive numerically the probability densities P o,e (d) that a grain of mean level spacing d shows pairing.

We use a random walk in the ensemble of impurity configurations to generate a Brownian motion model for energy levels in disordered conductors. Treating arc-length along the random walk as fictitous time, the resulting Langevin equation relates spectral statistics to eigenfunction correlations. Solving this equation at energy scales large compared with the mean level spacing, we obtain the spectral form factor, and its parametric dependence. PACS numbers: 05.45+b, 71.25.-s, 72.15.Rn (Submitted to PRL 29 March 1996) Statistical properties of the spectra of finite quantum systems have been a focus for research in three successive contexts: nuclear physics [1], the semiclassical limit of quantum mechanics [2], and studies of mesoscopic conductors [3,4]. A unifying idea is that the energies of individual eigenstates are frequently neither calculable nor interesting: instead, the concern should be with eigenvalue correlations, which typically are independent of many details of the Hamiltonian.Disordered mesoscopic conductors bring to this field both new behaviour and a natural ensemble for a statistical description -the ensemble of impurity configurations. For weak disorder, new behaviour arises because there can be a broad window in time, and hence a corresponding energy interval, between the scale, t el , for electron scattering from impurities and that for diffusion across the system, t erg ∼ L 2 /D (where L and D are the system size and diffusion constant). And at the mobility edge, specific, critical spectral statistics are expected [5,6]. The ensemble average provides the departure point for established perturbative [4] and non-perturbative [3] calculations of spectral correlations in disordered metals.In this paper we describe an alternative approach, in which the energy level distribution is averaged along a random walk through the ensemble. There are several precedents for study of levels as a function of position in the space of Hamiltonians. Most recently, a number of authors [7], in particular Szafer, Simons and Altshuler [8], have investigated parametric statistics: level correlations between different points on a smooth path in this space. Earlier, in connection with the semiclassical limit, Pechukas [9] used motion along such a path, with coordinate λ, to generate a dynamics for the one-dimensional gas formed by levels on the energy axis. And originally, in the context of random matrix theory (RMT), Dyson [10], employed a random walk through the matrix ensemble as the foundation for Brownian dynamics of levels, with arc-length, τ , being the fictitious time. Pechukas' and Dyson's ideas are linked, as shown schematically in Fig. 1, by the usual relation [10] between the end-to-end distance and the length of a random walk, λ 2 ∼ τ . In previous work on level dynamics [9][10][11][12], eigenfunction correlations have played no role, either (in the semiclassical limit) by assumption, or (for random matrices) by construction, RMT having no preferred basis. In contrast, for disordered metals, the basis o...

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