Lowering and raising operators for the free Meixner class of orthogonal polynomials

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line

On Inverse Scattering at a Fixed Energy for Potentials with a Regular Behaviour at Infinity

We study the inverse scattering problem for electric potentials and magnetic fields in \ere^d, d\geq 3, that are asymptotic sums of homogeneous terms at infinity. The main result is that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at some positive energy.Comment: This is a slightly edited version of the previous pape

New global stability estimates for monochromatic inverse acoustic scattering

We give new global stability estimates for monochromatic inverse acoustic scattering. These estimates essentially improve estimates of [P. Hahner, T. Hohage, SIAM J. Math. Anal., 33(3), 2001, 670-685] and can be considered as a solution of an open problem formulated in the aforementioned work

Approximate quantum cloaking and almost trapped states

We describe families of potentials which act as approximate cloaks for matter waves, i.e., for solutions of the time-independent Schr\"odinger equation at energy $E$, with applications to the design of ion traps. These are derived from perfect cloaks for the conductivity and Helmholtz equations, by a procedure we refer to as isotropic transformation optics. If $W$ is a potential which is surrounded by a sequence $\{V_n^E\}_{n=1}^\infty$ of approximate cloaks, then for generic $E$, asymptotically in $n$ (i) $W$ is both undetectable and unaltered by matter waves originating externally to the cloak; and (ii) the combined potential $W+V_n^E$ does not perturb waves outside the cloak. On the other hand, for $E$ near a discrete set of energies, cloaking {\it per se} fails and the approximate cloaks support wave functions concentrated, or {\it almost trapped}, inside the cloaked region and negligible outside. Applications include ion traps, almost invisible to matter waves or customizable to support almost trapped states of arbitrary multiplicity. Possible uses include simulation of abstract quantum systems, magnetically tunable quantum beam switches, and illusions of singular magnetic fields.Comment: Revised, with new figures. Single column forma

Discrete series representations for sl(2|1), Meixner polynomials and oscillator models

We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number beta>0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter gamma. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the momentum operator can be continuous or infinite discrete, depending on the value of gamma. We determine the position wavefunctions both in the continuous and discrete case, and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2)\subset sl(2|1), it can be seen why the case gamma=1 corresponds to the paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in the sl(2|1) model yields the canonical oscillator.Comment: (some minor misprints were corrected in this version