Spin-dependent Bohm trajectories for hydrogen eigenstates

The Bohm trajectories for several hydrogen atom eigenstates are determined, taking into account the additional momentum term that arises from the Pauli current. Unlike the original Bohmian result, the spin-dependent term yields nonstationary trajectories. The relationship between the trajectories and the standard visualizations of orbitals is discussed. The trajectories for a model problem that simulates a 1s-2p transition in hydrogen are also examined.Comment: 11 pages, 3 figure

Collage theorem-based approaches for solving inverse problems for differential equations : A review of recent developments

In this short survey, we review the current status of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. This involves an examination of several methods which are based on the so-called Collage Theorem, a simple consequence of Banach's Fixed Point Theorem, and its extensions

Supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian Screened Coulomb potential via Hamiltonian hierarchy inspired variational method

The supersymmetric solutions of PT-symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schrodinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.Comment: 13 page

Generalized fractal transforms and self-similarity : recent results and applications

Most practical as well as theoretical works in image processing and mathematical imaging consider images as real-valued functions, u: X \u2192 Rg, where X denotes the base space or pixel space over which the images are defined and Rg 82 R is a suitable greyscale space. A variety of function spaces F(X) may be considered depending on the application. Fractal image coding seeks to approximate an image function as a union of spatially-contracted and greyscale-modified copies of subsets of itself, i.e., u 48 Tu, where T is the so-called Generalized Fractal Transform (GFT) operator. The aim of this paper is to show some recent developments of the theory of generalized fractal transforms and how they can be used for the purpose of image analysis (compression, denoising). This includes the formulation of fractal transforms over various spaces of multifunctions, i.e., set-valued and measure-valued functions. The latter may be useful in nonlocal image processing

Random measure-valued image functions, fractal transforms and self-similarity

We construct a complete metric space (Y,dY) of random measure-valued image functions. This formalism is an extension of previous work on measure-valued image functions

Fourier transforms of measure-valued images, self-similarity and the inverse problem

After recalling the notion of a complete metric space (Y,dY)(Y,dY) of measure-valued images over a base (or pixel) space X, we define a complete metric space (F,dF)(F,dF) of Fourier transforms of elements \u3bc 08Y\u3bc 08Y. We also show that a fractal transform T:Y\u2192YT:Y\u2192Y induces a mapping M on the space FF. The action of M on an element U 08FU 08F is to produce a linear combination of frequency-expanded copies of M. Furthermore, if T is contractive in Y, then M is contractive on FF: as expected, the fixed point View the MathML sourceU\uaf of M is the Fourier transform of \u3bc 08

Fractal-based measure approximation with entropy maximization and sparsity constraints

Let (X, d) denote a complete metric space. An N-map iterated function system with probabilities (IFSP) is a set of N contraction maps wi : X ! X with associated probabilities pi. The IFSP, denoted as (w, p), defines a contractive Markov operator M, on the space of of probability measures M(X) equipped with the Monge-Kantorovich metric dMK. The unique fixed point \uaf\u3bc = M\uaf\u3bc is referred to as the invariant measure of the N-map IFSP. Here we consider the following inverse problem: Given a target measure \u3bc, find an IFSP (w, p) with invariant measure \uaf\u3bc sufficiently close to \u3bc, i.e., dMK(\u3bc, \uaf\u3bc) < \u1eb. From Banach\u2019s Theorem, the problem may converted into finding an IFSP with Markov operator M that minimizes the collage error dMK(\u3bc,M\u3bc). Nevertheless, the determination of optimal wi and pi is still a formidable problem. It was simplified in [1] by employing a fixed, infinite set of maps wi satisfying a refinement condition on (X, d). This problem was then translated into a moment matching problem that becomes a quadratic programming (QP) problem in the pi. In this paper we extend the method developed in [1] along two different directions. First, we search for a set of probabilities pi that not only minimizes the collage error but also maximizes the entropy of the iterated function system. Second, we include an extra term in the minimization process which takes into account the sparsity of the set of probabilities. In our new formulations, collage error minimization can be understood as a multi-criteria problem: i.e., collage error, entropy and sparsity. We consider two different methods of solution: (i) scalarization, which reduces the multi-criteria program to a single-criteria program by combining all objective functions with different trade-off weights and (ii) goal programming, involving the minimization of the distance between each objective function and its goal. Numerical examples show how the two above methods work in practic