An economical method to calculate eigenvalues of the Schroedinger Equation

The method is an extension to negative energies of a spectral integral equation method to solve the Schroedinger equation, developed previously for scattering applications. One important innovation is a re-scaling procedure in order to compensate for the exponential behaviour of the negative energy Green's function. Another is the need to find approximate energy eigenvalues, to serve as starting values for a subsequent iteration procedure. In order to illustrate the new method, the binding energy of the He-He dimer is calculated, using the He-He TTY potential. In view of the small value of the binding energy, the wave function has to be calculated out to a distance of 3000 a.u. Two hundred mesh points were sufficient to obtain an accuracy of three significant figures for the binding energy, and with 320 mesh points the accuracy increased to six significant figures. An application to a potential with two wells separated by a barrier, is also made.Comment: 19 pages, 3 figures, submitted to Eur. J. Phy

Comparison of numerical methods for the calculation of cold atom collisions

Three different numerical techniques for solving a coupled channel Schroedinger equation are compared. This benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same diagonal Lennard-Jones potential, one of positive and the other of negative energy. The coupling potential is of an exponential form. The methods are i) a recently developed spectral type integral equation method based on Chebyshev expansions, ii) a finite element expansion, and iii) a combination of an improved Numerov finite difference method and a Gordon method. The computing time and the accuracy of the resulting phase shift is found to be comparable for methods i) and ii), achieving an accuracy of ten significant figures with a double precision calculation. Method iii) achieves seven significant figures. The scattering length and effective range are also obtained.Comment: 22 pages, 3 figures, submitted to J. Comput. Phys. documentstyle [thmsa,sw20aip]{article} in .te

A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms

In the Hartree-Fock approximation the Pauli exclusion principle leads to a Schroedinger Eq. of an integro-differential form. We describe a new spectral noniterative method (S-IEM), previously developed for solving the Lippman-Schwinger integral equation with local potentials, which has now been extended so as to include the exchange nonlocality. We apply it to the restricted case of electron-Hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to three other methods. One is a non-iterative solution (NIEM) of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The third one is based on the Singular Value Decomposition of the exchange term followed by iterations over the remainder. The S-IEM method turns out to be more accurate by many orders of magnitude than any of the other three methods described above for the same number of mesh points.Comment: 29 pages, 4 figures, submitted to Phys. Rev.

An Inversion Approach to Ultrasonic Imaging through Reflective Interfaces in Multi-Layered Structures

The increased use of composite materials and adhesively bonded joints has resulted in the need for the development of inspection techniques appropriate for multi-layered structures. Normal incidence ultrasonic pulse-echo imaging has been and continues to be a principal technique for the detection of interface condition. Ideally, only a single reflection from each interface in the layered structure would be received and the ultrasonic image would be based upon a single parameter intrinsic to the material, such as the reflection coefficient. The reflection coefficient is, in turn, primarily determined by the relative change in ultrasonic impedance across the interface. In the absence of a complete inversion procedure by which the reflection coefficient may be calculated from pulse echo-data, the reflected signal amplitude is used to form the ultrasonic image. Unfortunately, the reflection amplitude, as indicated in Figure 1(a), often decreases rapidly due to the presence of reflective, overlying interfaces.</p

Semiseparable integral operators and explicit solution of an inverse problem for the skew-self-adjoint Dirac-type system

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational matrix function and $D=D^* \geq 0$ is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case

Convergence of sequential and asynchronous nonlinear paracontractions

Elsner L, Koltracht I, Neumann M. Convergence of sequential and asynchronous nonlinear paracontractions. Numerische Mathematik. 1992;62(1):305-319.We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented