Modern Physics Letters B

The inversion of scattering data to obtain the structure of a thin specimen illuminated by electrons is discussed. The inversion problem has two aspects, the reconstruction of one or more exit surface wave functions and the subsequent retrieval of the projected structure of the specimen. The second step is complicated by the fact that electrons interact strongly with matter and multiple scattering complicates the inversion problem unless the specimen is very thin, particularly in the case of crystalline specimens.

Based on the relation between the m-function and the spectral function we construct an inverse quantum scattering procedure at fixed energy which can be applied to spherical radial potentials vanishing beyond a fixed radius a. To solve the Gelfand–Levitan–Marchenko integral equation for the transformation kernel, we determine the input symmetrical kernel by using a minimum norm method with moments defined by the input set of scattering phase shifts. The method applied to the box and Gauss potentials needs further practical developments regarding the treatment of bound states.

The use of one-dimensional inverse quantum scattering theory for the design of semiconductor heterostructures is reviewed within the contexts of (i) the single-band constant effective mass model (ii) the single-band variable effective mass model and (iii) the multi-band pseudopotential model. As a fourth application a different inverse method is developed for the multi-band k · p model. This method is based on the one-dimensional coupled-channel Marchenko method. Preliminary numerical results are presented for a two-band k · p model of an InAs/GaSb/InAs single quantum well in which the strength of the interband coupling is approximately constant.

We consider three-dimensional inverse scattering with fixed energy for which the spherically symmetrical potential is nonvanishing only in a ball. We give exact upper and lower bounds for the phase shifts. We provide a variational formula for the Weyl–Titchmarsh m-function of the one-dimensional Schrödinger operator defined on the half-line.

We consider the Sturm–Liouville operator on the unit interval. We obtain two-sided a priori estimates of potential in terms of Dirichlet and Neumann eigenvalues and eigenvalues for 2 types of mixed boundary conditions.

Consider the fixed-ℓ inverse scattering problem. We show that the zeros of the regular solution of the Schrödinger equation, r_n(E), which are monotonic functions of the energy, determine a unique potential when the domain of the energy is such that the r_n(E) range from zero to infinity. This suggests that the use of the mixed data of phase-shifts {δ(ℓ₀, k), k ≥ k₀} ∪ {δ(ℓ, k₀), ℓ ≥ ℓ₀}, for which the zeros of the regular solution are monotonic in both domains, and range from zero to infinity, offers the possibility of determining the potential in a unique way.

Solution of the Cox–Thompson inverse scattering problem at fixed energy^1–3 is reformulated resulting in semi-analytic equations. The new set of equations for the normalization constants and the nonphysical (shifted) angular momenta are free of matrix inversion operations. This simplification is a result of treating only the input phase shifts of partial waves of a given parity. Therefore, the proposed method can be applied for identical particle scattering of the bosonic type (or for certain cases of identical fermionic scattering). The new formulae are expected to be numerically more efficient than the previous ones. Based on the semi-analytic equations an approximate method is proposed for the generic inverse scattering problem, when partial waves of arbitrary parity are considered.

Two body data alone cannot determine the potential uniquely, one needs three-body data as well. A method is presented here which simultaneously fits local or nonlocal potentials to two-body and three-body observables. The interaction of composite particles, due to the Pauli effect and the indistinguishability of the constituent particles, is genuinely nonlocal. As an example, we use a Pauli-correct nonlocal fish-bone type optical model for the α – α potential and derive the fitting parameters such that it reproduces the two-α and three-α experimental data.

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: (1) Property C and applications, (2) Stable inversion of fixed-energy 3D scattering data and its error estimate, (3) Inverse scattering with “incomplete” data, (4) Inverse scattering for inhomogeneous Schrödinger equation, (5) Krein's inverse scattering method, (6) Invertibility of the steps in Gel'fand–Levitan, Marchenko, and Krein inversion methods, (7) The Newton–Sabatier and Cox–Thompson procedures are not inversion methods, (8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, (10) Inverse obstacle scattering with fixed-frequency data, (11) Inverse scattering with data at a fixed energy and a fixed incident direction, (12) Creating materials with a desired refraction coefficient and wave-focusing properties.

The article gives a concise review on methods for solving the inverse scattering problem of coupled reaction channels in quantum mechanics at fixed energy and presents specific applications to nuclear reactions. Two methods are considered, namely the Newton–Sabatier method and an approximate method based on the first Born approximation. Examples with artificial coupling potentials are shown to provide successful tests of both the methods. Further developments of the coupled channel inverse problem at fixed energy are needed for manifold applications in various fields of physics.

This work deals with the inverse scattering problems for the two-dimensional Schrödinger equation

Starting from a system of N radial Schrödinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N × N exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on N(N + 1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. For N = 2, previous results are reviewed, in particular regarding the number of bound states and resonances of the potential. A schematic inverse problem with one resonance is considered.

In these expository notes we explain the role of geometric optics in wave propagation on domains or manifolds with corners or edges. Both the propagation of singularities, which describes where solutions of the wave equation may be singular, and the diffractive improvement under non-focusing hypotheses, which states that in certain places the diffracted wave is more regular than a priori expected, is described. In addition, the wave equation on differential forms with natural boundary conditions, which in particular includes a formulation of Maxwell's equations, is studied.

It was an important examination to give a review talk at the previous Conference on Inverse Quantum Scattering (1996, Lake Balaton) about computer visualization of this science in front of its fathers — creators, B. M. Levitan and V. A. Marchenko. We have achieved a new understanding that the discovered main rules of transformations of a single wave function bump, e.g., for the ground bound states of one dimensional quantum systems are applicable to any state of any potential with arbitrary number of bumps from finite to unlimited ones as scattering states and bound states embedded into continuum. It appeared that we need only to repeat the rule mentally the necessary number of times. That uttermost simplification and unification of physical notion of spectral, scattering and decay control for any potential have got an obligatory praise from B. M. Levitan at the conference and was a mighty stimulus for our further research After that we have written both Russian (2002) and improved English editions of “Submissive Quantum Mechanics. New Status of the Theory in Inverse Problem Approach”¹ (appeared at the very end of 2007). This book was written for correction of the present defect in quantum education throughout the world. Recently the quantum IP intuition helped us to discover a new concept of permanent wave resonance with potential spatial oscillations.² This means the constant wave swinging frequency on the whole energy intervals of spectral forbidden zones destroying physical solutions and deepening the theory of waves in periodic potentials. It also shows the other side of strengthening the fundamentally important magic structures. A ‘new language’ of wave bending will be presented to enrich our quantum intuition, e.g., the paradoxical effective attraction of barriers and repulsion of wells in multichannel systems, etc.