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The problem of a shallow donor impurity located at the centre of a symmetrical paraboloidal quantum dot (SPQD) is solved exactly. The Schrödinger equation is separated in the paraboloidal coordinate system. Three different cases are discussed for the radial-like equations. For a bound donor, the energy is negative and the solutions are described by Whittaker functions. For a non-bound donor, the energy is positive and the solutions become coulomb wave functions. In the last case, the energy is equal to zero and the solutions reduce to Bessel functions. Using the boundary conditions at the dot surfaces, the variations of the donor kinetic and potential energies versus the size of the dot are obtained. The problem of a shallow donor impurity in a Hemiparaboloidal Quantum dot (HPQD) is also studied. It is shown that the wave functions of a HPQD are specific linear combinations of those of a SPQD.

Localization; Lyapunov Exponent; Transfer Matrix; Functional Equation. PACS numbers: 71.23.An 1. Introduction The present interest in one-dimensional disordered electron systems is mainly focused on the effect of cor- related disorder, see e. g. [1 – 4], on Anderson localiza- tion in Bose-Einstein condensates [5], or on the effect of dynamic disorder as by phonons [6]. In this paper, we give an exemplary direct insight into the properties of a random system by deriving analytical expressions of the Lyapunov exponent. This is achieved for asymp- totic regions of high and low

Dryden, vol. 6, esp. 862-76. 22 Francis Cairns, Virgil's Augustan Epic (Cambridge: Cambridge UP 1989) 92-108, and 5-6, 71 23 'An Horatian Ode upon Cromwel's Return from Ireland,' The Poems and Letters of Andrew Marvell ed. H.M. Margoliouth, 3d ed. 2 vols. (Oxford: Clarendon 1971) i: 91-4 24 Hobbes, Works 10: iv 25 Abraham Cowley, The Collected Works of Abraham Cowley, ed. Thomas O. Calhoun, Laurence Heyworth, and Allan Pritchard, i vol. to date (Newark: U of Delaware P 1989- ) i: 376-7 26 Raymond Williams, Marxism and Literature (Oxford: Oxford UP 1977) 122 9

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