arXiv subjects

Matti Selg

Publications and source records attributed to Matti Selg.

6 recordsLinked to original sources

arXiv2015

An improved algorithm for reconstructing reflectionless potentials

A fully algebraic approach to reconstructing one-dimensional reflectionless potentials is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful properties of special determinants - the alternants - have been exploited. The main formula takes an especially simple form if one aims to reconstruct a symmetric reflectionless potential. Several examples are presented to illustrate the efficiency of the method.

quant-ph↗

arXiv2015

Inverted approach to the inverse scattering problem: complete solution of the Marchenko equation for a model system

An example of full solution of the inverse scattering problem on the half line is presented. For this purpose, a simple analytically solvable model system (Morse potential) is used, which is expected to be a reasonable approximation to a real potential. First one calculates all spectral characteristics for the fixed model system. This way one gets all the necessary input data (otherwise unobtainable) to implement powerful methods of the inverse scattering theory. In this paper, the multi-step procedure to solve the Marchenko integral equation is described in full details. Excellent performance of the method is demonstrated and its combination with the Marchenko differential equation is discussed. In addition to the main results, several important analytic properties of the Morse potential are unveiled. For example, a simple analytic algorithm to calculate the phase shift is derived.

quant-ph↗

arXiv2007

Potential energy curves and transition moments for the excimer states 0u+ (3P1) and 1u (3P2) of Ar2

Exactly solvable rererence potentials of several smoothly joined Morse-type components were constructed for the lowest two excimer states of Ar2 molecule. The parameters of the potentials have been ascertained by fitting to the experimental data, and they are reliable in a wide range of nuclear separations. A large number of quantum mechanical Franck-Condon factors for bound-free and bound-bound transitions have been calculated and compared with the observed spectroscopic features. The fitting procedure also involved dipole transition moments, which have been adjusted to the known radiative lifetimes of the vibrational levels. The resulting potential energy curves accurately reproduce the first and the second emission continua of Ar2* as well as the oscillatory spectrum related to their inner turning point region. The numbering and the positions of the vibrational levels reported by Herman et al. [P. R. Herman, P. E. LaRocque, and B. P. Stoicheff, J. Chem. Phys. 89, 4535 (1988)] have been confirmed.

physics.chem-ph↗

arXiv2006

Reference potential approach to the inverse Schrödinger problem: explicit demonstration of Levinson theorem and a solution scheme for Krein equation

A recently proposed reference potential approach to the inverse Schrödinger problem is further developed. As previously, theoretical developments are demonstrated on example of diatomic xenon molecule in its ground electronic state. An exactly solvable reference potential for this quantum system is used, which enables to solve the related energy eigenvalue problem exactly. Moreover, the full energy dependence of the phase shift can also be calculated analytically, and as a particular result, full agreement with Levinson theorem has been achieved and explicitly demonstrated. In principle, this important spectral information can be reused to calculate an improved potential for the system, and such possibilities are discussed in the paper. Aiming at this goal, one may calculate an auxiliary potential with no bound states, whose spectral density for positive energies is exactly the same as that of the reference potential. To this end, one may solve Krein equation, which in the present context is simpler than using Gelfand-Levitan method. General solution of Krein equation can be expressed as a Neumann series. Convergence of this series of multi-dimensional integrals at distances not close to the origin is hard to achieve without a simple asymptotic formula for calculating the kernel of Krein equation. As proven in this paper, such an asymptotic formula exists, and its parameters can be easily ascertained.

quant-ph↗

arXiv2005

Reference potential approach to the quantum-mechanical inverse problem: II. Solution of Krein equation

A reference potential approach to the one-dimensional quantum-mechanical inverse problem is developed. All spectral characteristics of the system, including its discrete energy spectrum, the full energy dependence of the phase shift, and the Jost function, are expected to be known. The technically most complicated task in ascertaining the potential, solution of a relevant integral equation, has been decomposed into two relatively independent problems. First, one uses Krein method to calculate an auxiliary potential with exactly the same spectral density as the initial reference potential, but with no bound states. Thereafter, using Gelfand-Levitan method, it is possible to introduce, one by one, all bound states, along with calculating another auxiliary potential of the same spectral density at each step. For the system under study (diatomic xenon molecule), the kernel of the Krein integral equation can be accurately ascertained with the help of solely analytic means. At small distances the calculated auxiliary potential with no bound states practically coincides with the initial reference potential, which is in full agreement with general theoretical considerations. Several possibilities of solving the Krein equation are proposed and the prospects of further research discussed.

quant-ph↗

arXiv2005

Reference potential approach to the quantum-mechanical inverse problem: I. Calculation of phase shift and Jost function

Elegant and mathematically rigorous methods of the quantum inverse theory are difficult to put into practice because there is always some lack of needful input information. In this situation, one may try to construct a reference potential, whose spectral characteristics would be in a reasonable agreement with the available data of the system's properties. Since the reference potential is fixed, it is always possible to calculate all its spectral characteristics, including phase shift for scattering states and Jost function, the main key to solve the inverse problem. Thereafter, one can calculate a Bargmann potential whose Jost function differs from the initial one only by a rational factor. This way it is possible, at least in principle, to construct a more reliable potential for the system. The model system investigated in this paper is diatomic xenon molecule in ground electronic state. Its reference potential is built up of several smoothly joined Morse type components, which enables to solve the related energy eigenvalue problem exactly. Moreover, the phase shift can also be calculated in part analytically, and the Jost function can be acertained very accurately in the whole range of positive energies. Full energy dependence of the phase shift has been determined and its excellent agreement with the Levinson theorem demonstrated. In addition, asymptotically exact analytic formulas for the phase shift and the Jost function, independent of each other, are obtained and their physical background elucidated.

quant-ph↗